TEL AVIV UNIVERSITYRaymond and Beverly Sackler Faculty of Exact Sciences

Blavatnik School of Computer Science

Algebraic Techniques in Combinatorial andComputational Geometry

A thesis submitted toward a degree of

Doctor of Philosophy

by

Noam Solomon

This work was carried out under the supervision of

Prof. Micha Sharir

Submitted to the Senate of Tel-Aviv UniversitySeptember 2017

To Sapir, my love

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Abstract

In this thesis we study several problems in combinatorial geometry, mainly in incidence geometry.We study a variety of incidence questions that involve points and other geometric objects, startingwith lines in three and four dimensions, moving to algebraic curves, and then also to algebraicsurfaces in three dimensions. We develop and use several “infrastructural” tools in algebra andalgebraic geometry for tackling these problems, tools that should also be useful for many othercombinatorial problems too. We also apply these bounds in several other problems in combinato-rial geometry.

In the first part of the thesis we consider the problem of obtaining tight asymptotic bounds onthe number of incidences between points and lines in higher dimensions, extending the founda-tional bound of Szemeredi and Trotter from 1983 for the planar case, and the more recent ground-breaking result of Guth and Katz (in 2010) for the three-dimensional case. The latter work intro-duced methods from advanced algebra and especially from algebraic geometry, which were notused in combinatorics before. This enabled Guth and Katz to (almost) settle the distinct distancesproblem of Erdos, of obtaining a lower bound for the number of distinct distances determined byany set of n points in the plane, a problem which stubbornly stood open for over 60 years, despitevery bold attempts to solve it. The work of Guth and Katz has given significant added momen-tum to incidence geometry, making many problems, including those studied in this thesis, deemedhopeless before the breakthrough, amenable to the new techniques.

We extend the study of point-line incidences to four dimensions, and then to points lyingon two- and three-dimensional varieties. We also found a more elementary proof of the Guth-Katz point-line incidence bound in three dimensions. We also derive lower bounds for incidencesbetween points and lines on a 3-dimensional quadratic surface in R4, and obtain Ramsey-typeresults involving the contact graph between lines in R3.

In the second part of the thesis, we extend our study of point-line incidences to the study ofincidences between points and algebraic curves in three and higher dimensions. One particularcase of this study results in a new bound on the number of incidences between points and circlesin R3.

We then study incidences between points and constant-degree algebraic surfaces in three di-

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mensions, such as planes, spheres, etc. As a result of this study, we obtain several improved boundsfor the number of distinct and repeated distances in a set of points lying on a two-dimensional va-riety in R3.

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Table of Contents

Acknowledgments vii

1: Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Incidences between points and lines in R4 . . . . . . . . . . . . . . . . . 10

1.2.2 Incidences between points and lines in R3, with applications to Ramsey-type theorems for lines in R3 . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.3 Improved bounds on incidences of lines with points on a variety . . . . . 13

1.2.4 Incidences with curves in three and higher dimensions . . . . . . . . . . 14

1.2.5 Incidences with surfaces in three dimensions . . . . . . . . . . . . . . . 20

1.3 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 Lines on varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.2 Generalized Bezout’s theorem . . . . . . . . . . . . . . . . . . . . . . . 28

1.3.3 Generically finite morphisms and the Theorem of the Fibers . . . . . . . 29

1.3.4 Flecnode polynomials and ruled surfaces in three and four dimensions . . 30

1.3.5 Flat points and the second fundamental form . . . . . . . . . . . . . . . 34

1.3.6 Finitely and infinitely ruled surfaces in four dimensions, and u-resultants 36

I Incidences between points and lines 39

2: Incidences between points and lines in R4 41

3: Highly incidental patterns on a quadratic hypersurface in R4 97

II Incidences between points and lines in R3, with applications 105

4: Incidences between points and lines in R3 107

5: Ramsey-type theorems for lines in 3-space 129

III Incidences between points and lines on varieties 147

6: Incidences between points and lines on two- and three-dimensionalvarieties 149

IV Incidences between points and curves and points and surfaces 191

7: Incidences with curves in Rd 193

8: Incidences with curves and surfaces in three dimensions, with appli-cations to distinct and repeated distances 211

V Conclusion 249

9: Discussion 251

9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.2 Future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

References 257

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Acknowledgments

I first want to express my deepest gratitude to Professor Micha Sharir, my advisor and friend.Micha has been an incredible source of inspiration to me, and a true mentor. I am both humbledand proud to have been his student, and will continue to collaborate with him in the future. Thankyou so much Micha!

I am also indebted to my supporting family, my mother Ma’ayana, my father Arie, my brotherShay, my sister Michal, my sister-in-law Hagar, and my baby nephew Yoavi.

Finally, I am so happy to dedicate this thesis to the love of my life, Sapir.

Part of this research was carried out while I was visiting, and enjoying the warm hospitalityand stimulating environment of, the Institute for Pure and Applied Mathematics (IPAM) at UCLA,in the spring of 2014.

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1 Introduction

1.1 Background

Combinatorial geometry is a field that studies combinatorial problems that have some geometricaspect. It was pioneered and developed by Paul Erdos, starting at the beginning of the 20th century.While such problems (sometimes referred to as Erdos-type problems) are often easy to state, someof them are very difficult, have a deep underlying theory, and remain (or have remained) open formany decades.

In the past decade the landscape of combinatorial geometry has considerably changed, dueto two groundbreaking papers by Guth and Katz ([55] in 2008 and [56] in 2010). They haveintroduced reasonably simple techniques from algebraic geometry that facilitated successful solu-tions of several major problems in combinatorial geometry. Their first paper obtained a completesolution to the joints problem, a problem involving incidences between points and lines in threedimensions which has been proposed by Chazelle et al. [28] in 1992. The second Guth–Katz paperpresented a nearly complete solution to the classical problem of Erdos [43] on distinct distances

in the plane, which was a major open problem since 1946. Both problems (especially the secondone) have been extensively studied over the years, using more traditional, and progressively morecomplex methods of combinatorial geometry, but with only partial and incomplete results.

This “marriage” of Algebraic Geometry with Combinatorial and Computational Geometry

poses major challenges for both areas, including distilling existing machinery in algebraic geom-etry and developing new tools, geared towards the new “client” area, and the application of thistoolbox to numerous basic problems in combinatorial and computational geometry.

In this thesis, we develop additional bridges between the two disciplines, develop additionalalgebraic machinery, and apply this machinery to a successful solution of several problems inCombinatorial Geometry. We assume basic knowledge of Algebra and Algebraic Geometry, andwhenever more advanced knowledge is needed, we will elaborate and give the relevant back-ground. Most of this advanced machinery is presented in Section 1.3, as well as in the variouschapters.

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Review of developments preceding this thesis. The main topic of study in the thesis is inci-

dence geometry in three, four, and higher dimensions. We first review the earlier developments inthis field, and then go on to discuss our results and original contribution.

In its simplest, original form, the joints problem, posed in [28] in 1992, is to obtain a sharpupper bound on the number of points that can be incident to at least three non-coplanar lines, inany set of n lines in three dimensions; these points are called joints. Simple constructions showthat the number of joints can be Ω(n3/2) and the goal was to obtain a matching upper bound. After15 years of frustrating research, the best upper bound obtained using the “traditional” machinery,was O(n1.623) [46].

Then, in 2008, Guth and Katz [55] established the tight upper bound O(n3/2), thus solvingthe problem completely. They used in the proof several reasonably simple tools from algebraicgeometry, and we mention here two of them: (i) Given a set P of m points in R3, one can find atrivariate polynomial f of degree D = O(m1/3) that vanishes at all the points of P. (An appropriategeneralization holds in any dimension d ≥ 1, except that the degree of the resulting polynomialdrops to O(m1/d).) (ii) Given two trivariate polynomials f and g with no common factor, and withcorresponding zero sets Z( f ), Z(g), the number of lines that are fully contained in Z( f )∩Z(g)

is at most deg( f ) · deg(g); see Corollary 1.3.4. (This can be regarded as an extended variant ofBezout’s theorem (see, e.g., Section 1.3 and Fulton [49]).)

Here is a brief, rough, and informal description of the analysis of Guth and Katz. Given a setL of n lines in R3, they “force”, in a preliminary pruning and sampling step, most of the jointsof L to lie on the zero set Z( f ) of a polynomial f of degree D ≤ cn1/2, with a sufficiently smallconstant c, and then only consider lines of L that are also fully contained in Z( f ) (the other linesdo not generate too many joints). Now a joint incident to three non-coplanar lines, all containedin Z( f ), must be a singular point of f , and lines that contain more than D such joints must consistexclusively of singular points (each of the three first-order derivatives of f must vanish identicallyon such a line). Lines that contain fewer than D joints contribute at most nD = O(n3/2) joints, sothey can be ignored. Now, assuming f to be irreducible, and applying the preceding result (ii) tof and one of its partial derivatives, say fx, we conclude that the number of such “critical” lines isat most D2 ≤ c2n, which we can make smaller than, say, n/2. An inductive argument on n thencompletes the proof.

The actual proof in [55] is more involved and technical. It has been greatly simplified in twosubsequent papers by Kaplan, Sharir and Shustin [67] and by Quilodran [86]; the bound has alsobeen extended, in both papers, to any dimension d ≥ 3, where now a joint is a point incident to atleast d lines, not all in a common hyperplane; the worst-case bound is Θ(nd/(d−1)).

Incidences. Although the joints problem might appear, on the face of it, just a curiosity, laterdevelopments, as being reviewed here, show that it is in fact a significant pillar in the study ofincidences between points and lines, curves, hyperplanes or surfaces, as well as of several otherrelated fields, both combinatorial and algorithmic. We briefly mention the highlights of this topic,

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extensively studied during the past 30 years, and considered as one of the main active areas in com-binatorial geometry; see Pach and Sharir [83] for a bit outdated but still relevant comprehensivesurvey.

In the simplest form of the problem, we are given a set P of m (distinct) points and a set L of n

(distinct) lines in the plane, and we wish to obtain sharp upper and lower bounds for the maximumpossible value of I(P,L), the number of incidences between the points of P and the lines of L,where an incidence is a pair (p, `) ∈ P×L with p ∈ `.

The 1983 celebrated theorem of Szemeredi and Trotter [118] asserts that I(P,L)=O(m2/3n2/3+

m+n), and that this bound is tight in the worst case. Many extensions of the problem have beenstudied, in which one considers, instead of lines, other curves in the plane (e.g., circles), or linesand other curves in higher dimensions, or planes, hyperplanes, or more general surfaces in higherdimensions. In most of these extensions, though, tight bounds on the maximum number of inci-dences are not known. Incidence problems, besides being a fascinating topic of study in its ownright, show up in many applications in combinatorial geometry, including Erdos’s famous repeated

distances problem (see below), and many other problems concerning repeated patterns in a pointset.

Moreover, in general, there is a close connection between combinatorial and algorithmic ques-tions in geometry, which has been a major and recurring theme during the past 30 years, because(i) sharp combinatorial bounds are needed to estimate the efficiency of algorithms that computethe relevant structures, and (ii) both types of studies tend to use the same or very similar tools.In the specific case at hand, incidences have deep links to numerous problems in computationalgeometry, with several common tools that they share (most notably, space partitioning techniques,discussed in detail below). As some simple illustrations, the Szemeredi–Trotter bound is more orless the same as the best running time of an algorithm for performing n halfplane range queries ona set of m points in the plane, where the goal is to count the number of points in each halfplane, orfor counting intersections between m red line segments and n blue line segments in the plane, aswell as for many other algorithmic problems of a similar nature.

In addition, connections between incidences and the continuous variants of the Kakeya problem

[119] have been observed for some time, and have served as a major motivation for Guth and Katz(following an earlier breakthrough progress by Dvir [34] for the case of finite fields) to study thejoints problem, as indicated by the title of their first paper “Algebraic methods in discrete analogsof the Kakeya problem”.

If one considers incidences between m points and n lines in higher dimensions, say in d = 3dimensions, the problem, on first sight, seems totally uninteresting. Indeed, one can project thepoints and lines onto some generic plane, observe that incidences are preserved in the projection,and apply the Szemeredi–Trotter bound. Since the bound is worst-case tight in the plane, it con-tinues to be so in any higher dimension. The joints problem, in retrospect, was an attempt toremove the triviality from this extension, by forcing the input lines, in a sense, to be “truly three-dimensional”. As follows from the results of Guth and Katz (and even from the weaker previous

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results), one does indeed get improved bounds in truly three-dimensional scenes, when the amountof coplanarity of the input points and lines is kept under control.

Back to joints. Elekes, Kaplan and Sharir [39] have extended the study of [55] to consider notjust the number of joints but also the number of incidences between the joints of L and the lines ofL; that is, each joint is now counted with multiplicity equal to the number of lines incident to it. Infact, the analysis of [39] applies to any set of m points and any set of n lines in R3, provided that(a) each point is incident to at least three lines, and (b) no plane contains more than some specifiednumber of points. There is some “algebraic magic” behind the first condition (a); it has to do withthe fact that the second fundamental form of the vanishing polynomial f must also vanish at suchpoints (when they are non-singular; see [39, 55], and Section 1.3.5). The second condition, orsome alternative one, is needed to avoid situations in which, say, all the given points and lines liein a common plane, which, in view of the preceding discussion, is crucial in order to “bypass”the Szemeredi–Trotter incidence bound. Under the assumptions in [39], with no plane containingmore than n points, the maximum possible number of incidences is Θ(m1/3n) when m≥ n, and isequal to the Szemeredi–Trotter bound otherwise.

Distinct distances. The next development took place in an attempt to apply the new machineryto the planar distinct distances problem of Erdos [43]. In this celebrated problem the goal is toestablish a sharp lower bound on the minimum possible number of distinct distances between theelements of a set S of n points in the plane. Erdos noticed that the set of vertices of the

√n×√n

integer grid generates only O(n/√

logn) distinct distances, and conjectured this to be also thelower bound, namely, that any set of n points in the plane determines at least Ω(n/

√logn) distinct

distances. Again, traditional techniques, becoming progressively more sophisticated during the 65years since the original problem statement, have been unable to settle the conjecture, and the bestlower bound that was achieved, by Katz and Tardos [69], was Ω(n0.8641).

Nevertheless, about 15 years ago, Gy. Elekes had come up, in an unpublished note, with aningenious program to reduce the planar distinct distances problem to an incidence problem be-tween points and curves in three dimensions. To tackle the latter problem, though, he needed acouple of fairly deep conjectures, which neither he nor anybody else knew how to solve at thattime. If these conjectures could be established, they would have lead to the almost tight lowerbound Ω(n/ logn) on the number of distinct distances. In Elekes and Sharir [40], written by Sharirafter Elekes’s passing away in 2008, Elekes’s program has been laid out and developed, and thenew algebraic machinery has been applied to it, but this fell short of settling Elekes’s conjectures,as the algebraic machinery, available from the 2008 Guth–Katz paper and the follow-up ones, wasstill too weak.

This was taken care of in the second breakthrough of Guth and Katz [56], in November 2010,where they introduced new algebraic machinery, based on the polynomial ham sandwich theorem

of Stone and Tukey [116] from 1942, which allowed them to establish Elekes’s conjectures and

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thereby obtain the aforementioned lower bound Ω(n/ logn) for distinct distances. Specifically,their main result, an extension of the main conjecture of Elekes, is: Given N lines in three di-mensions, the number of points that are incident to at least k ≥ 3 of these lines is O(N3/2/k2),provided that no plane contains more than N1/2 lines. The case k = 2 is also treated in [56]. Thereone needs to assume that no plane or regulus (doubly ruled quadric) contains more than N1/2 lines,and the analysis is based on algebraic properties of ruled surfaces, established by Monge, Salmonand Cayley in the 19th century [81, 90]. It turns out that the question of k ≥ 3 can be formulatedas a question about incidences between points and lines in three dimensions. Specifically, Guthand Katz showed:1

Theorem 1.1.1 (Guth and Katz [56]). Let P be a set of m distinct points and L a set of n distinct

lines in R3, and let s≤ n be a parameter, such that no plane contains more than s lines of L. Then

I(P,L) = O(

m1/2n3/4 +m2/3n1/3s1/3 +m+n).

The application of the polynomial ham sandwich theorem in [56] results in a so-called poly-

nomial partitioning scheme, a new tool that appears to be very powerful in combinatorial andcomputational geometry, nicely complementing and strengthening the 20-years-old arsenal of ge-ometric partitions based on cuttings [29] and on simplicial partitions [76]. Roughly, it states that,given a set P of m points in Rd , and a parameter t < m, one can find a d-variate polynomial f , ofdegree D = O(t1/d), such that each (open) connected component (“cell”) of Rd \Z( f ) contains atmost m/t points of P; the number of cells is O(Dd) =O(t). This partitioning of P is not exhaustive,as some (perhaps many, or all) points of P may lie on Z( f ), and they require a special treatment,depending on the specific problem at hand. Handling these points in a systematic manner appearsto be a missing fundamental ingredient of the infrastructure of the new paradigm that has not yetbeen fully resolved (see below for an elaboration of this issue).

The power of the new polynomial partitioning technique has quickly been recognized by thecommunity, and has already lead to many interesting new results, and bears a lot of potential forthe future to come. It is also one of the major tools used in this thesis.

Incidences with curves and surfaces. It is only natural to replace lines by other (simple) ge-ometric objects, like circles or other algebraic curves, or, in three or higher dimensions, planes,spheres and other algebraic surfaces. We give here a brief account of a few such incidence results,before and after the “revolution”.

Points and curves, the planar case. The case of incidences between points and curves has arich history, starting with the aforementioned case of points and lines in the plane [31, 116, 117],where the worst-case tight bound on the number of incidences is Θ(m2/3n2/3 +m+ n), where m

is the number of points and n is the number of lines. Still in the plane, Pach and Sharir [83]

1 This bound is not explicitly stated in [56], but it follows directly from the bounds that are established there.

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extended this bound to incidence bounds between points and curves with k degrees of freedom.These are curves with the property that, for each set of k points, there are only µ = O(1) curvesthat pass through all of them, and each pair of curves intersect in at most µ points; µ is called themultiplicity (of the degrees of freedom).

Theorem 1.1.2 (Pach and Sharir [83]). Let P be a set of m points in R2 and let C be a set of n

bounded-degree algebraic curves in R2 with k degrees of freedom and with multiplicity µ . Then

I(P,C ) = O(

mk

2k−1 n2k−22k−1 +m+n

),

where the constant of proportionality depends on k and µ .

Remarks. (1) The result of Pach and Sharir holds for more general families of curves, not neces-sarily algebraic, but, since algebraicity will be assumed in higher dimensions, we assume it alsoin the plane.

(2) Recently, Sheffer et al. [109] extended the result of Pach and Sharir to the complex plane, byshowing that if P is a set of m points in C2 and C is a set of n bounded-degree algebraic curves inC2 with k degrees of freedom and with multiplicity µ , then

I(P,C ) = O(

mk

2k−1+ε n2k−22k−1 +m+n

),

for any ε > 0, where the constant of proportionality depends on ε,k and µ .

Except for the case k = 2 (lines have two degrees of freedom), the bound is not known, andstrongly suspected not to be tight in the worst case. Indeed, in a series of papers during the2000’s [4, 11, 75], an improved bound has been obtained for incidences with circles, parabolas, orother families of curves with certain properties (see [4] for the precise formulation). Specifically,for a set P of m points and a set C of n circles, or parabolas, or similar curves [4], we have

I(P,C ) = O(

m2/3n2/3 +m6/11n9/11 log2/11(m3/n)+m+n). (1.1)

Some further (slightly) improved bounds, over the bound in Theorem 1.1.2, for more generalfamilies of curves in the plane, have been obtained by Chan [25, 26] and by Bien [18]. They are,however, considerably weaker than the bound in (1.1).

Recently, Sharir and Zahl [107] have considered families of constant-degree algebraic curvesin the plane that belong to an s-dimensional family of curves. This means that each curve in thatfamily can be represented by a constant number of real parameters, so that, in this parametricspace, the points representing the curves lie in an s-dimensional algebraic variety F of someconstant degree (to which we refer as the “complexity” of F ). For example, lines or unit circlesform 2-dimensional families, and arbitrary circles form a 3-dimensional family. See [107] formore details.

Theorem 1.1.3 (Sharir and Zahl [107]). Let C be a set of n algebraic plane curves that belong

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to an s-dimensional family F of curves of maximum constant degree E, no two of which share a

common irreducible component, and let P be a set of m points in the plane. Then, for any ε > 0,

the number I(P,C ) of incidences between the points of P and the curves of C satisfies

I(P,C ) = O(

m2s

5s−4 n5s−65s−4+ε +m2/3n2/3 +m+n

),

where the constant of proportionality depends on ε , s, E, and the complexity of the family F .

Except for the factor O(nε), this is a significant improvement over the bound in Theorem 1.1.2(for s ≥ 3), in cases where the assumptions in Theorem 1.1.3 imply (as they often do) that C hask = s degrees of freedom. Concretely, when k = s, we obtain an improvement, except for the factornε , for the entire “meaningful” range n1/s ≤m≤ n2, in which the bound is superlinear. The factornε makes the bound in [107] slightly weaker only when m is close to the lower end n1/s of thatrange. Note also that, for s = 3, this bound almost coincides with the one in (1.1).

Incidences with curves in three dimensions. The seminal work of Guth and Katz [56], as givenin Theorem 1.1.1, has lead to many recent works on incidences between points and lines or othercurves in three and higher dimensions; see [24, 57, 98, 102, 103] for a sample of these results.

Of particular significance is the recent work of Guth and Zahl [57] on the number of 2-richpoints in a collection of curves in R3, namely, points incident to at least two of the given curves.For the case of lines, Guth and Katz [56] have shown that the number of such points is O(n3/2),when no plane or regulus contains more than n1/2 lines. Guth and Zahl obtain the same asymptoticbound for general algebraic curves, under analogous (but stricter) restrictive assumptions. Forexample, by taking circles instead of lines, Guth and Zahl’s assumption is that no plane or spherecontains more than O(n1/2) circles. In the general case, the assumption is that no surface that isdoubly ruled by curves in a given family of curves contains more than O(n1/2) such curves (weelaborate on this notion below).

The study in this thesis requires the extension to three dimensions of the notions of havingk degrees of freedom and of being an s-dimensional family of curves. The definitions of theseconcepts, as given above for the planar case, extend, in a fairly straightforward manner, to three(or higher) dimensions, as will be discussed in more detail later on.

We note that these two concepts do not coincide anymore in three or higher dimensions. Forexample, lines in three dimensions have two degrees of freedom, but they form a 4-dimensionalfamily of curves (this is the number of parameters needed to specify a line in R3). See Section 1.2.4for more details concerning this discrepancy.

Points and surfaces. Many of the earlier works on point-surface incidences in three dimensionshave only considered special classes of surfaces, most notably planes and spheres (see below).The case of more general surfaces has barely been considered, till the work of Zahl [125], who hasstudied the general case of incidences between m points and n bounded-degree algebraic surfaces

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in R3 that have k degrees of freedom. More precisely, in analogy with the case of curves, oneneeds to assume that for any k points there are at most µ = O(1) of the given surfaces that passthrough all of them. Zahl’s bound is O

(m

2k3k−1 n

3k−33k−1 +m+n

), with the constant of proportionality

depending on k, µ , and the maximum degree of the surfaces.

By Bezout’s theorem, if we require every triple of the given surfaces to have finite intersection,the number of intersection points would be at most E3, where E is the degree of the surfaces. Inparticular, E3 +1 points would then have at most two of the given surfaces passing through all ofthem. In many instances, though, the actual number of degrees of freedom can be shown to bemuch smaller. In general, though, one has to allow for the possibility that three (or more) surfacesintersect in a common curve (for example, many planes can intersect in a common line, or manyspheres can intersect in a common circle). Handling these general situations ia a major themestudied in Chapter 8 of this thesis.

Zahl’s bound was later generalized by Basu and Sombra [16] to incidences between points andbounded-degree hypersurfaces in R4 satisfying certain analogous conditions.

Points and planes. Initial partial results on point-plane incidences in three dimensions havebeen obtained by Edelsbrunner, Guibas and Sharir [36]. More recently, Apfelbaum and Sharir [6](see also Brass and Knauer [21] and Elekes and Toth [41]) have shown that if the incidence graphG(P,H), for a set P of m points and a set H of n planes, does not contain a copy of Kr,s, for constantparameters r and s, then I(P,H) = O(m3/4n3/4 + m + n). In more generality, Apfelbaum andSharir [6] have shown that if I = I(P,H) is significantly larger than this bound, then G(P,H) mustcontain a large complete bipartite subgraph P′×H ′, such that |P′| · |H ′|= Ω(I2/(mn))−O(m+n).Moreover, as also shown in [6] (slightly improving a similar result of Brass and Knauer [21]),G(P,H) can be expressed as the union of complete bipartite graphs Pi×Hi so that ∑i(|Pi|+ |Hi|) =O(m3/4n3/4 +m+ n). (This is a specialization to the case d = 3 of a similar result of [6, 21] inany dimension d, and is a special case of the more general analysis of point-surface incidences inthree dimensions in this thesis, as will be shortly reviewed.)

We note that Fox et al. [47] present a more general framework that includes incidences prob-lems of many kinds, and yields, for incidences between points and planes, almost the same bound,namely O(m3/4+ε n3/4+ε +m+ n), for any ε > 0, where the constant of proportionality dependson ε .

Points and spheres. Earlier works on the special case of point-sphere incidences in three dimen-sions go back to Chung [30] and to Clarkson et al. [31], and continue with the work of Aronov etal. [9]. Later, Agarwal et al. [1] have bounded the number of non-degenerate spheres with respectto a given point set; this bound was subsequently improved by Apfelbaum and Sharir [7].2

2 Given a finite point set P ⊂ R3 and a constant 0 < η < 1, a sphere σ ⊂ R3 is called η-degenerate (with respect to P),if there exists a circle c ⊂ σ such that |c∩P| ≥ η |σ ∩P|. This definition extends a similar earlier definition for planes(and lines) in Elekes and Toth [41].

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The aforementioned recent work of Zahl [125] can be applied in the case of spheres if oneassumes that no three, or any larger but constant number, of the spheres intersect in a commoncircle. In this case the family has k = 3 degrees of freedom — any three points determine aunique circle that passes through all of them, and, by assumption, only O(1) spheres containthat circle. Zahl’s bound then becomes O(m3/4n3/4 +m+ n). In particular, this bound holds forcongruent (unit) spheres (where three such spheres can never contain a common circle). The caseof incidences with unit spheres have also been studied in Kaplan et al. [65], with the same upperbound; see also [105]. This bound slightly improves an older upper bound in [28].

If many spheres of the family can intersect in a common circle, the bound does no longerhold. The only earlier work that handled this situation is by Apfelbaum and Sharir [6], where itwas assumed that the given spheres are non-degenerate. In this case the bound obtained in [6] isO(m8/11n9/11 +m+n). Interestingly, this is also the bound that Zahl’s result would have yieldedif the spheres had k = 4 degrees of freedom, which however they only “almost have”: four genericpoints determine a unique sphere that passes through all of them, but four co-circular points deter-mine an infinity of such spheres.

Distinct and repeated distances in three dimensions. The case of spheres is of particular in-terest, because it arises, in a standard and natural manner, in the analysis of distinct and repeated

distances determined by n points in three dimensions. After Guth and Katz’s almost complete so-lution of the number of distinct distances in the plane [56], the three-dimensional case has movedto the research forefront. The prevailing conjecture, due to Erdos, is that the lower bound isΩ(n2/3) (which is the best possible in the worst case, because the n points of the n1/3×n1/3×n1/3

integer grid determine only O(n2/3) distinct distances. However, the current record, obtained bycombining the results of Solymosi and Vu [114] with that of Guth and Katz [56], is Ω∗(n3/5) (anotation hiding polylogarithmic factors), and the problem of closing that gap seems much harderthan the two-dimensional problem.

A standard reduction to point-sphere incidences goes as follows. Let P denote the set of n

points, let t denote the number of distinct distances determined by P, and let δ1, . . . ,δt denote theactual distances. Define a set S of nt spheres, whose centers are in P, and whose radii are δi, fori = 1, . . . , t. Then I(P,S) = n(n−1), and an upper bound on I(P,S) in terms of n and t immediatelyimplies a lower bound on the number t of distinct distances.

Obtaining lower bounds for distinct distances using circles (in the plane) or spheres (in higherdimensions) in the manner just sketched has in general been suboptimal when compared with moreeffective methods (such as in [56]), but, as we will show in Chapter 8, it can still be used to obtainnew lower bounds (larger than Ω(n2/3)) in certain favorable special cases, such as the case whenthe points lie on an algebraic variety of constant degree.

The status of the case of unit, or repeated distances is also far from being satisfactory. Herethe reduction to point-circle or point-sphere incidences is even more straightforward and natural.That is, let P denote the set of n points and let S denote the set of unit circles (in the plane) or

9

unit spheres (in higher dimensions) whose centers lie in P. Then, I(P,S) is equal to twice thenumber of unit distances. The planar case is “stuck” with the upper bound O(n4/3) of Spenceret al. [115] from the 1980’s, which is also an immediate consequence of the Pach-Sharir’s bound(Theorem 1.1.2) with k = 2 degrees of freedom. The proof has been greatly simplified in [31, 117],but the bound has not been improved (and is actually known to be tight for certain non-Euclideannorms [122]). In the plane, the best known lower bound, noted by Erdos, is n1+c/ log logn. Theupper bound O(n4/3) also holds for points on the 2-sphere, and there, surprisingly, it is tight inthe worst case (when the repeated distance is 1, say, and the radius of the sphere is 1/

√2) [45],

but it is strongly believed that in the plane the correct bound is equal or close to the near-linearlower bound mentioned above. In three dimensions, the aforementioned bound of Zahl [125], alsoreconstructed in Kaplan et al. [65], with k = 3 degrees of freedom, immediately implies the upperbound O(n3/2) on the number of repeated distances (a slight improvement over the earlier boundof Clarkson et al. [31]), and the best known lower bound is still only Ω(n4/3 log logn) [43]. A veryrecent small improvement of the upper bound has been announced in Zahl [128].

1.2 Our results

The thesis consists of seven studies, presented in four parts and reviewed in the following fourrespective subsections.

1.2.1 Incidences between points and lines in R4

In Chapter 2, we extend the study of Guth and Katz [56], from point-line incidences in threedimensions to point-line incidences in four dimensions, giving worst-case tight or nearly tightbounds on the number of such incidences. This much harder question requires the development ofadditional tools and techniques from algebraic geometry, most of which are reviewed in Section 1.3below, and a variety of methods for interfacing them with the problem at hand. Loosely speaking,we study the patterns in which lines can touch one another when they are “thrown” into fourdimensions, where a major subproblem is to understand these patterns when the lines lie in some3-dimensional algebraic surface.

In a preliminary work [99], we proved that, for each ε > 0, there exists an integer cε , so thatthe following holds. Let P be a set of m distinct points and L a set of n distinct lines in R4, and letq,s≤ n be parameters, such that (i) for any polynomial f ∈R[x,y,z,w] of degree ≤ cε , its zero setZ( f ) does not contain more than q lines of L, and (ii) no 2-plane contains more than s lines of L.Then,

I(P,L)≤ Aε

(m2/5+ε n4/5 +m1/2+ε n2/3q1/12 +m2/3+ε n4/9s2/9

)+A(m+n), (1.2)

where Aε depends on ε , and A is an absolute constant.

We have subsequently improved and tightened the bound. The improved result, stated next, ispresented in Chapter 2, and has appeared in [102].

10

Theorem 1.2.1. Let P be a set of m distinct points and L a set of n distinct lines in R4, and let

q,s ≤ n be parameters, such that (i) each hyperplane or quadratic hypersurface contains at most

q lines of L, and (ii) each 2-flat contains at most s lines of L. Then

I(P,L)≤ 2c√

logm(

m2/5n4/5 +m)+A

(m1/2n1/2q1/4 +m2/3n1/3s1/3 +n

), (1.3)

where A and c are suitable absolute constants. When m ≤ n6/7 or m ≥ n5/3, we have the sharper

bound

I(P,L)≤ A(

m2/5n4/5 +m+m1/2n1/2q1/4 +m2/3n1/3s1/3 +n). (1.4)

In general, except for the factor 2c√

logm, the bound is tight in the worst case, for any values of

m,n, and for corresponding suitable ranges of q and s.

The term m2/3n1/3s1/3 comes from the planar Szemeredi–Trotter bound [118], and is unavoid-able, as it can be attained if we densely “pack” points and lines into 2-flats, in patterns that attainthe Szemeredi–Trotter bound.

Likewise, the term m1/2n1/2q1/4 comes from the bound of Guth and Katz [56] in three dimen-sions (as in Theorem 1.1.1), and is again unavoidable, as it can be attained if we densely “pack”points and lines into hyperplanes, in patterns that attain the bound in three dimensions.

Our solution employs fairly heavy machinery from algebraic geometry, some going back morethan 150 years. For example, an 1846 theorem due to Cayley and Salmon (and, independentlyobtained by Monge) [81, 90], states that an algebraic surface in three dimensions can fully containonly a bounded number of lines, unless it it is “ruled” by lines. The study of ruled surfaces,including the way they are embedded in four and higher dimensions, is a central theme in our study.Another related theorem that we use, from 1901, due to Segre and Severi [94, 95], characterizeshypersurfaces in complex 4-space that are “infinitely ruled” by lines. This machinery is partlypresented in Section 1.3, and partly in Chapter 2.

Quadrics may increase incidences. In a follow-up work with Ruixiang Zhang [111], presentedin Chapter 3, we show that the restrictions made in Theorem 1.2.1 are essential, i.e., droppingthem would result in a larger number of incidences. Specifically, we show that the condition inassumption (i) of Theorem 1.2.1 that quadrics do not contain too many lines, cannot be dropped,in certain cases, by proving the following theorem.

Theorem 1.2.2 (Solomon and Zhang [111]). For integers m,n, there is a configuration of m points

and n lines in R4, such that all the points (resp., lines) are contained (resp., fully contained) in

S := (x1,x2,x3,x4) ∈ R4 | x1 = x22 + x2

3− x24,

and (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in a com-

mon hyperplane is O(n/m1/3), and (iii) the number of incidences between the points and lines is

11

Ω(m2/3n1/2 +m+n).

In particular, when n9/8 < m < n3/2, this results in a larger number of incidences than thebound in Theorem 1.2.1.

1.2.2 Incidences between points and lines in R3, with applications to Ramsey-type theorems for

lines in R3

In Chapter 4, we give a fairly elementary and simple proof of the Guth-Katz bound in three di-mensions, namely, we show that the number of incidences between m points and n lines in R3, sothat no plane contains more than s lines, is

O(

m1/2n3/4 +m2/3n1/3s1/3 +m+n)

(in the precise statement, the constant of proportionality of the first and third terms depends, in arather weak manner, on the relation between m and n).

As already mentioned, the original proof in [56] uses fairly involved machinery from algebraicand differential geometry, so it is highly desirable to simplify the proof, in the interest of better un-derstanding the geometric structure of the problem, and of providing new tools for tackling similarproblems. Such an approach has also recently been undertaken in a follow-up study of Guth [51].In Chapter 4, we present a different and simpler derivation, with better bounds than those in [51],and without the restrictive assumptions made there. Our result has a potential for applications toother incidence problems in higher dimensions, and, in a sense, the four-dimensional bound thatwe present in Chapter 2 is one such application. The results of Chapter 4 have appeared in [100].

Ramsey-type theorems for lines in R3. In Chapter 5, we prove, jointly with Michael Payne andJean Cardinal [24], geometric Ramsey-type statements on collections of lines in 3-space. Thesestatements give guarantees on the size of a clique or an independent set in (hyper)graphs inducedby incidence relations between lines, points, and reguli in 3-space. Among other things, we provethe following statements.

1. The intersection graph of n lines in R3 has a clique or an independent set of size Ω(n1/3).

2. Every set of n lines in R3 has a subset of Ω(√

n) lines that are all stabbed by one line, or asubset of Ω

((n/ logn)1/5

)lines such that no 6-subset is stabbed by a common line.

3. Every set of n lines in general position in R3 has a subset of Ω(n2/3) lines that lie on acommon regulus, or a subset of Ω(n1/3) lines such that no 4-subset is contained in a regulus.

We also refer to a recent work of Pach et al. [84], that studies geometric questions related to thegeometry of lines and segments in R3.

12

The results of Chapter 5 have appeared in [24].

1.2.3 Improved bounds on incidences of lines with points on a variety

When the points lie on some two-dimensional variety in 3-space or three-dimensional variety in4-space, whose degree is not too large, we show, in Chapter 6 of the thesis, that the numberof point-line incidences is substantially smaller than the bounds in Theorem 1.1.1 and in The-orem 1.2.1, respectively. In fact, our results do not assume that the two- or three-dimensionalvariety is embedded in 3-space or 4-space, respectively, and they also hold when the varieties areembedded in higher-dimensional Euclidean spaces. Concretely, our first main result, for this setup,is the following theorem.

Theorem 1.2.3. (a) The real case: Let P be a set of m distinct points and L a set of n distinct

lines in Rd , for any d ≥ 3, and let 2≤ s≤ D be two integer parameters, so that all the points and

lines lie in a common two-dimensional algebraic variety V of degree D that does not contain any

2-flat, and so that no 2-flat contains more than s lines of L. Then

I(P,L) = O(

m1/2n1/2D1/2 +m2/3D2/3s1/3 +m+n). (1.5)

(b) The complex case: Under exactly the same assumptions, when the ambient space is Cd , for

any d ≥ 3, we have

I(P,L) = O(

m1/2n1/2D1/2 +m2/3D2/3s1/3 +D3 +m+n). (1.6)

In both cases, when D and s are constants, we get the linear bound O(m+n).

Our next main result for this setup is the following theorem.

Theorem 1.2.4. (a) The real case: Let P be a set of m distinct points and L a set of n distinct

lines in Rd , for any d ≥ 4, and let s and D be parameters, such that (i) all the points of P lie on a

three-dimensional algebraic variety of degree D, without 3-flat or 3-quadric components, and (ii)

no 2-flat contains more than s lines of L. Then

I(P,L) = O(

m1/2n1/2D+m2/3n1/3s1/3 +nD+m). (1.7)

(b) The complex case: Under exactly the same assumptions, when the ambient space is Cd , for

any d ≥ 4, we have

I(P,L) = O(

m1/2n1/2D+m2/3n1/3s1/3 +D6 +nD+m). (1.8)

In both cases, when D and s are constants, we get the linear bound O(m+n).

13

A major feature of this analysis is that the bounds also hold over the complex field, as in (1.6)and (1.8) (with a small added term that is negligible when the degree of the ambient variety issmall). In contrast, the more general point-line incidences bounds in Theorem 1.1.1 and 1.2.1are only known to hold over the reals. The interplay between the real and complex cases will bediscussed in detail in Section 1.3.

The results of Chapter 6 have appeared in [103].

1.2.4 Incidences with curves in three and higher dimensions

The results presented so far involve incidences with lines. Our next set of results, presented inPart IV, involve incidences between points and constant-degree algebraic curves in three andhigher dimensions.

Two different studies address this problem. We first study the general problem, in any dimen-sion. This work, joint with Adam Sheffer, is presented in Chapter 7 and has appeared in [97].The second study is presented in Chapter 8 and has appeared in [106]. It only studies the three-dimensional case, but obtains significantly refined results (over the general results in Chapter 7).Chapter 8 also contains a study of incidences between points and surfaces in R3, which will bereviewed later in this subsection.

Incidences with algebraic curves in Rd . In Chapter 7, we prove, jointly with Adam Sheffer, abound on the number of incidences between points and algebraic curves in Rd . Specifically, weprove that the number of incidences between m points and n bounded-degree algebraic curves withk degrees of freedom in Rd is

O

(m

kdk−d+1+ε n

dk−ddk−d+1 +

d−1

∑j=2

mk

jk− j+1+ε nd( j−1)(k−1)

(d−1)( jk− j+1) q(d− j)(k−1)

(d−1)( jk− j+1)j +m+n

), (1.9)

for any ε > 0, where the constant of proportionality depends on k,ε and d, provided that no j-dimensional surface of degree ≤ c j(k,d,ε), a constant parameter depending on k, d, j, and ε ,contains more than q j input curves, and that the q j’s satisfy certain mild conditions. The notion ofk degrees of freedom, defined in the planar case in the context of Theorem 1.1.2, easily extends,for curves, to any higher dimension. That is, we require that at most µ = O(1) curves of the givenfamily pass through any k specified points, and that any pair of curves intersect in at most µ points.We will return to this notion in the review of Chapter 8 given below.

This bound generalizes the planar incidence bound of Pach and Sharir (Theorem 1.1.2) to Rd .It also generalizes (with certain restrictions) several other results, including some of the resultspresented in the thesis. For example, for the case of lines in any dimension d, we have k = 2, and

14

the bound in (1.9) is

O

(m

2d+1+ε n

dd+1 +

d−1

∑j=2

m2

j+1+ε nd( j−1)

(d−1)( j+1) q(d− j)

(d−1)( j+1)j +m+n

).

This almost reconstructs the bound of Guth and Katz [56] for d = 3, and the results of Chapter 2for d = 4. A similar “almost reconstruction” applies to incidences with circles in three dimensions(where k = 3), as compared with the results of Sharir et al. [98] and those in Chapter 8 (reviewedbelow).

In spite of its general applicability, the bound in (1.9) suffers from several handicaps, alsodiscussed in more detail later on: (i) It has the factors mε in several of its terms. (ii) The assump-tions that it requires, that no j-dimensional surface of degree ≤ c j(k,d,ε) contains more than q j

curves, is fairly restrictive; these degree bounds, although being constants, can be rather large. (iii)The “lower-dimensional” bounds in the sum in (1.9) are not sharp when the q j’s are small, e.g.,when comparing them to the results in [56] and in Chapter 2 for the case of lines. For example,in d = 3 dimensions, the term in (1.9) is O(m2/3+ε n1/2q1/6

2 ), as opposed to O(m2/3n1/3q1/32 ) in

Theorem 1.1.1, and similar discrepancies occur in d = 4 dimensions.

In spite of these handicaps, the result is fairly general and powerful. It partly generalizes aresult of Guth [51] in three dimensions (Guth’s three-dimensional bound has a better dependencyon q2). It also improves a d-dimensional general incidence bound by Fox et al. [47], in the spe-cial case of incidences with algebraic curves. This bound is also related to works by Dvir andGopi [35] and by Hablicsek and Scherr [59] concerning rich lines in high-dimensional spaces.Our bound is a general-purpose bound in Rd , and as such, is not known to be tight in most cases.In Chapter 8, we significantly improve this bound, and get rid of the handicaps noted above, in thethree-dimensional case.

Incidences with curves in three dimensions. We now proceed to review the results in Chap-ter 8, the final chapter in the thesis. This review is longer than those of the previous chapters,because we also need to discuss several technical assumptions and results, on which our newbounds depend.

Preliminaries. This chapter contains two sets of results, one involving incidences with curves inthree dimensions, and one involving incidences with surfaces in three dimensions. We begin withthe case of curves.

In order to state our results, we first define the notions of k degrees of freedom (already men-tioned above), of constructibility, and of surfaces infinitely ruled by curves.

k degrees of freedom. Let C0 be an infinite family of irreducible algebraic curves of constantdegree E in R3. Formally, in complete analogy with the planar case, we say that C0 has k degrees

15

of freedom with multiplicity µ , where k and µ are constants, if (i) for every tuple of k points in R3

there are at most µ curves of C0 that are incident to all k points, and (ii) every pair of curves of C0

intersect in at most µ points. As in [83], the bounds that we derive depend more significantly on k

than on µ—see below.

We remark that the notion of k degrees of freedom gets more involved for surfaces, and raisesseveral annoying technical issues. For example, how many points does it take to define, say, asphere (up to a fixed multiplicity)? Clearly, four generic points do the job (they define a uniquesphere passing through all four of them), but four co-circular points do not.

While it seems possible to come up with some sort of working definition, we bypass thisissue in this thesis, by defining this notion, for a family F of surfaces, only with respect to agiven surface V , by saying that F has k degrees of freedom with respect to V if the family of theirreducible components of the curves σ ∩V | σ ∈F, counted without multiplicity, has k degreesof freedom, in the sense just defined. In the case of spheres, for example, this definition gives fourdegrees of freedom when V is neither a plane nor a sphere, but only three when V is a plane or asphere.

Constructibility. In the statements of the point-curve incidence theorems, we also assume thatC0 is a constructible family of curves. This notion generalizes the notion of being algebraic, and isdiscussed in detail in Guth and Zahl [57]. Informally, a set Y ⊂Cd is constructible if it is a Booleancombination of algebraic sets. The formal definition goes as follows (see, e.g., Harris [61, Lecture3]). For z ∈ C, define v(0) = 0 and v(z) = 1 for z 6= 0. Then Y ⊆ Cd , for some fixed d, is aconstructible set if there exist a finite set of polynomials f j : Cd → C, for j = 1, . . . ,JY , and asubset BY ⊂ 0,1JY , so that x ∈ Y if and only if (v( f1(x)), . . . ,v( fJY (x))) ∈ BY .

The constructible sets form a Boolean algebra. This means that finite unions and intersectionsof constructible sets are constructible, and the complement of a constructible set is constructible.Another fundamental property of constructible sets is that, over C, the projection of a constructibleset is constructible; this is known as Chevalley’s theorem (see Harris [61, Theorem 3.16] and Guthand Zahl [57, Theorem 2.3]). If Y is a constructible set, we define the complexity of Y to bemin(deg f1+ · · ·+deg fJY ), where the minimum is taken over all representations of Y , as describedabove. As just observed, constructibility of a family C0 of curves extends the notion of C0 beings-dimensional. One of the main motivations for using the notion of constructible sets (rather thanjust s-dimensionality) is the fact, established by Guth and Zahl [57, Proposition 3.3], that the setC3,E of irreducible curves of degree at most E in complex 3-dimensional space (either affine orprojective) is a constructible set of constant complexity that depends only on E.

Remark. The definition of constructibility is given over the complex field C. This is in accordancewith most of the basic algebraic geometry tools, which have been developed over the complexfield; we will elaborate further about it below.

16

Surfaces infinitely ruled by curves. Back in three dimensions, a surface V is (singly, doubly, orinfinitely) ruled3 by some family Γ of curves of degree at most E, if each point p ∈ V is incidentto (at least one, at least two, or infinitely many) curves of Γ that are fully contained in V . Asalready discussed, the connection between ruled surface theory and incidence geometry goes backto the pioneering work of Guth and Katz [56] and shows up in many subsequent works. A detailedreview of ruled surfaces is given in Section 1.3. See also Guth’s recent survey [53] and recentbook [54], and Kollar [71] for details.

For now, we only consider the case of infinitely ruled surfaces. We recall that the only surfacesthat are infinitely ruled by lines are planes (see, e.g., Fuchs and Tabachnikov [48, Corollary 16.2]),and that the only surfaces that are infinitely ruled by circles are spheres and planes (see, e.g.,Lubbes [74, Theorem 3] and Schicho [93]). It should be noted that, in general, for this definitionto make sense, it is important to require that the degree E of the ruling curves be much smaller thandeg(V ). Otherwise, every variety V is infinitely ruled by, say, the curves V ∩h, for hyperplanes h,having the same degree as V . A challenging open problem is to characterize all the surfaces thatare infinitely ruled by algebraic curves of degree at most E (or by certain special classes thereof).However, the following result of Guth and Zahl provides a useful necessary condition for thisproperty to hold.

Theorem 1.2.5 (Guth and Zahl [57]). Let V be an irreducible surface, and suppose that it is

doubly ruled by curves of degree at most E. Then deg(V )≤ 100E2.

In particular, an irreducible surface that is infinitely ruled by curves of degree at most E isdoubly ruled by these curves, so its degree is at most 100E2. Therefore, if V is irreducible ofdegree D larger than this bound, V cannot be infinitely ruled by curves of degree at most E.

Incidences with curves in R3.

Theorem 1.2.6 (Curves in R3). Let P be a set of m points and C a set of n irreducible algebraic

curves of constant degree E, taken from a constructible family C0, of constant complexity, with k

degrees of freedom (and some multiplicity µ) in R3, such that no surface that is infinitely ruled by

curves of C0 contains more than q curves of C , for a parameter q< n. Then

I(P,C ) = O(

mk

3k−2 n3k−33k−2 +m

k2k−1 n

k−12k−1 q

k−12k−1 +m+n

), (1.10)

where the constant of proportionality depends on k, µ , E, and the complexity of the family C0.

Remarks. (1) In certain favorable situations, such as in the cases of lines or circles, discussedabove, the surfaces that are infinitely ruled by curves of C0 have a simple characterization. In suchcases the theorem has a stronger flavor, as its assumption on the maximum number of curves on

3 Here we use the simple definition, requiring ruledness et every point of the surface. As noted earlier, this can be donewithout loss of generality.

17

a surface has to be made only for this concrete kind of surfaces. For example, as already noted,for lines (resp., circles) we only need to require that no plane (resp., no plane or sphere) containsmore than q of the curves. In general, as mentioned, characterizing infinitely ruled surfaces by aspecific family of curves is a difficult task. Nevertheless, we can overcome this issue by replacingthe assumption in the theorem by a more restrictive one, requiring that no surface that is infinitelyruled by curves of degree at most E contain more than q curves of C . By Theorem 1.2.5, anyinfinitely ruled surface of this kind must be of degree at most 100E2. Hence, an even simpler(albeit weaker) formulation of the theorem is to require that no surface of degree at most 100E2

contains more than q curves of C . This can indeed be much weaker: In the case of circles, say,instead of making this requirement only for planes and spheres, we now have to make it for everysurface of degree at most 400.

(2) In several recent works (including the one presented in Chapter 7 of the thesis; see [51, 97, 98]),the assumption in the theorem is replaced by a much more restrictive assumption, that no surface ofdegree at most cε contains more than q given curves, where cε is a constant that depends on anotherprespecified parameter ε > 0 (where ε affects the resulting incidence bound), and is typically verylarge (and increases as ε becomes smaller). Getting rid of such an ε-dependent constant (and ofthe ε in the exponent) is a significant feature of Theorem 1.2.6.

(3) Theorem 1.2.6 generalizes the incidence bound of Guth and Katz [56], obtained for the caseof lines. In this case, lines have k = 2 degrees of freedom, they certainly form a constructible (infact, a 4-dimensional) family of curves, and, as just noted, planes are the only surfaces in R3 thatare infinitely ruled by lines. Thus, in this special case, both the assumptions and the bound inTheorem 1.2.6 are identical to those in Guth and Katz [56]. That is, if no plane contains more thanq input lines, the number of incidences is O(m1/2n3/4 +m2/3n1/3q1/3 +m+n).

Improving the bound. The bound in Theorem 1.2.6 can be further improved, if we also throwinto the analysis the dimensionality s of the family C0. Actually, as will follow from the proof,the dimensionality that will be used is only that of any subset of C0 whose members are fullycontained in some variety that is infinitely ruled by curves of C0. As just noted, such a varietymust be of constant degree (at most 100E2, or smaller as in the cases of lines and circles), and theadditional constraint that the curves be contained in the variety can typically be expected to reducethe dimensionality of the family.

For example, if C0 is the collection of all circles in R3, then, since the only surfaces that areinfinitely ruled by circles are spheres and planes, the subfamily of all circles that are contained insome sphere or plane is only 3-dimensional (as opposed to the entire C0, which is 6-dimensional).

We capture this setup by saying that C0 is a family of reduced dimension s if, for each surfaceV that is infinitely ruled by curves of C0, the subfamily of the curves of C0 that are fully containedin V is s-dimensional. In this case we obtain the following variant of Theorem 1.2.6.

Theorem 1.2.7 (Curves in R3). Let P be a set of m points and C a set of n irreducible algebraic

18

curves of constant degree E, taken from a constructible family C0 with k degrees of freedom (and

some multiplicity µ) in R3, such that no surface that is infinitely ruled by curves of C0 contains

more than q of the curves of C , and assume further that C0 is of reduced dimension s. Then

I(P,C ) = O(

mk

3k−2 n3k−33k−2

)+Oε

(m2/3n1/3q1/3 +m

2s5s−4 n

3s−45s−4 q

2s−25s−4+ε +m+n

), (1.11)

for any ε > 0, where the first constant of proportionality depends on k, µ , s, E, and the maximum

complexity of any subfamily of C0 consisting of curves that are fully contained in some surface

that is infinitely ruled by curves of C0, and the second constant also depends on ε .

Remarks. (1) Theorem 1.2.7 is an improvement of Theorem 1.2.6 when s ≤ k and m > n1/k, incases where q is sufficiently large so as to make the second term in (1.10) dominate the first term;for smaller values of m the bound is always linear. This is true except for the term qε , whichaffects the bound only when m is very close to n1/k (when s = k). When s > k we get a thresholdexponent β = 5s−4k−2

ks−4k+2s (which becomes 1/k when s = k), so that the bound in Theorem 1.2.7 isstronger (resp., weaker) than the bound in Theorem 1.2.6 when m> nβ (resp., m< nβ ), again, upto the extra factor qε .

(2) The bounds in Theorems 1.2.6 and 1.2.7 improve, in three dimensions, the result in Chapter 7,in three significant ways: (i) The leading terms in both bounds are essentially the same, but thepresent bound is sharper, in that it does not include the factor O(nε) appearing in Chapter 7. (ii)The assumption here, concerning the number of curves on a low-degree surface, is much weakerthan the one made in Chapter 7, where it was required that no surface of some (constant butpotentially very large) degree cε , that depends on ε , contains more than q curves of C (See alsoRemark (2) following Theorem 1.2.6). (iii) The two variants of the non-leading terms here aresignificantly smaller than those in Chapter 7, and, in a certain sense (that will be elaborated inChapter 8) are best possible.

Point-circle incidences in R3. Theorem 1.2.7 yields a new bound for the case of incidencesbetween points and circles in R3, which improves over the previous bound of Sharir, Sheffer, andZahl [98]. Here, as already discussed, we have k = s = 3 for the case of circles (s = 3 is thedimension of the family of the circles contained in some sphere or plane), so the theorem yieldsthe bound

I(P,C ) = O(

m3/7n6/7 +m2/3n1/3q1/3 +m6/11n5/11q4/11+ε +m+n),

for any ε > 0, where q is the maximum number of the given circles that are coplanar or cospherical.In fact, the extension of the planar bound (1.1) to higher dimensions, due to Aronov et al. [8],asserts that, for any set C of circles in any dimension, we have

I(P,C ) = O(

m2/3n2/3 +m6/11n9/11 log2/11(m3/n)+m+n), (1.12)

19

which is slightly better than the general bound of Sharir and Zahl [107] (given in Theorem 1.1.3).If we use this bound, instead of that in Theorem 1.1.3, in the proof of Theorem 1.2.7 (specializedfor the case of circles; see details in Chapter 8), we get the following slight improvement.

Theorem 1.2.8. Let P be a set of m points and C a set of n circles in R3, so that no plane or

sphere contains more than q circles of C . Then

I(P,C ) = O(

m3/7n6/7 +m2/3n1/3q1/3 +m6/11n5/11q4/11 log2/11(m3/q)+m+n).

Here too we have the three improvements noted in Remark (2) following Theorem 1.2.7. Inparticular, in the sense of part (iii) of that remark, the new bound is “best possible” with respect tothe best known bound (1.12) for the planar or spherical cases; again, see the chapter for details.

1.2.5 Incidences with surfaces in three dimensions

We now review the final set of results in this thesis, involving incidences between points andconstant-degree algebraic surfaces in three dimensions. These results are also presented in Chap-ter 8, and have appeared in [106].

Incidence graph decomposition, for points on a variety and surfaces. In the case of point-surface incidences, the incidence graph between the points and surfaces can contain large completebipartite graphs, each involving points on some curve and surfaces containing this curve (unlikeearlier studies, reviewed in Section 1.1, we do not have to rule out this possibility, which makes ourapproach more general). Our bounds estimate the total size of the vertex sets in such a completebipartite graph decomposition of the incidence graph. In favorable cases, our bounds translate intoactual incidence bounds. Overall, here too our results provide a “grand generalization” of manyof the previous studies of (special instances of) this problem.

Our first main result on point-surface incidences deals with the special case where the pointsof P lie on some algebraic variety V of constant degree. Besides being of independent interest,this is a major ingredient of the analysis for the general case of an arbitrary set of points in R3 andsurfaces.

In the statements of the following theorems we assume that the set S of the given surfacesis taken from some infinite family F that either has k degrees of freedom with respect to V

(with some multiplicity µ), as defined earlier, for suitable constant parameters k (and µ), or isof reduced dimension s with respect to V , for some constant parameter s, meaning that the familyΓ := σ ∩V | σ ∈ F is an s-dimensional family of curves (this is reminiscent of the notion ofreduced dimension defined above for curves).

Theorem 1.2.9. Let P be a set of m points on some algebraic surface V of constant degree D in

R3, and let S be a set of n algebraic surfaces in R3 of maximum constant degree E, taken from

some family F of surfaces, which either has k degrees of freedom with respect to V (with some

20

multiplicity µ), or is of reduced dimension s with respect to V , for some constant parameters

k (and µ) or s. We also assume that the surfaces in S do not share any common irreducible

component (which certainly holds when they are irreducible). Then the incidence graph G(P,S)

can be decomposed as

G(P,S) =⋃

γ

(Pγ ×Sγ), (1.13)

where the union is over all irreducible components of curves γ of the form σ ∩V , for σ ∈ S, and,

for each such γ , Pγ = P∩ γ and Sγ is the set of surfaces in S that contain γ .

If F has k degrees of freedom then

∑γ

|Pγ |= O(

mk

2k−1 n2k−22k−1 +m+n

), (1.14)

and if F is s-dimensional then we have, for any ε > 0,

∑γ

|Pγ |= O(

m2s

5s−4 n5s−65s−4+ε +m2/3n2/3 +m+n

), (1.15)

where the constants of proportionality depends on D, E, and the complexity of the family F , and

either on k and µ in the former case, or on ε and s in the latter case.

Moreover, in both cases we have ∑γ |Sγ |=O(n), where the constant of proportionality depends

on D and E.

Remark. A major feature of this result is that it does not impose any restrictions on the incidencegraph, such as requiring it not to contain some fixed complete bipartite graph Kr,r, for r a constant,as is done in the preceding studies [16, 65, 125]. We re-iterate that, to allow for the existence oflarge complete bipartite graphs, the bounds in (1.14) and (1.15), as well as the bound ∑γ |Sγ | =O(n), are not on the number of incidences (that is, on the number of edges in G(P,S), which couldbe as high as mn) but on the overall size of the vertex sets of the subgraphs in the complete bipartitegraph decomposition of G(P,S). This leads to the same asymptotic bound on |G(P,S)| itself, if oneassumes that this graph does not contain Kr,r as a subgraph, for a constant r.

This kind of compact representation of incidences has already been used in the previous studies(mentioned in Section 1.1) of Brass and Knauer [21], Apfelbaum and Sharir [6], and our recentwork [104], albeit only for the special cases of planes or spheres.

Another way of bypassing the possible presence of large complete bipartite graphs in G(P,S),used in several earlier works [1, 6, 41], is to assume that the surfaces in S are non-degenerate.These studies only considered the cases of planes and spheres (or of hyperplanes and spheres inhigher ) [1, 41]. For spheres, for example, this means that no more than some fixed fraction of thepoints of P on any given sphere can be cocircular. Although large complete bipartite graphs canexist in G(P,S) in this case, the non-degeneracy assumption allows us to control, in a sharp form,the number of incidences (and shows that the resulting complete bipartite graphs are not so large

21

after all). It would be interesting (and, as we believe, doable) to extend our analysis to the case of(suitably defined) more general non-degenerate surfaces. These remarks also apply to the generalcase (involving points anywhere in R3), given in Theorem 1.2.14 below.

A mixed incidence bound (for points on most varieties and general surfaces). Our secondresult on point-surface incidences is an improvement of Theorem 1.2.9, still for the case wherethe points of P lie on some algebraic variety V of constant degree, where we now also assumethat V is not infinitely ruled by the intersection curves of pairs of members of the given familyF of surfaces. In this case we obtain an improved, “mixed” bound, in which G(P,S) can be splitinto two subgraphs, G0(P,S) and G1(P,S), where the bound in (1.14) or in (1.15) now holds for|G0(P,S)|, i.e., for the actual number of incidences that it represents, and where G1(P,S) admitsa complete bipartite graph decomposition, as above, for which the sum of the vertex sets is only4

O(m+n). The actual bound is slightly sharper—see below.

Specializing the theorem to the case of spheres, as is done below, leads to interesting implica-tions to distinct and repeated distances in three dimensions.

Theorem 1.2.10. Let P be a set of m points on some irreducible algebraic surface V of constant

degree D in R3, and let S be a set of n algebraic surfaces in R3 of constant degree E, which do

not share any common irreducible component, taken from some infinite constructible family F of

surfaces that either has k degrees of freedom with respect to V (with some multiplicity µ) or is

s-dimensional with respect to V , for some constant parameters k (and µ) or s. Assume further that

V is not infinitely ruled by the family C0 of the irreducible components of the intersection curves

of pairs of surfaces5 in F . Then the incidence graph G(P,S) can be decomposed as

G(P,S) = G0(P,S)∪⋃

γ

(Pγ ×Sγ), (1.16)

where the union is over all irreducible curves γ contained in (one-dimensional) intersections of

the form σ ∩σ ′ ∩V , for σ 6= σ ′ ∈ S, and, for each such γ , Pγ ⊆ P∩ γ (for some points on some

curves, these incident pairs are moved to, and are contained in G0(P,S)), and Sγ is the set (of size

at least two) of surfaces in S that contain γ .

Moreover, if F has k degrees of freedom with respect to V (with some multiplicity µ) then

|G0(P,S)|= O(

mk

2k−1 n2k−22k−1 +m+n

), (1.17)

and if F is s-dimensional with respect to V then, for any ε > 0,

|G0(P,S)|= O(

m2s

5s−4 n5s−65s−4+ε +m2/3n2/3 +m+n

), (1.18)

4 In fact, many “bad” things must happen for G1(P,S) to be nontrivial, and in many situations one would expect G1(P,S)to be empty; see below.

5 A stricter assumption is that V is not infinitely ruled by algebraic curves of degree at most E2, which will hold if weassume that each irreducible component of V has degree larger than 100E4.

22

where the constants of proportionality depends on D, E, and the complexity of the family F , and

either on k and µ in the former case, or on ε and s in the latter case. In either case we also have

∑γ

|Pγ |= O(m), and ∑γ

|Sγ |= O(n),

where the constants of proportionality depend on D, E, and the complexity of the family F , and

either on k (and µ) in the former case, or on ε and s in the latter case.

Remarks. (1) As already alluded to, we note that, typically, one would expect the completebipartite decomposition part of (1.16) to be empty or trivial. To really be significant, (a) manysurfaces of S would have to intersect in a common curve, and, in cases where the multiplicity ofthese curves is not that large, (b) many curves of this kind would have to be fully contained in V

(and also to contain a non-constant number of points of P). Thus, in many cases, in which (a) and(b) do not hold, the bounds in (1.17) or in (1.18) in Theorem 1.2.10 are for the overall number ofincidences. Note also that both Theorem 1.2.9 and Theorem 1.2.10 yield a decomposition of (thewhole or a portion of) G(P,S) into complete bipartite subgraphs. The major difference is that thebound on the overall vertex set size of these graphs is (relatively) large in Theorem 1.2.9, but it isonly linear in m and n (if at all nonzero) in Theorem 1.2.10.

(2) We note that if V is infinitely ruled by our curves the results break down. For a simple example,take m points and N lines in the plane which form Θ(m2/3N2/3) incidences between them. Nowpick any surface V in R3, say the paraboloid z = x2 + y2 for specificity, and lift up each of the N

lines to a vertical parabola on V . Clearly, V is infinitely ruled by such parabolas, and we get asystem of m points and n parabolas with Θ(m2/3N2/3) incidences between them. It is also easyto turn this construction into a point-surface incidence structure, in which ∑γ |Pγ | is equal to thisbound, which is larger than the lower bound O(m+N) asserted in the theorem. The line y = ax+b

in the plane is lifted to the parabola γa,b = (x,y,z) ∈ R3 : y = ax+ b,z = x2 + y2 contained inthe paraboloid V . Define a family S of quadratic surfaces parameterized by a,b,c0,c1,c2 ∈ R bySa,b,c0,c1,c2 := (x,y,z) ∈R3 | (z−x2−y2)+(y−ax−b)(c0 +c1x+c2y) = 0. For any c0,c1,c2 ∈R, the quadric Sa,b,c0,c1,c2 contains the parabola γa,b, i.e., many surfaces in S intersect in a commonparabola.

Incidences between points on a variety and spheres. A particular case of interest is when S

is a set of spheres. The intersection curves of spheres are circles, and, as already noted, the onlysurfaces that are infinitely ruled by circles are spheres and planes. Hence, to apply Theorem 1.2.10,we need to assume that the constant-degree surface V that contains the points of P has no planar orspherical components, thereby ensuring that V is not infinitely ruled by circles. Clearly, as alreadynoted, spheres in R3 have four degrees of freedom, and they form a four-dimensional family ofsurfaces, with respect to any such variety. We can therefore apply Theorem 1.2.10, with s = 4, andconclude:

23

Theorem 1.2.11. Let P be a set of m points on some algebraic surface V of constant degree D

in R3, which has no linear or spherical components, and let S be a set of n spheres, of arbitrary

radii, in R3. The incidence graph G(P,S) can be decomposed as

G(P,S) = G0(P,S)∪⋃

γ∈Γ

(Pγ ×Sγ), (1.19)

where Γ is the set of circles that are contained in V and in at least two spheres of S, and such that,

for each γ ∈ Γ, Pγ = P∩ γ and Sγ is the set of all spheres in S that contain γ . We have

|G0(P,S)|= O(

m1/2n7/8+ε +m2/3n2/3 +m+n), (1.20)

∑γ

|Pγ |= O(m), and ∑γ

|Sγ |= O(n),

for any ε > 0, where the constant of proportionality depends on D and ε .

Remarks. (1) Since V does not contain a planar or spherical component, the number of circles inΓ is O(D2), as follows by Guth and Zahl [57]. That is, the union in (1.20) is only over a constantnumber of circles. On the other hand, there might also be incidence edges contained in completebipartite graphs corresponding to circles that are not contained in V , whose number might be quitelarge. These incidences are recorded in G0(P,S) and their number is bounded in (1.20).

(2) Zahl’s study [125] yields the bound |G(P,S)|=O(m3/4n3/4+m+n), under the assumption thatG(P,S) does not contain Kr,3, for some (arbitrary) constant r (that is, assuming that every triple ofspheres intersect in at most r points of P). Our bound is better for m> n1/2 (ignoring the nε factorin our bound). Overall, on one hand, for this rather restrictive assumption, Zahl’s result is moregeneral, as it does not require the points to lie on a constant-degree variety, but on the other handit is more restrictive, due to its assumption on G(P,S), which we do not make.

We also note that if we assume that G(P,S) does not contain any Kr,r, for r > 3 a constant, thebound in the second part of (1.20) becomes a bound on the number of incidences, so, under thissomewhat weaker assumption (than that of Zahl), we improve Zahl’s bound for points on a varietyand for m> n1/2.

The bound in (1.20) further improves when either (i) the centers of the spheres of S lie on V

(or on some other constant-degree variety), or (ii) the spheres of S have the same radius. In bothcases, S is only three-dimensional, so the bound improves to

|G0(P,S)|= O(

m6/11n9/11+ε +m2/3n2/3 +m+n), (1.21)

for any ε > 0. When both conditions hold—the spheres are congruent and their centers lie onV —S is only two-dimensional with respect to V , and the bound improves still further to

|G0(P,S)|= O(

m2/3n2/3+ε +m+n).

24

Using a slightly refined machinery, developed in [105], the latter bound can actually be improvedfurther to

|G(P,S)|= O(m2/3n2/3 +m+n). (1.22)

Applications of Theorem 1.2.11 and (1.21), (1.22):

Distinct distances. As already mentioned, and as will be detailed in the proofs of the followingresults, the new bounds on point-sphere incidences have immediate applications to the study ofdistinct and repeated distances determined by a set of n points in R3, when the points (or a subsetthereof—see below) lie on some fixed-degree algebraic variety. Specifically, for distinct distances,we have the following results.

Theorem 1.2.12. (a) Let P be a set of n points on an algebraic surface V of constant degree D in

R3, with no linear or spherical components. Then the number of distinct distances determined by

P is Ω(n7/9−ε), for any ε > 0, where the constant of proportionality depends on D and ε .

(b) Let P1 be a set of m points on a surface V as in (a), and let P2 be a set of n arbitrary points in

R3. Then the number of distinct distances determined by pairs of points in P1×P2 is

Ω

(min

m4/7−ε n1/7−ε , m1/2n1/2, m

),

for any ε > 0, where the constant of proportionality depends on D and ε .

Remark. In a recent work [105] (not included in this thesis), we have obtained slightly improvedbounds, replacing the ε in the exponents by a polylogarithmic factor, using a more refined spacedecomposition technique.

While we believe that the bounds in the theorem are not tight, we note that the bounds inboth (a) and (b) (with, say, m = n) are significantly larger than the conjectured best-possible lowerbound Ω(n2/3) for arbitrary point sets in R3.

Repeated distances. As another application, we bound the number of unit (or repeated) distancesinvolving points on a surface V , as above.

Theorem 1.2.13. (a) Let P be a set of n points on some algebraic surface V of constant degree D

in R3, which does not contain any planar or spherical components. Then P determines O(n4/3)

unit distances, where the constant of proportionality depends on D.

(b) Let P1 be a set of m points on a surface V as in (a), and let P2 be a set of n arbitrary points in

R3. Then the number of unit distances determined by pairs of points in P1×P2 is

O(

m6/11n9/11+ε +m2/3n2/3 +m+n),

for any ε > 0, where the constant of proportionality depends on D and ε .

25

In part (a) we extend, to the case of general constant-degree algebraic surfaces, the knownbound O(n4/3), which is worst-case tight when V is a sphere [45]. Part (b) gives (say, for the casem = n) an intermediate bound between O(n4/3) and the best known upper bound O(n3/2) for anarbitrary set of points in R3 [65, 125].

Another thing to notice is that, for distinct distances, the situation is quite different when V is(or contains) a plane or a sphere, in which case the bound goes up to Ω(n/ logn) [56, 119] (seealso Sheffer’s survey [108] for details).

Incidence graph decomposition (for arbitrary points and surfaces). Our final main result onpoint-surface incidences deals with the general setup involving a set S of algebraic surfaces and anarbitrary set of points in R3. The analysis in this general setup proceeds by a recursive argument,based on the polynomial partitioning technique of Guth and Katz [56], in which Theorem 1.2.9plays a central role. This result extends a recent result in our preliminary work [105, Theorem 1.4]from spheres to general surfaces, and extends a recent result of Zahl [125], for general algebraicsurfaces, to the case where no constraints are imposed on G(P,S).

Theorem 1.2.14. Let P be a set of m points in R3, and let S be a set of n surfaces from some

s-dimensional family6 F of surfaces, of constant maximum degree E in R3. Then the incidence

graph G(P,S) can be decomposed as

G(P,S) = G0(P,S)∪⋃

γ

(Pγ ×Sγ), (1.23)

where the union is now over all curves γ of intersection of at least two of the surfaces of S, and,

for each such γ , Pγ = P∩ γ and Sγ is the set (of size at least two) of surfaces in S that contain γ .

Moreover, we have, for any ε > 0,

J(P,S) := ∑γ

(|Pγ |+ |Sγ |

)= O

(m

2s3s−1 n

3s−33s−1+ε +m+n

), and |G0(P,S)|= O(m+n),

(1.24)where the constants of proportionality depend on ε , s, D, E, and the complexity of the family F .

1.3 Algebraic Preliminaries

In this section we collect and adapt a large part of the machinery from algebraic geometry thatwe need for our analysis. Some supplementary machinery is developed within the analysis in thesubsequent chapters. The proofs are deferred to later chapters in the thesis.

In what follows, to facilitate the application of standard techniques in algebraic geometry, itwill be more convenient to work over the complex field C, and in complex (affine or projective)spaces. This is in accordance with most of the basic algebraic geometry tools, which have been

6 Here we use the general notion of s-dimensionality, not confined to points on a variety.

26

developed over the complex field. Some care has to be exercised when applying them over thereals. Some of the tools that we need to use (See, e.g., Theorem 1.3.2, and the results of Guth andZahl [57]) apply over the complex field, but not over the reals. On the other hand, when we applythe partitioning method of [56] (as in the proofs of Theorems 1.2.6 and 1.2.14) or when we useTheorem 1.1.3, we (have to) work over the reals.

It is a fairly standard practice in algebraic geometry that handles a real algebraic variety V ,defined by real polynomials, by considering its complex counterpart VC, namely the set of com-plex points at which the polynomials defining V vanish. The rich toolbox that complex algebraicgeometry has developed allows one to derive various properties of VC, which, with some care, canusually be transported back to the real variety V .

This issue arises time and again in this thesis. Roughly speaking, we approach it as follows.We apply the polynomial partitioning technique to the given sets of points and of lines, curvesor surfaces, in the original real (affine) space, as we should. Within the cells of the partitioningwe then apply some field-independent argument, based either on induction or on some ad-hoccombinatorial argument. Then we need to treat points that lie on the zero set of the partitioningpolynomial. We can then switch to the complex field, when it suits our purpose, noting that thisstep preserves all the real incidences; at worst, it might add additional incidences involving thenon-real portions of the variety and of the curves or surfaces. Hence, the bounds that we obtainfor this case transport, more or less verbatim, to the real case too.

1.3.1 Lines on varieties

We begin with several basic notions and results in differential and algebraic geometry that we willneed (see, e.g., Ivey and Landsberg [63], and Landsberg [73] for more details). For a vector spaceV (over R or C), let PV denote its projectivization. That is, PV = V \ 0/ ∼, where v ∼ w iffw = αv for some non-zero constant α .

An algebraic variety is the common zero set of a finite collection of polynomials. We call itaffine, if it is defined in the affine space, or projective, if it is defined in the projective space, interms of homogeneous polynomials. For an (affine) algebraic variety X , and a point p∈ X , let TpX

denote the (affine) tangent space of X at the point p. A point p is non-singular if dimTpX = dimX

(see Hartshorne [62, Definition I.5 and Theorem I.5.1]). For a point p ∈ X , let Σp denote the setof the complex lines passing through p and contained in X , and let Ξp denote the union of theselines (here X is implicit in these notations). For p fixed, the lines in Σp can be represented by theirdirections, as points in PTpX . In Hartshorne [62, Ex.I.2.10], Ξp is also called the (affine) cone overΣp. Clearly, Ξp ⊆ TpX .

Consider the special case where X is a hypersurface in C4, i.e., X = Z( f ), for a non-linearpolynomial f ∈ C[x,y,z,w], which we assume to be irreducible, where

Z( f ) = p ∈ C4 | f (p) = 0

27

is the zero set of Z( f ). A line `v = p+ tv | t ∈ C passing through p in direction v is said toosculate to Z( f ) to order k at p, if the Taylor expansion of f around p in direction v vanishes toorder k, i.e., if f (p) = 0,

∇v f (p) = 0, ∇2v f (p) = 0, . . . , ∇

kv f (p) = 0, (1.25)

where ∇v f (which for uniformity we also denote as ∇1v f ), ∇2

v f , . . . ,∇kv f are, respectively, the first,

second, and higher order derivatives of f , up to order k, in direction v (where v is regarded as avector in projective 3-space, and the derivatives are interpreted in a scale-invariant manner—weonly care whether they vanish or not). That is, ∇v f = ∇ f · v, ∇2

v f = vT H f v, where H f is theHessian matrix of f , and ∇i

v f is similarly defined, for i> 2, albeit with more complicated explicitexpressions. For simplicity of notation, put Fi(p;v) := ∇i

v f (p), for i≥ 1.

In fact, one can extend the definition of osculation of lines to arbitrary varieties in any dimen-sion (see, e.g., Ivey and Landsberg [63]). For a variety X , a point p ∈ X , and an integer k ≥ 1,let Σk

p ⊂ PTpX denote the variety of the lines that pass through p and osculate to X to order k atp; as before, we represent the lines in Σk

p, for p fixed, by their directions, as points in the corre-sponding projective space. For each k ∈ N, there is a natural inclusion Σp ⊆ Σk

p. In analogy withthe previous notation, we denote by Ξk

p the union of the lines that pass through p with directionsin Σk

p. We let F(X) denote the variety of lines (fully) contained in X ; this is known as the Fano

variety of X , and it is a subvariety of the (2d−2)-dimensional Grassmannian manifold of lines inPd(C); see Harris [61, Lecture 6, page 63] for details, and [61, Example 6.19] for an illustration,and for a proof that this is indeed a variety. We will sometimes denote F(X) also as Σ (or Σ(X)), toconform with the notation involving osculating lines. We also let Σk denote the variety of the linesosculating to order k at some point of X , and can be thought of as the union of the Σk

p over p ∈ X .When representing lines in Σ or Σk we can no longer use the local representation by directions,and instead represent them, in the customary manner, as points within the Grassmanian manifold.Here too Σk can be shown to be a variety (within the Grassmannian manifold) and F(X)⊆ Σk foreach k. We also have, for any p ∈ Z, Σp ⊆ F(X) and Σk

p ⊆ Σk.

Genericity. We recall that a property is said to hold generically (or generally) for polynomialsf1, . . . , fn, of some prescribed degrees, if there are nonzero polynomials g1, . . . ,gk in the coeffi-cients of the fi’s, such that the property holds for all f1, . . . , fn for which none of the polynomialsg j is zero (see, e.g., Cox et al. [33, Definition 3.6]). In this case we say that the collection f1, . . . , fn

is general or generic, with respect to the property in question, namely, with respect to the vanishingof the polynomials g1, . . . ,gk that define that property.

1.3.2 Generalized Bezout’s theorem

An affine (resp. projective) variety X ⊂ Cd (resp. X ⊂ Pd(C)) is called irreducible if, wheneverV is written in the form V = V1∪V2, where V1 and V2 are affine (resp., projective) varieties, then

28

either V1 =V or V2 =V .

Theorem 1.3.1 (Cox et al. [32, Theorem 4.6.2, Theorem 8.3.6]). Let V be an affine (resp., projec-

tive) variety. Then V can be written as a finite union

V =V1∪·· ·∪Vm,

where Vi is an irreducible affine (resp., projective) variety, for i = 1, . . . ,m.

If one also requires that Vi 6⊆Vj for i 6= j, then this decomposition is unique, up to a permutation(see, e.g., [32, Theorem 4.6.4, Theorem 8.3.6]), and is called the minimal decomposition of V intoirreducible components.

We next state a generalized version of Bezout’s theorem, as given in Fulton [49]. It will be amajor technical tool in our analysis.

Theorem 1.3.2 (Fulton [49, Proposition 2.3]). Let V1, . . . ,Vs be subvarieties of Pd , and let Z1, . . . ,Zr

be the irreducible components of⋂s

i=1 Vi. Then

r

∑i=1

deg(Zi)≤s

∏j=1

deg(Vj).

A simple application of Theorem 1.3.2 yields the following useful result.

Lemma 1.3.3. A curve C ⊂ P4 of degree D can contain at most D lines.

This immediately yields the following result, derived in Guth and Katz [55] (see also [39]) ina somewhat different manner.

Corollary 1.3.4. Let f and g be two trivariate polynomials without a common factor. Then

Z( f ,g) := Z( f )∩Z(g) contains at most deg( f ) ·deg(g) lines.

1.3.3 Generically finite morphisms and the Theorem of the Fibers

The following results can be found, e.g., in Harris [61, Chapter 11].

For a map π : X→Y of projective varieties, and for y∈Y , the variety π−1(y) is called the fiber

of π over y.

The following result is a slight paraphrasing of Harris [61, Proposition 7.16] and also appearsin Sharir and Solomon [103, Theorem 6.3]

Theorem 1.3.5 (Harris [61, Proposition 7.16]). Let f : X→Y be the map induced by the standard

projection map π : Pd→ Pr (which retains r of the coordinates and discards the rest), where r< d,

X ⊂ Pd and Y ⊂ Pr are projective varieties, X is irreducible, and Y is the image of X. Then the

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general fiber7 of the map f is finite if and only if dim(X) = dim(Y ). In this case, the number of

points in a general fiber of f is constant.

An important technical tool for our analysis is the following so-called Theorem of the Fibers.

Theorem 1.3.6 (Harris [61, Corollary 11.13]). Let X be a projective variety and π : X → Pd be

a polynomial map (i.e., the coordinate functions x0 π, . . . ,xd π are homogeneous polynomials);

let Y = π(X) denote its image. For any p ∈ Y , let λ (p) = dim(π−1(p)). Then λ (p) is an upper

semi-continuous function of p in the Zariski topology8 on Y ; that is, for any m, the locus of points

p ∈ Y such that λ (p) ≥ m is closed in Y . Moreover, if X0 ⊂ X is any irreducible component,

Y0 = π(X0) its image, and λ0 the minimum value of λ (p) on Y0, then

dim(X0) = dim(Y0)+λ0.

1.3.4 Flecnode polynomials and ruled surfaces in three and four dimensions

Ruled surfaces in three dimensions. We first review several basic properties of ruled two-dimensional surfaces in R3 or in C3. Most of these results are considered folklore in the literature,although we have been unable to find concrete rigorous proofs (in the “modern” jargon of algebraicgeometry).

For a modern approach to ruled surfaces, there are many references; see, e.g., Hartshorne [62,Section V.2], or Beauville [17, Chapter III]. The theory goes back to the 19th century, as presentedin Salmon’s monograph [90], and later by Edge [37]. Guth and Katz’s paper [56] presents severalimportant properties of ruled surfaces in three dimensions, and more expanded reviews are givenin Guth’s recent book [54] and survey [53], and in Kollar’s paper [71].

We say that a real (resp., complex) surface X is ruled by real (resp., complex) lines if everypoint p ∈ X in a Zariski-open dense set is incident to a real (resp., complex) line that is fullycontained in X ; see, e.g., [90] or [37] for further details on ruled surfaces. This definition isslightly weaker than the classical definition, where it is required that every point of X be incidentto a line contained in X (e.g., as in [90]). It has been used in recent works, see, e.g., [56, 71].Similarly to the proof of Lemma 3.4 in Guth and Katz [56], a limiting argument implies that thetwo definitions are equivalent.

We note that some care has to be exercised when dealing with ruled surfaces, because ruled-ness may depend on the underlying field. Specifically, it is possible for a surface defined by realpolynomials to be ruled by complex lines, but not by real lines. For example, the sphere de-fined by x2 + y2 + z2− 1 = 0, regarded as a real variety, is certainly not ruled by lines, but as a

7 The meaning of this statement is that the assertion holds for the fiber at any point outside some lower-dimensionalexceptional subvariety.

8 The Zariski closure of a set Y is the intersection of all varieties X that contain Y . Y is Zariski closed if it is equal to itsclosure (and is therefore a variety), and is Zariski open if its complement is Zariski closed. See [62] for further details.

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complex variety it is ruled by (complex) lines. (Indeed, each point (x0,y0,z0) on the sphere isincident to the (complex) line (x0 +αt,y0 +β t,z0 + γt), for t ∈ C, where α2 +β 2 + γ2 = 0 andαx0 +βy0 + γz0 = 0, which is fully contained in the sphere.)

In three dimensions, a two-dimensional irreducible ruled surface can be either singly ruled,or doubly ruled (notions that are elaborated below), or a plane (the only infinitely ruled surface;again see below).

The discussion so far pertains only to surfaces that are ruled by lines, but in general one canalso consider surfaces ruled by other families of curves (e.g., by circles), where the definition ofruledness extends to these cases in a straightforward manner. We will consider ruledness by moregeneral families of curves in Chapter 8. For notational convenience, though, ruled surface, withoutany extra qualifications refers to a surface ruled by lines.

Reguli. A regulus is the surface (in R3 or C3) spanned by all lines that meet three pairwise skewlines in 3-space.9 For an elementary proof that a doubly ruled surface over R must be a regulus,we refer the reader to Fuchs and Tabachnikov [48, Theorem 16.4]. As the following lemma shows,the only doubly ruled surfaces are reguli, where a regulus is the union of all lines that meet threepairwise skew lines. There are only two kinds of reguli, both of which are quadrics—hyperbolicparaboloids and hyperboloids of one sheet; see, e.g., Fuchs and Tabachnikov [48] for more details.

Lemma 1.3.7. Let V be an irreducible ruled surface in R3 or in C3 which is not a plane, and let

C ⊂V be an algebraic curve, such that every non-singular point p ∈V \C is incident to exactly

two lines that are fully contained in V . Then V is a regulus.

Singly ruled surfaces. Ruled surfaces that are neither planes nor reguli are called singly ruled

surfaces (a terminology justified by Theorem 1.3.8, given below). A line `, fully contained in anirreducible singly ruled surface V , such that every point of ` is “doubly ruled”, i.e., every point on` is incident to another line fully contained in V , is called an exceptional line of V . A point pV ∈V

that is incident to infinitely many lines fully contained in V is called an exceptional point of V .

The following result is another folklore result in the theory of ruled surfaces, used in manystudies (such as Guth and Katz [56]). It justifies the terminology “singly-ruled surface”, by show-ing that the surface is generated by a one-dimensional family of lines, and that each point on thesurface, with the possible exception of points lying on some curve, is incident to exactly one gen-

erator (see below). It also shows that there are only finitely many exceptional lines; the propertythat their number is at most two (see [56]) is presented later. We give (in Chapter 6) a detailed andrigorous proof, to make our presentation as self-contained as possible; we are not aware of anysimilarly detailed argument in the literature.

9 Technically, in some definitions (cf., e.g., Edge [37, Section I.22]) a regulus is a one-dimensional family of generatorlines of the actual surface, i.e., a curve in the Plucker or Grassmannian space of lines, but we use here the alternativenotion of the surface spanned by these lines.

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Theorem 1.3.8. (a) Let V be an irreducible ruled two-dimensional surface of degree D> 1 in R3

(or in C3), which is not a regulus. Then, except for finitely many exceptional lines, the lines that

are fully contained in V are parameterized by an irreducible algebraic curve Σ0 (in the parametric

Plucker space P5 that represents lines in 3-space), and thus yield a 1-parameter family of genera-tor lines `(t), for t ∈ Σ0, that depend continuously on the real or complex parameter t. Moreover,

if t1 6= t2, and `(t1) 6= `(t2), then there exist sufficiently small and disjoint neighborhoods ∆1 of t1and ∆2 of t2, such that all the lines `(t), for t ∈ ∆1∪∆2, are distinct.

(b) There exists a one-dimensional curve C ⊂ V , such that any point p in V \C is incident to

exactly one line fully contained in V .

Exceptional lines on a singly ruled surface. In view of Theorem 1.3.8, every point on a singlyruled surface V is incident to at least one generator. Hence an exceptional (non-generator) line isa line `⊂V such that every point on ` is incident to a generator (which is different from `).

Lemma 1.3.9. Let V be an irreducible ruled surface in R3 or in C3, which is neither a plane

nor a regulus. Then (i) V contains at most two exceptional lines, and (ii) V contains at most one

exceptional point.

Following Theorem 1.3.8, we refer to irreducible ruled surfaces that are neither planes norreguli as singly ruled. A line `, fully contained in an irreducible singly ruled surface V , such thatevery point of ` is incident to another line fully contained in V , is called an exceptional line ofV (these are the lines mentioned in Theorem 1.3.8(a)). If there exists a point pV ∈ V , which isincident to infinitely many lines fully contained in V , then pV is called an exceptional point of V .By Guth and Katz [56], V can contain at most one exceptional point pV (in which case V is a conewith pV as its apex), and (as also asserted in the theorem) at most two exceptional lines.

Flecnodes in three dimensions and the Cayley–Salmon–Monge Theorem. We first recall theclassical theorem of Cayley and Salmon, also due to Monge. Consider a polynomial f ∈ C[x,y,z]of degree D ≥ 3. A flecnode of f is a point p on the zero set Z( f ) of f , for which there exists aline that is incident to p and osculates to Z( f ) at p to order three. That is, if the direction of theline is v then f (p) = 0, and ∇v f (p) = ∇2

v f (p) = ∇3v f (p) = 0, where ∇v f ,∇2

v f ,∇3v f are, respec-

tively, the first, second, and third-order derivatives of f in the direction v. The flecnode polynomial

of f , denoted FL f , is the polynomial obtained by eliminating v, via resultants (see, e.g., Cox etal. [33]), from these three homogeneous equations (where p is regarded as a fixed parameter). (SeeSalmon [90], and the relevant applications thereof in [39, 56], for details concerning flecnode poly-nomials in three dimensions; see also Ivey and Landsberg [63] for a more modern generalizationof this concept.)

The Cayley–Salmon theorem [90], independently obtained by Monge [81], asserts that anirreducible surface Z( f ) is ruled by lines if and only if FL f vanishes identically on Z( f ).

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Theorem 1.3.10 (Cayley and Salmon [90], Monge [81]). Let f ∈C[x,y,z] be an irreducible poly-

nomial of degree D≥ 3. Then Z( f ) is ruled by (complex) lines if and only if Z( f )⊆ Z(FL f ).

A simple proof of the Cayley–Salmon–Monge theorem can be found in Terry Tao’s blog [120].

As shown in Salmon [90, Chapter XVII, Section III], the degree of FL f is at most 11D− 24.By construction, the flecnode polynomial of f vanishes on all the flecnodes of f , and in particularon all the lines fully contained in Z( f ). We will also be using the following result, established byGuth and Katz [55]; see also [39]. It is in fact an immediate consequence of Corollary 1.3.4.

Proposition 1.3.11. Let f be a trivariate irreducible polynomial of degree D. If Z( f ) fully contains

more than 11D2−24D lines then Z( f ) is ruled by (possibly complex) lines.

The flecnode polynomial in four dimensions. The notions of flecnodes and of the flecnodepolynomial can be extended to four dimensions. Informally, the four-dimensional flecnode poly-

nomial FL4f of a 4-variate polynomial f is defined analogously to the three-dimensional variant

FL f , and captures the property that a point on Z( f ) is incident to a line that osculates to Z( f ) upto the fourth order. Specifically, let f ∈ C[x,y,z,w] be a polynomial of degree D ≥ 4. A flecnode

of f is a point p ∈ Z( f ) for which there exists a line that passes through p and osculates to Z( f )

to order four at p. Therefore, if the direction of the line is v = (v0,v1,v2,v3), then it osculates toZ( f ) to order four at p if f (p) = 0 and

Fi(p;v) := ∇iv f (p) = 0, for i = 1,2,3,4, (1.26)

where ∇iv f is the ith order derivative of f in the direction v.

The four-dimensional flecnode polynomial of f , denoted FL4f , is the polynomial obtained by

eliminating v from the four equations in the system (1.26). Note that these four polynomials arehomogeneous in v (of respective degrees 1, 2, 3, and 4). We thus have a system of four equationsin eight variables, which is homogeneous in the four variables v0,v1,v2,v3. Eliminating thosevariables results in a single polynomial equation in p= (x,y,z,w). Using standard techniques, as inCox et al. [33], the resulting polynomial FL4

f is the multipolynomial resultant Res4(F1,F2,F3,F4)

of F1,F2,F3,F4, regarding these as polynomials in v (where the coefficients are polynomials inp). By definition, FL4

f vanishes at all the flecnodes of f . The following results are immediateconsequences of the theory of multipolynomial resultants, presented in Cox et al. [33].

Lemma 1.3.12. Given a polynomial f ∈C[x,y,z,w] of degree D≥ 4, its flecnode polynomial FL4f

has degree O(D).

Lemma 1.3.13. Given a polynomial f ∈ C[x,y,z,w] of degree D≥ 4, every line that is fully con-

tained in Z( f ) is also fully contained in Z(FL4f ).

Ruled surfaces in four dimensions. Flecnode polynomials are a major tool for characterizingruled surfaces also in four dimensions. This is manifested in the following theorem of Landsberg

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[73] (which extends Theorem 1.3.10 to four dimensions, and is even more general than this), is acrucial tool for our analysis. It is established in [73] as a considerably more general result, but weformulate here a special instance that suffices for our needs.

Theorem 1.3.14 (Landsberg [73]). Let f ∈ C[x,y,z,w] be a polynomial of degree D ≥ 4. Then

Z( f ) is ruled by (complex) lines if and only if Z( f )⊆ Z(FL4f ).

When f is of degree≤ 3, we have the following simpler situation, whose easy proof is omitted.

Lemma 1.3.15. For every polynomial f ∈ C[x,y,z,w] of degree ≤ 3, Z( f ) is ruled by (possibly

complex) lines.

1.3.5 Flat points and the second fundamental form

The three-dimensional case Following the notations in Guth and Katz [55] and in [39] (see alsoPressley [85] and Ivey and Landsberg [63] for more basic references), we call a non-singular pointp of Z( f ), for f ∈C[x,y,z], linearly flat, if it is incident to at least three distinct lines that are fullycontained in Z( f ) (and thus also in the tangent plane TpZ( f )). The condition for a point p to belinearly flat was worked out in [55] (see also [39]), and is as follows. Let p be a non-singular pointof Z( f ), and let f (2) denote the second-order Taylor expansion of f at p. That is, we have, for anydirection vector v and t ∈ C,

f (2)(p+ tv) = t∇ f (p) · v+ 12 t2vT H f (p)v. (1.27)

If p is linearly flat, there exist three lines `1, `2, `3, contained in the tangent plane TpZ( f ), such thatvT H f (p)v = 0, when v is the direction of the lines `1, `2 and `3. (clearly, the first term ∇ f (p) · valso vanishes for these directions). Using a suitable coordinate frame within TpZ( f ), we can regardvT H f (p)v as a quadratic trivariate homogeneous polynomial, and thus vanishes on the entire lines`1, `2 and `3.

Since vT H f (p)v vanishes on three lines inside TpZ( f ), a (generic) line `, fully contained inTpZ( f ) and not passing through p, intersects these lines at three distinct points, at which vT H f v

vanishes. Since this is a quadratic polynomial, it must vanish identically on `. Thus, vT H f v is zerofor all vectors v ∈ TpZ( f ), and thus f (2) vanishes identically on TpZ( f ). In this case, we say thatp is a flat point of Z( f ). Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f )

(albeit not necessarily vice versa. 10)

We next express the set of flat points of Z( f ) as the zero set of three polynomials. In order forvT H f v to vanish identically on TpZ( f ), it is necessary and sufficient that vanish on the three vectors(∇ f (p)×e j)

T H f (p)(∇ f (p)×e j)= 0, for j = 1,2,3, where e j are the standard basis vectors in R3,

10 For example, for the surface in R3 defined by the zero set of f = x+ y+ z+ x3, the point 0 = (0,0,0) ∈ Z( f ) is flat(because the second order Taylor expansion of f near 0 is the plane x+ y+ z = 0), but is not linearly flat, since there isno line incident to 0 and contained in Z( f ).

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for j = 1,2,3 and × stand for the vector product in R3. Therefore, a regular point p ∈ Z( f ) is flatif and only if the three polynomials Π j(p) defined by Π j(p) = (∇ f (p)× e j)

T H f (p)(∇ f (p)× e j)

vanish at p.

The four-dimensional case. We continue with the four-dimensional analog. Extending theabove notation to four dimensions, we call a non-singular point p of Z( f ), for f ∈ C[x,y,z], lin-

early flat, if it is incident to at least three distinct 2-flats that are fully contained in Z( f ) (andthus also in the tangent hyperplane TpZ( f )). The condition for a point p to be linearly flat can beworked out as follows, suitably extending the three-dimensional case.

Let p be a non-singular point of Z( f ), and let f (2) denote the second-order Taylor expansionof f at p. That is, we have, as in the three-dimensional case, for any direction vector v and t ∈ C,

f (2)(p+ tv) = t∇ f (p) · v+ 12 t2vT H f (p)v. (1.28)

If p is linearly flat, there exist three 2-flats π1, π2, π3, contained in the tangent hyperplane TpZ( f ),such that vT H f (p)v = 0, for all v ∈ π1,π2,π3 (clearly, the first term ∇ f (p) ·v also vanishes for anysuch v). Using a suitable coordinate frame within TpZ( f ), we can regard vT H f (p)v as a quadratictrivariate homogeneous polynomial.

Since vT H f (p)v vanishes on three 2-flats inside TpZ( f ), a (generic) line `, fully contained inTpZ( f ) and not passing through p, intersects these 2-flats at three distinct points, at which vT H f v

vanishes. Since this is a quadratic polynomial, it must vanish identically on `. Thus, vT H f v is zerofor all vectors v ∈ TpZ( f ), and thus f (2) vanishes identically on TpZ( f ). In this case, we say thatp is a flat point of Z( f ). Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f )

(albeit not necessarily vice versa, as in three dimensions).

We next express the set of flat points of Z( f ) as the zero set of a certain collection of polyno-mials. To do so, we define three canonical 2-flats, on which we test the vanishing of the quadraticform vT H f v. (The preceding analysis shows that, for a linearly flat point, it does not matter whichtriple of 2-flats is used for testing the linear flatness, as long as they are distinct.) These will be the2-flats

πxp := TpZ( f )∩x = xp, π

yp := TpZ( f )∩y = yp, and π

zp := TpZ( f )∩z = zp. (1.29)

These are indeed distinct 2-flats, unless TpZ( f ) is orthogonal to the x-, y-, or z-axis. Denote byZ( f )axis the subset of non-singular points p∈ Z( f ), for which TpZ( f ) is orthogonal to one of theseaxes, and assume in what follows that p ∈ Z( f ) \Z( f )axis. We can ignore points in Z( f )axis byassuming that the coordinate frame of the ambient space is generic, to ensure that none of our(finitely many) input points has a tangent hyperplane that is orthogonal to any of the axes.

Lemma 1.3.16. Let p be a non-singular point of Z( f )\Z( f )axis. Then p is a flat point of Z( f ) if

and only if p is a flat point of each of the varieties Z( f |x=xp),Z( f |y=yp),Z( f |z=zp).

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Recall from Elekes et al. [39] that p is flat for f |x=xp if and only if Π1j := Π j( f |x=xp)

vanishes at p, for j = 1,2,3, where Π j(h) = (∇h× e j)T Hh(∇h× e j), and where e1,e2,e3 denote

the unit vectors in the respective y-, z-, and w-directions, and the symbol × stands for the vectorproduct in x = xp, regarded as a copy of C3. In fact, when xp is also considered as a variable(call it x then), we get that, as in the three-dimensional case, each of Π1

j , for j = 1,2,3, is apolynomial in x,y,z,w of (total) degree 3D− 4. Similarly, the analogously defined polynomialsΠ2

j := Π j( f |y=yp),Π3j := Π j( f |z=zp), for j = 1,2,3, vanish at p if and only if p is a flat point of

f |y=yp and f |z=zp. By Lemma 1.3.16, we conclude that a non-singular point p∈ Z( f )\Z( f )axis

is flat if and only if Πij(p) = 0, for 1≤ i, j ≤ 3.

We say that a line ` ⊂ Z( f ) is a singular line of Z( f ), if all of its points are singular. We saythat a line `⊂ Z( f ) is a flat line of Z( f ) if it is not a singular line of Z( f ), and all of its non-singularpoints are flat. An easy observation is that a flat line can contain at most D−1 singular points ofZ( f ) (these are the points on ` where all four first-order partial derivatives of f vanish). Similarly,a non-singular line is flat if (and only if) it is incident to at least 3D−3 flat points.

The second fundamental form. We use the following notations and results from differentialgeometry; see Pressley [85] and Ivey and Landsberg [63] for details. For a variety X (in anydimension), the differential dγ of the Gauss mapping γ that maps each point p ∈ X to its tangentspace TpX , is called the second fundamental form of X . In four dimensions, say, for X = Z( f ), andfor any non-singular point p ∈ Z( f ), the second fundamental form, locally near p, can be writtenas (see [63])

∑1≤i, j≤3

ai jduidu j,

where x = x(u1,u2,u3) is a parametrization of Z( f ), locally near p, and ai j = xuiu j · n, wheren = n(p) = ∇ f (p)/‖∇ f (p)‖ is the unit normal to Z( f ) at p. Since the second fundamental formis the differential of the Gauss mapping, it does not depend on the specific local parametrizationof f near p. An important property of the second fundamental form is that it vanishes at everynon-singular flat point p ∈ Z( f ) (see, e.g., Pressley [85] and Ivey and Landsberg [63]).

Lemma 1.3.17. If a line ` ⊂ Z( f ) is flat, then the tangent space TpZ( f ) is fixed for all the non-

singular points p ∈ `.

1.3.6 Finitely and infinitely ruled surfaces in four dimensions, and u-resultants

As already mentioned, in three dimensions the only infinitely ruled surfaces in three are planes (andin fact every triply ruled surface is a plane). The situation is more interesting in four dimensions.Recall again the definition of Ξp, for a polynomial f and a point p ∈ Z( f ), which is the union ofall (complex) lines passing through p and fully contained in Z( f ), and that of Σp, as the set ofdirections (considered as points in PTpZ( f )) of these lines.

Fix a line ` ∈ Ξp, and let v = (v0,v1,v2,v3) ∈ P3 represent its direction. Since ` ⊂ Z( f ), the

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four terms Fi(p;v) = ∇iv f (p), for i = 1,2,3,4, must vanish at p. These terms, which we denote

shortly as Fi(v) at the fixed p, are homogeneous polynomials of respective degrees 1,2,3, and 4 inv = (v0,v1,v2,v3). (Note that when D≤ 3, some of these polynomials are identically zero.)

The following discussion provides a (partial) algebraic characterization of points p ∈ Z( f ) forwhich |Σp| is infinite in four dimensions; that is, points that are incident to infinitely many linesthat are fully contained in Z( f ). We refer to this situation by saying that Z( f ) is infinitely ruled

at p. To be precise, here we only characterize points that are incident to infinitely many lines thatosculate to Z( f ) to order three. The passage from this to the full characterization will be doneduring the analysis in the Chapter 2.

u-resultants. The algebraic tool that we use for this purpose are u-resultants. Specifically, fol-lowing and specializing Cox et al. [33, Chapter 3.5, page 116], define, for a vector u=(u0,u1,u2,u3)∈P3,

U(p;u0,u1,u2,u3) = Res4

(F1(p;v),F2(p;v),F3(p;v),u0v0 +u1v1 +u2v2 +u3v3

),

where Res4(·) denotes, as earlier, the multipolynomial resultant of the four respective (homoge-neous) polynomials, with respect to the variables v0,v1,v2,v3. For fixed p, this is the so-calledu-resultant of F1(v),F2(v),F3(v).

Theorem 1.3.18. The function U(p;u0,u1,u2,u3) is a homogeneous polynomial of degree 6 in

the variables u0,u1,u2,u3, and is a polynomial of degree O(D) in p = (x,y,z,w). For fixed p ∈Z( f ), U(p;u0,u1,u2,u3) is identically zero as a polynomial in u0,u1,u2,u3, if and only if there are

infinitely many (complex) directions v = (v0,v1,v2,v3), such that the corresponding lines p+ tv |t ∈ C osculate to Z( f ) to order three at p.

Remark. Theorem 1.3.18 shows that the subset of Z( f ) consisting of the points incident to in-finitely many lines that osculate to Z( f ) to order three is contained in a subvariety of Z( f ), whichis the intersection of Z( f ) with the common zero set of the coefficients of U (considered as poly-nomials in x,y,z,w).

Corollary 1.3.19. Fix p ∈ Z( f ). The polynomial U(p;u0,u1,u2,u3) is identically zero, as a poly-

nomial in u0,u1,u2,u3, if and only if there are more than six (complex) lines osculating to Z( f ) to

order 3 at p.

More to come. We end here the preliminary overview of the algebraic geometry machinery thatwe use in this thesis. Additional tools, as well as proofs of some of the results reviewed here, willbe presented in the relevant subsequent chapters of the thesis.

37

38

Part I

Incidences between points and lines

39

2 Incidences between points and linesin R4

41

Discrete Comput Geom (2017) 57:702–756DOI 10.1007/s00454-016-9822-2

Incidences Between Points and Lines in R4

Micha Sharir1 · Noam Solomon1

Received: 9 January 2016 / Revised: 15 August 2016 / Accepted: 26 August 2016 /Published online: 14 September 2016© Springer Science+Business Media New York 2016

Abstract We show that the number of incidences between m distinct points and ndistinct lines inR

4 is O(2c√logm(m2/5n4/5+m)+m1/2n1/2q1/4+m2/3n1/3s1/3+n),

for a suitable absolute constant c, provided that no 2-plane contains more than s inputlines, and no hyperplane or quadric contains more than q lines. The bound holdswithout the factor 2c

√logm whenm ≤ n6/7 orm ≥ n5/3. Except for the factor 2c

√logm ,

the bound is tight in the worst case.

Keywords Combinatorial geometry · Incidences · The polynomial method ·Algebraic geometry · Ruled surfaces

1 Introduction

Let P be a set ofm distinct points in R4 and let L be a set of n distinct lines in R

4. LetI (P, L) denote the number of incidences between the points of P and the lines of L;that is, the number of pairs (p, ) with p ∈ P, ∈ L , and p ∈ . If all the points of

Editor in Charge: János Pach

Part of this research was performed while the authors were visiting the Institute for Pure and AppliedMathematics (IPAM), which is supported by the National Science Foundation. An earlier version of thisstudy appears in Proc. 30th Annu. ACM Sympos. Comput. Geom., 2014, 189–197, and the present versionis also available in arXiv:1411.0777v1.

Micha Sharirmichas@tau.ac.il

Noam Solomonnoam.solom@gmail.com

1 School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel

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P and all the lines of L lie in a common plane, then the classical Szemerédi–Trottertheorem [42] yields the worst-case tight bound

I (P, L) = O(m2/3n2/3 + m + n). (1)

This bound clearly also holds in R4 (or in any other dimension), by projecting the

given lines and points onto some generic plane. Moreover, the bound will continueto be worst-case tight by placing all the points and lines in a common plane, in aconfiguration that yields the planar lower bound.

In the recent groundbreaking paper of Guth and Katz [15], an improved bound hasbeen derived for I (P, L), for a set P ofm points and a set L of n lines in R

3, providedthat not too many lines of L lie in a common plane.1 Specifically, they showed:

Theorem 1.1 (Guth and Katz [15]) Let P be a set of m distinct points and L a set ofn distinct lines in R

3, and let s ≤ n be a parameter, such that no plane contains morethan s lines of L. Then

I (P, L) = O(m1/2n3/4 + m2/3n1/3s1/3 + m + n). (2)

This bound is tight in the worst case.

In this paper, we establish the following analogous and sharper result in four dimen-sions.

Theorem 1.2 Let P be a set of m distinct points and L a set of n distinct lines in R4,

and let q, s ≤ n be parameters, such that (i) each hyperplane or quadric contains atmost q lines of L, and (ii) each 2-flat contains at most s lines of L. Then

I (P, L) ≤ 2c√logm(m2/5n4/5 + m) + A(m1/2n1/2q1/4 + m2/3n1/3s1/3 + n), (3)

where A and c are suitable absolute constants. When m ≤ n6/7 or m ≥ n5/3, we getthe sharper bound

I (P, L) ≤ A(m2/5n4/5 + m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + n). (4)

In general, except for the factor 2c√logm, the bound is tight in the worst case, for any

values of m, n, and for corresponding suitable ranges of q and s.

The proof of Theorem 1.2 will be by induction on m. To facilitate the inductiveprocess, we extend the theorem as follows. We say that a hyperplane or a quadric Hin R

4 is q-restricted for a set of lines L and for an integer parameter q, if there existsa polynomial gH of degree at most O(

√q), such that each of the lines of L that is

contained in H , except for at most q lines, is contained in some irreducible componentof H ∩ Z(gH ) that is ruled by lines and is not a 2-flat (see below for details). In other

1 The additional requirement in [15], that no regulus contains toomany lines, is not needed for the incidencebound given below.

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words, a q-restricted hyperplane or quadric contains in principle at most q lines of L ,but it can also contain an unspecified number of additional lines, all fully contained inruled (non-planar) components of the zero set of some polynomial of degree O(

√q).

We then have the following more general result.

Theorem 1.3 Let P be a set of m distinct points and L a set of n distinct lines inR4, and let q and s ≤ n be parameters, such that (i ′) each hyperplane or quadric is

q-restricted, and (ii) each 2-flat contains at most s lines of L. Then,

I (P, L) ≤ 2c√logm(m2/5n4/5 + m) + A(m1/2n1/2q1/4 + m2/3n1/3s1/3 + n), (5)

where the parameters A and c are as in Theorem 1.2. As in the preceding theorem,when m ≤ n6/7 or m ≥ n5/3, we get the sharper bound

I (P, L) ≤ A(m2/5n4/5 + m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + n). (6)

Moreover, except for the factor 2c√logm, the bound is tight in the worst case, as above.

The requirement that a hyperplane or quadric H be q-restricted extends (i.e., is aweaker condition than) the simpler requirement that H contains at most q lines of L .Hence, Theorem 1.2 is an immediate corollary of Theorem 1.3.

A few remarks are in order.

(a) Only the range√n ≤ m ≤ n2 is of interest; outside this range, regardless of the

dimension of the ambient space, we have the well known and trivial upper boundI (P, L) = O(m + n), an immediate consequence of (1).

(b) The term m1/2n1/2q1/4 comes from the bound of Guth and Katz [15] in threedimensions (as in Theorem 1.1), and is unavoidable, as it can be attained if wedensely “pack” points and lines into hyperplanes, in patterns that realize the boundin three dimensions within each hyperplane; see Sect. 4 for details.

(c) Likewise, the termm2/3n1/3s1/3 comes from the planar Szemerédi–Trotter bound(1), and is too unavoidable, as it can be attained if we densely pack points andlines into 2-planes, in patterns that realize the bound in (1); again, see Sect. 4.

(d) Ignoring these terms, and the term n, which is included only to cater for the casem <

√n, the two terms m2/5n4/5 and m “compete” for dominance; the former

dominates when m = O(n4/3) and the latter when m = (n4/3). Thus the boundin (5) is qualitatively different within these two ranges.

(e) The thresholdm = n4/3 also arises in the related problem of joints (points incidentto at least four lines not in a common hyperplane) in a set of n lines in 4-space;see [20,29], and a remark below.

By a standard argument, the theorem implies the following corollary.

Corollary 1.4 Let L be a set of n lines in R4, satisfying the assumptions (i’) and (ii)

in Theorem 1.3, for given parameters q and s. Then, for any k = (2c√log n), the

number m≥k of points incident to at least k lines of L satisfies

m≥k = O(2 4

3 c√log nn4/3

k5/3+ nq1/2

k2+ ns

k3+ n

k

).

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Remark (i) It is instructive to compare Corollary 1.4 with the analysis of joints ina set L of n lines. In R

d , a joint of L is a point incident to at least d lines of L ,not all in a common hyperplane. As shown in [20,29], the maximum number ofjoints of such a set is O(nd/(d−1)), and this bound is worst-case tight. In fourdimensions, this bound is O(n4/3), which corresponds to the numerator of thefirst term of the bound in Corollary 1.4.

(ii) The other terms cater to configurations involving co-hyperplanar or coplanarlines. For example, when q = n, the second term is O(n3/2/k2), in accordancewith the bound obtained in Guth and Katz [15] in three dimensions, and whens = n, the third and fourth terms comprise (an equivalent formulation of) thebound (1) of Szemerédi and Trotter [42] for the planar case.

(iii) A major interesting and challenging problem is to extend the bound of Corol-lary 1.4 for any value of k. In particular, is it true that the number of intersection

points of the lines (this is the case k = 2) is O(243 c

√log nn4/3 + nq1/2 + ns)? We

conjecture that this is indeed the case.(iv) Another challenging problem is to improve our bound, so as to get rid of, or at

least reduce the factor 2c√logm . As stated in the theorems, this can be achieved

when m ≤ n6/7 or m ≥ n5/3.

Additional remarks and open issues are given in the concluding Sect. 5.

Background Incidence problems have been a major topic in combinatorial and com-putational geometry for the past thirty years, starting with the Szemerédi-Trotterbound [42] back in 1983. Several techniques, interesting in their own right, have beendeveloped, or adapted, for the analysis of incidences, including the crossing-lemmatechnique of Székely [41], and the use of cuttings as a divide-and-conquer mechanism(e.g., see [3]). Connections with range searching and related problems in computa-tional geometry have also been noted, and studies of the Kakeya problem (see, e.g.,[43]) indicate the connection between this problem and incidence problems. See Pachand Sharir [27] for a comprehensive (albeit a bit outdated) survey of the topic.

The landscape of incidence geometry has dramatically changed in the past sevenyears, due to the infusion, in two groundbreaking papers by Guth and Katz [14,15](the first of which was inspired by a similar result of Dvir [6] for finite fields), ofnew tools and techniques drawn from algebraic geometry. Although their two directgoals have been to obtain a tight upper bound on the number of joints in a set of linesin three dimensions [14], and an almost tight lower bound for the classical distinctdistances problem of Erdos [15], the new tools have quickly been recognized as usefulfor incidence bounds of various sorts. See [10,21,22,38,40,47,48] for a sample ofrecent works on incidence problems that use the new algebraic machinery.

The simplest instances of incidence problems involve points and lines. Szemerédiand Trotter completely solved this special case in the plane [42]. Guth and Katz’ssecond paper [15] provides a worst-case tight bound in three dimensions, under theassumption that no plane contains toomany lines; seeTheorem1.1.Under this assump-tion, the bound in three dimensions is significantly smaller than the planar bound(unless one ofm, n is significantly smaller than the other), and the intuition is that thisphenomenon should also show up aswemove to higher dimensions. Unfortunately, the

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analysis becomes more involved in higher dimensions, and requires the developmentor adaptation of progressively more complex tools from algebraic geometry. Most ofthese tools still appear to be unavailable, and their absence leads either to interesting(new) open problems in the area, or to the need to adapt existing machinery to fit intothe new context.

The present paper is a first step in this direction, which considers the four-dimensional case. It does indeed derive a sharper, nearly optimal bound, assumingthat the configuration of points and lines is “truly four-dimensional”, in the precisesense spelled out in Theorems 1.2 and 1.3.

We also note that studying incidence problems in four (or higher) dimensions hasalready taken place in several contemporary works, such as in Solymosi and Tao [40],Zahl [48], and Basu and Sombra [1] (and in work in progress by Solymosi and deZeeuw). These works, though, consider incidences with higher-dimensional varieties,and the study of incidences involving lines, presented in this paper, is new. (Thereare several ongoing studies, including a companion work joint with Sheffer, that aimto derive weaker but more general bounds involving incidences between points andcurves in higher dimensions.) For very recent related studies, see Dvir and Gopi [7]and Hablicsek and Scherr [16].

Our study of point-line incidences in four dimensions has lead us to adapt moreadvanced tools in algebraic geometry, such as tools involving surfaces that are ruled bylines or by flats, including Severi’s 1901 work [34], as well as the more recent worksof Landsberg [19,25] on osculating lines and flats to algebraic surfaces in higherdimensions.

In a preliminary version of this study [35], we have obtained a weaker and moreconstrained bound.Adiscussion of the significant differences between this preliminarywork and the present one is given in the overview of the proof, which comes next.

Overview of the proof 2 The analysis follows the general approach of Guth andKatz [15], albeit with many significant adaptations and modifications. We use induc-tion on m = |P|, but we begin the description by ignoring this aspect (for a while).We apply the polynomial partitioning technique of Guth and Katz [15], with somepolynomial f ∈ R[x, y, z, w] of suitable degree D, and obtain a partition of R

4 intoO(D4) cells, each containing at most O(m/D4) points of P .

In our first phase, we use

D = O(m2/5/n1/5), for m = O(n4/3),

and D = O(n/m1/2), for m = (n4/3). (7)

There are three types of incidences that may arise: an incidence between a point insome cell of the partition and a line crossing that cell, an incidence between a pointon the zero set Z( f ) of f and a line not fully contained in Z( f ), and an incidencebetween a point on Z( f ) and a line fully contained in Z( f ). The above choices of Dmake it a fairly easy task to bound the number of incidences of the first two types, and

2 In this overviewwe assume some familiarity of the reader with the new “polynomial method” of Guth andKatz, and with subsequent applications thereof. Otherwise, the overview can be skipped on first reading.

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the hard part is to estimate the number of incidences of the third kind, as we have nocontrol on the number of points and lines contained in Z( f )—in the worst case all thepoints and lines could be of this kind.

At the “other end of the spectrum,” choosing D to be a constant (as done in ourpreliminary aforementioned study of this problem [35] and in other recent studies ofrelated problems [13,38,40]) simplifies considerably the handling of incidences onZ( f ), but then the analysis of incidences within the cells of the partition becomesmore involved, as the sizes of the subproblems within each cell are too large. In theworks just cited (as well as in this paper), this is handled via induction, but the price ofa naive inductive approach is three-fold: First, the bound becomes weaker, involvingadditional factors of the form O(mε), for any ε > 0 (with a constant of proportionalitythat depends on ε). Second, the requirement that no hyperplane or quadric containsmore than q lines of L has to be replaced by themuchmore restrictive requirement, thatno variety of degree at most cε contains more than q input lines, where cε is a (fairlylarge) constant that depends on ε (and becomes larger as ε gets smaller). Finally, thesharp “lower-dimensional” terms, such as m1/2n1/2q1/4 and m2/3n1/3s1/3 in our case(recall that both are worst-case tight), do not pass through the induction successfully,so they have to be replaced by weaker terms; see the preliminary version [35] for suchweaker terms, and [38] for a similar phenomenon in a different incidence problem inthree dimensions.

We note that a recent study by Guth [13] reexamines the point-line incidence prob-lem in R

3 and presents an alternative and simpler analysis (than the original one in[15]), in which he uses a constant-degree partitioning polynomial, and manages tohandle successfully the relevant lower-dimensional term m2/3n1/3s1/3 through theinduction, but the analysis still incurs the extra mε factors in the bound, and needsthe restrictive assumption that no algebraic surface of some large constant maximumdegree cε contains too many lines. In a companion paper [36], we provide yet anothersimpler derivation (which is somewhat sharper than Guth’s) of such an incidencebound in three dimensions.

Our approach is to use two different choices of the degree of the partitioning poly-nomial. We first choose the large value of D specified above, and show that the boundin the right-hand side of (5) accounts for the incidences within the partition cells, forthe incidences between points on Z( f ) and lines not fully contained in Z( f ), and formost of the cases involving incidences between points and lines on the zero set Z( f ).We are then left with “problematic” subsets of points and lines on Z( f ), which aredifficult to analyze when the degree is large. (Informally, this happens when the lineslie in certain ruled two-dimensional subvarieties of Z( f ).) To handle them, we retainonly these subsets, discard the partitioning, and start afresh with a new partitioningpolynomial of a much smaller, albeit still non-constant degree. As the degree is nowtoo small, we need induction to bound the number of incidences within the partitioncells. A major feature that makes the induction work well is that the first partitioningstep ensures that the surviving set of lines that is passed to the induction is such thateach hyperplane or quadric is now O(D2)-restricted, with respect to the set of sur-viving lines, and each 2-flat contains at most O(D) lines of that set (where D is thelarge degree used in the first partitioning step). As a consequence, the induction worksbetter, and “retains” the lower-dimensional terms m1/2n1/2q1/4 and m2/3n1/3s1/3.

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(In fact, it does not touch them at all, because q and s are not passed to the inductionstep.) We still pay a small price for this approach, involving the extra factor 2c

√logm

in the “leading terms” m2/5n4/5 + m (but not in the “lower-dimensional” terms); thisextra factor is needed to make the induction work, and is a consequence of using apartitioning polynomial of small degree. When m is “not too close to” n4/3, as spec-ified in the theorems, induction, and the use of a second partitioning polynomial, arenot needed, and a direct analysis yields the sharper bound in (6), without this extrafactor.

The idea of using a “small” degree for the partitioning polynomial is not new, andhas been applied also in [38,48]. However, the induction process in [38] results inweaker lower-dimensional terms, which we avoid here with the use of two differentpartitionings. We note that we have recently applied this approach in the aforemen-tioned study of point-line incidences in three dimensions [36], with a simpler analysis(than that in [13,15]) and an improved bound (than the one in [13]).

The main (and hard) part of the analysis is still in handling incidences within Z( f )in the first partitioning step, where the degree of f is large. (Similar issues arise inthe second step too, but the bounds there are generally sharper than those obtained inthe first step, simply because the degree is smaller.) This is done as follows. We firstignore the singular points on Z( f ). They will be handled separately, as points lying onthe zero sets of polynomials of smaller degree (namely, partial derivatives of f ). Wealso assume that f is irreducible, by considering each irreducible factor of the originalf separately (see Sect. 3 for details). This step results in a partition of the points ofP and the lines of L among several varieties, each defined by an irreducible factor off or of some derivative of f , so that it suffices to bound the number of incidencesbetween points and lines assigned to the same variety. The number of “cross-variety”incidences is shown to be only O(nD), a bound that we are “happy” to pay.

We next define (a four-dimensional variant of) the flecnode polynomial g := FL4fof f (see Salmon [32] for the more classical three-dimensional variant, which is usedin Guth and Katz [14,15]), which vanishes at those points p ∈ Z( f ) that are incidentto a line that osculates to Z( f ) (i.e., agrees with Z( f ) near p) up to order four (and inparticular to lines that are fully contained in Z( f )); see below for precise definitions.We show that g = FL4f is a polynomial of degree O(D). If g ≡ 0 on Z( f ) thenZ( f ) is ruled by lines3 (as follows from Landsberg’s work [25], which provides ageneralization of the classical Cayley–Salmon theorem [15,32]). We handle this caseby first reducing it to the case where Z( f ) is “infinitely ruled” by lines, meaningthat most of its points are incident to infinitely many lines that are contained in Z( f )(otherwise, we can show, using Bézout’s theorem, that most points are incident to atmost 6 lines, for a total of O(m) incidences), and then by using the aforementionedresult of Severi [34] from 1901, which shows that in this case Z( f ) is ruled by 2-flats(each point on Z( f ) is incident to a 2-flat that is fully contained in Z( f )), unless Z( f )is a hyperplane or a quadric. This allows us to reduce the problem to several planarincidence problems, which are reasonably easier to handle.

3 That is, every point p ∈ Z( f ) is incident to a line that is fully contained in Z( f ); see Salmon [8,15,24,32,37] for definitions.

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The other case is where the common zero set Z( f, g) of f and g is two-dimensional.In this case, we decompose Z( f, g) into its irreducible components, and show thatthe number of incidences between points of P and lines fully contained in irreduciblecomponents that are not 2-flats is

minO(mD2 + nD), O(m + nD4). (8)

Both terms are too large for the standard “large” values of D, but they are non-trivialto establish, and are useful tools for slightly improving the bound and simplifying theanalysis considerably when D is not too large—see below. The derivation of thesebounds is based on a new study of point-line incidences within ruled two-dimensionalvarieties in 3-space, provided in a companion paper [37].

The irreducible components that are 2-flats are harder to handle, because their num-ber can be O(D2) (as follows from the generalized version of Bézout’s theorem [12]),a number that turns out to be too large for the purpose of our incidence bound, whena naive analysis (with a large value of D) is used, so some care is needed in this case.The difficult step in this part is when there are many points, each contained in at leastthree (and in general many) 2-flats fully contained in Z( f, g) (and thus in Z( f )).Non-singular points of this kind are called linearly flat points of Z( f ), naturally gen-eralizing Guth and Katz’s notion of linearly flat points in R

3 [15] (see also Kaplan etal. [10]). Linearly flat points are also flat points, i.e., points where the second funda-mental form of Z( f ) vanishes (e.g., see Pressley [28]). Flatness of a point p can beexpressed, again by a suitable generalization to four dimensions of the techniques in[10,15], by the vanishing of nine polynomials, each of degree ≤ 3D − 4, at p, whichare constructed from f and from its first and second-order derivatives. The problemcan then be reduced to the case where all the points and lines are flat (a line is flat,when not all of its points are singular points of Z( f ), and all of its non-singular pointsare flat). With a careful (and somewhat intricate) probing into the geometric proper-ties of flat lines, we can bound the number of incidences with flat lines by reducingthe problem into several incidence problems in three dimensions (specifically, withinhyperplanes tangent to Z( f ) at the flat points), and then using an extension of Guthand Katz’s bound (2) for each of these problems, where, in this application, we exploitthe fact that each hyperplane contains at most q lines, to obtain a better, q-dependentbound.

However, as noted, the terms O(mD2) (when n6/7 ≤ m ≤ n4/3) and O(nD4)

(when n4/3 ≤ m ≤ n5/3) are too large [for the choices of our “large” values of D in(7)].We retain and also use them in the second partitioning step, when the degree of thepartitioning polynomial is smaller, but finesse them, for the large D, by showing that,after pruning away points and lines whose incidences can be estimated directly [withinthe bound (6), not using the weaker bounds of (8)], we are left with subsets for whichevery hyperplane or quadric is O(D2)-restricted, and each 2-flat contains at mostO(D) lines. However, when m ≤ n6/7 or m ≥ n5/3, the terms O(mD2), O(nD4) arenot too large, and there is no need for this part of the analysis, and a direct application ofthe bounds in (8) yields the sharper bound in (6) and simplifies the proof considerably.

For the remaining range of m and n, we go on to our second partitioning step. Wediscard f and start afresh with a new partitioning polynomial h of degree E D.

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As already noted, bounding incidences within the partition cells becomes non-trivial,and we use induction, exploiting the fact that now the parameters q and s are replacedby O(D2) and O(D), respectively. On the flip side of the coin, bounding incidenceswithin Z(h) is now simpler, because E is smaller, and we can use the bounds in (8)(i.e., O(mE2+nE) or O(m+nE4)) to establish the bound in (5) for the “problematic”incidences.

The reason for using the weaker requirement that each hyperplane and quadricbe q-restricted, instead of just requiring that no hyperplane or quadric contain morethan q lines of L , is that we do not know how to bound the overall number of linesin a hyperplane or quadric H by O(D2), because of the potential existence of ruledcomponents of Z( f, g) within H , which can accommodate any number of lines. Amajor difference between this case and the analysis of ruled components in Guth andKatz’s study [15] is that here the overall degree of Z( f, g) is O(D2), as opposed to thedegree of Z( f ) being only D in [15]. This precludes the application of the techniquesof Guth and Katz to our scenario—they would lead to bounds that are too large.

We also note that our analysis of incidences within Z( f ) is actually carried out (inthe projective 4-space) over the complex field, which makes it simpler, and facilitatesthe application of numerous tools from algebraic geometry that are developed in thissetting. The passage from the complex projective setup back to the real affine one isstraightforward—the former is a generalization of the latter. The real affine setup isneeded only for the construction of a polynomial partitioning, which is meaninglessover C. Once we are within the variety Z( f ), we can switch to the complex projectivesetup, and reap the benefits noted above.

Note that, in spite of these improvements, Theorem 1.3 still has the peculiar feature,which is not needed in Guth and Katz [15] (for the incidence bound of Theorem 1.1),that also requires that every quadric be q-restricted (or, in the simpler version inTheorem1.2, contains atmostq lines of L). 4 In a recentwork in progress, Solomon andZhang [39] show that this requirement cannot be dropped, by providing a constructionof a quadric that contains many points and lines, where the number of incidencesbetween them is significantly larger than the bound in (5) (where q now only boundsthe number of lines in a hyperplane).

2 Algebraic Preliminaries

In this section we collect and adapt a large part of the machinery from algebraicgeometry that we need for our analysis. Some supplementary machinery is developedwithin the analysis iself.

In what follows, to facilitate the application of standard techniques in algebraicgeometry, it will be more convenient to work over the complex fieldC, and in complexprojective spaces. We do so even though Theorem 1.3 is stated (and will be proved)only for the real affine case. The passage between the two scenarios, in the proof of the

4 This is not quite the case: Guth and Katz also require that no regulus contains more than s (actually,√n)

lines, but this is made to bound the number of points incident to just two lines, and is not needed for theincidence bound in Theorem 1.1.

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theorem, will be straightforward, as discussed in the preceding overview. Concretely,the realness of the underlying field is needed only for the partitioning step itself, whichhas no (simple) parallel over C. However, after reducing the problem to points andlines contained in Z( f ), it is more convenient to carry out the analysis overC, to allowus to apply the algebraic machinery that we are going to present next.

2.1 Lines on Varieties

We begin with several basic notions and results in differential and algebraic geometrythat we will need (see, e.g., Ivey and Landsberg [19], and Landsberg [25] for moredetails). For a vector space V (over R or C), let PV denote its projectivization. Thatis, PV = V \ 0/ ∼, where v ∼ w iff w = αv for some non-zero constant α.

An algebraic variety is the common zero set of a finite collection of polynomials.We call it affine, if it is defined in the affine space, or projective, if it is defined inthe projective space, in terms of homogeneous polynomials. For an (affine) algebraicvariety X , and a point p ∈ X , let TpX denote the (affine) tangent space of X at thepoint p. A point p is non-singular if dim TpX = dim X (see Hartshorne [18, Def.I.5 and Thm. I.5.1]). For a point p ∈ X , let p denote the set of the complex linespassing through p and contained in X , and letp denote the union of these lines (hereX is implicit in these notations). For p fixed, the lines in p can be represented bytheir directions, as points in PTpX . In Hartshorne [18, Ex. I.2.10], p is also calledthe (affine) cone over p. Clearly, p ⊆ TpX .

Consider the special case where X is a hypersurface in C4, i.e., X = Z( f ), for a

non-linear polynomial f ∈ C[x, y, z, w], which we assume to be irreducible, where

Z( f ) = p ∈ C4 | f (p) = 0

is the zero set of Z( f ). A line v = p + tv | t ∈ C passing through p in directionv is said to osculate to Z( f ) to order k at p, if the Taylor expansion of f around p indirection v vanishes to order k, i.e., if f (p) = 0,

∇v f (p) = 0, ∇2v f (p) = 0, . . . , ∇k

v f (p) = 0, (9)

where∇v f (which for uniformity we also denote as∇1v f ),∇2

v f, . . . ,∇kv f are, respec-

tively, the first, second, and higher order derivatives of f , up to order k, in directionv (where v is regarded as a vector in projective 3-space, and the derivatives are inter-preted in a scale-invariant manner—we only care whether they vanish or not). Thatis, ∇v f = ∇ f · v, ∇2

v f = vT H f v, where H f is the Hessian matrix of f , and ∇ iv f

is similarly defined, for i > 2, albeit with more complicated explicit expressions. Forsimplicity of notation, put Fi (p; v) := ∇ i

v f (p), for i ≥ 1.In fact, one can extend the definition of osculation of lines to arbitrary varieties in

any dimension (see, e.g., Ivey and Landsberg [19]). For a variety X , a point p ∈ X ,and an integer k ≥ 1, let k

p ⊂ PTpX denote the variety of the lines that pass throughp and osculate to X to order k at p; as before, we represent the lines ink

p, for p fixed,by their directions, as points in the corresponding projective space. For each k ∈ N,

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there is a natural inclusionp ⊆ kp. In analogy with the previous notation, we denote

by kp the union of the lines that pass through p with directions in k

p. We let F(X)

denote the variety of lines (fully) contained in X ; this is known as the Fano varietyof X , and it is a subvariety of the (2d − 2)-dimensional Grassmannian manifold oflines in P

d(C); see Harris [17, Lect. 6, p. 63] for details, and [17, Ex. 6.19] for anillustration, and for a proof that this is indeed a variety. We will sometimes denoteF(X) also as (or (X)), to conform with the notation involving osculating lines.We also let k denote the variety of the lines osculating to order k at some point ofX , and can be thought of as the union of the k

p over p ∈ X . When representing linesin or k we can no longer use the local representation by directions, and insteadrepresent them, in the customary manner, as points within the Grassmanian manifold.Here too k can be shown to be a variety (within the Grassmannian manifold) andF(X) ⊆ k for each k. We also have, for any p ∈ Z , p ⊆ F(X) and k

p ⊆ k .

Genericity We recall that a property is said to hold generically (or generally) forpolynomials f1, . . . , fn , of some prescribed degrees, if there are nonzero polynomialsg1, . . . , gk in the coefficients of the fi ’s, such that the property holds for all f1, . . . , fnfor which none of the polynomials g j is zero (see, e.g., Cox et al. [4, Def. 3.6]). Inthis case we say that the collection f1, . . . , fn is general or generic, with respect tothe property in question, namely, with respect to the vanishing of the polynomialsg1, . . . , gk that define that property.

2.2 Generalized Bézout’s Theorem

An affine (resp. projective) variety X ⊂ Cd (resp. X ⊂ P

d(C)) is called irreducibleif, whenever V is written in the form V = V1 ∪ V2, where V1 and V2 are affine (resp.,projective) varieties, then either V1 = V or V2 = V .

Theorem 2.1 (Cox et al. [5, Thms. 4.6.2, 8.3.6]) Let V be an affine (resp., projective)variety. Then V can be written as a finite union

V = V1 ∪ · · · ∪ Vm,

where Vi is an irreducible affine (resp., projective) variety, for i = 1, . . . ,m.

If one also requires that Vi Vj for i = j , then this decomposition is unique, up to apermutation (see, e.g., [5, Thms. 4.6.4, 8.3.6]), and is called theminimal decompositionof V into irreducible components.

We next state a generalized version of Bézout’s theorem, as given in Fulton [12].It will be a major technical tool in our analysis.

Theorem 2.2 (Fulton [12, Prop. 2.3]) Let V1, . . . , Vs be subvarieties of Pd , and let

Z1, . . . , Zr be the irreducible components of⋂s

i=1 Vi . Then

r∑i=1

deg(Zi ) ≤s∏

j=1

deg(Vj ).

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A simple application of Theorem 2.2 yields the following useful result.

Lemma 2.3 A curve C ⊂ P4 of degree D can contain at most D lines.

Proof Let t denote the number of these lines, and let C0 ⊂ C denote their union.Intersect C0 with a generic hyperplane H . By Theorem 2.2, the number of intersectionpoints satisfies

t ≤ deg(C0) · deg(H) ≤ deg(C) · 1 = D,

as asserted. This immediately yields the following result, derived in Guth and Katz [14] (see

also [10]) in a somewhat different manner.

Corollary 2.4 Let f and g be two trivariate polynomials without a common factor.Then Z( f, g) := Z( f ) ∩ Z(g) contains at most deg( f ) · deg(g) lines.Proof This follows since Z( f, g) is a curve of degree at most deg( f ) · deg(g).

2.3 Generically Finite Morphisms and the Theorem of the Fibers

The following results can be found, e.g., in Harris [17, Chap. 11].For a map π : X → Y of projective varieties, and for y ∈ Y , the variety π−1(y) is

called the fiber of π over y.The following result is a slight paraphrasing of Harris [17, Prop. 7.16] and also

appears in Sharir and Solomon [37, Thm. 7]

Theorem 2.5 (Harris [17, Prop. 7.16]) Let f : X → Y be the map induced by thestandard projectionmapπ : P

d → Pr (which retains r of the coordinates and discards

the rest), where r < d, X ⊂ Pd and Y ⊂ P

r are projective varieties, X is irreducible,and Y is the image of X. Then the general fiber5 of the map f is finite if and onlyif dim(X) = dim(Y ). In this case, the number of points in a general fiber of f isconstant.

An important technical tool for our analysis is the following so-called Theorem ofthe Fibers.

Theorem 2.6 (Harris [17, Cor. 11.13]) Let X be a projective variety and π : X → Pd

be a polynomial map (i.e., the coordinate functions x0 π, . . . , xd π are homo-geneous polynomials); let Y = π(X) denote its image. For any p ∈ Y , letλ(p) = dim(π−1(p)). Then λ(p) is an upper semi-continuous function of p in theZariski topology6 onY ; that is, for anym, the locus of points p ∈ Y such thatλ(p) ≥ m

5 The meaning of this statement is that the assertion holds for the fiber at any point outside some lower-dimensional exceptional subvariety.6 The Zariski closure of a set Y is the intersection of all varieties X that contain Y . Y is Zariski closed ifit is equal to its closure (and is therefore a variety), and is Zariski open if its complement is Zariski closed.See [18] for further details.

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is closed in Y . Moreover, if X0 ⊂ X is any irreducible component, Y0 = π(X0) itsimage, and λ0 the minimum value of λ(p) on Y0, then

dim(X0) = dim(Y0) + λ0.

2.4 Flecnode Polynomials and Ruled Surfaces in Four Dimensions

Ruled surfaces in three dimensions We first review several basic properties of ruledtwo-dimensional surfaces in R

3 or in C3. Most of these results are considered folklore

in the literature, although we have been unable to find concrete rigorous proofs (inthe “modern” jargon of algebraic geometry). For the sake of completeness we providesuch proofs in a companion paper [37].

For a modern approach to ruled surfaces, there are many references; see, e.g.,Hartshorne [18, Sect. V.2], or Beauville [2, Chapter III]. We say that a real (resp.,complex) surface X is ruled by real (resp., complex) lines if every point p ∈ X in aZariski-open dense set is incident to a real (resp., complex) line that is fully containedin X ; see, e.g., [32] or [8] for further details on ruled surfaces. This definition isslightly weaker than the classical definition, where it is required that every point ofX be incident to a line contained in X (e.g., as in [32]). It has been used in recentworks, see, e.g., [15,24]. Similarly to the proof of Lemma 3.4 in Guth and Katz [15],a limiting argument implies that the two definitions are equivalent. We spell out thedetails in Lemma 6.1 in the appendix (see also Sharir and Solomon [37, Lem. 11]).

We note that some care has to be exercised when dealing with ruled surfaces,because ruledness may depend on the underlying field. Specifically, it is possible fora surface defined by real polynomials to be ruled by complex lines, but not by reallines. For example, the sphere defined by x2 + y2 + z2 − 1 = 0, regarded as a realvariety, is certainly not ruled by lines, but as a complex variety it is ruled by (complex)lines. (Indeed, each point (x0, y0, z0) on the sphere is incident to the (complex) line(x0+αt, y0+βt, z0+γ t), for t ∈ C, whereα2+β2+γ 2 = 0 andαx0+βy0+γ z0 = 0,which is fully contained in the sphere.)

In three dimensions, a two-dimensional irreducible ruled surface can be either singlyruled, or doubly ruled (notions that are elaborated below), or a plane. As the followinglemma shows, the only doubly ruled surfaces are reguli, where a regulus is the unionof all lines that meet three pairwise skew lines. There are only two kinds of reguli,both of which are quadrics—hyperbolic paraboloids and hyperboloids of one sheet;see, e.g., Fuchs and Tabachnikov [11] for more details.

The following (folklore) lemma provides a (somewhat stronger than usual) charac-terization of doubly ruled surfaces; see [37] for a proof.

Lemma 2.7 Let V be an irreducible ruled surface inR3 or inC

3 which is not a plane.If there exists an algebraic curve C ⊂ V , such that every non-singular point p ∈ V \Cis incident to exactly two lines that are fully contained in V , then V is a regulus.

When V is an irreducible ruled surface which is neither a plane nor a regulus, itmust be singly ruled, in the precise sense spelled out in the following theorem (seealso [15]); again, see [37, Thm. 10] for a proof.

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Theorem 2.8 (a) Let V be an irreducible ruled two-dimensional surface of degreeD > 1 in R

3 (or in C3), which is not a regulus. Then, except for at most two excep-

tional lines, the lines that are fully contained in V are parametrized by an irreduciblealgebraic curve 0 in the Plücker space P

5, and thus yield a 1-parameter family ofgenerator lines (t), for t ∈ 0, that depend continuously on the real or complexparameter t . Moreover, if t1 = t2, and (t1) = (t2), then there exist sufficiently smalland disjoint neighborhoods 1 of t1 and 2 of t2, such that all the lines (t), fort ∈ 1 ∪ 2, are distinct.(b) There exists a one-dimensional curve C ⊂ V , such that any point p in V \ C isincident to exactly one generator line of V .

Following this theorem, we refer to irreducible ruled surfaces that are neither planesnor reguli as singly ruled. A line , fully contained in an irreducible singly ruled surfaceV , such that every point of is incident to another line fully contained in V , is calledan exceptional line of V (these are the lines mentioned in Theorem 2.8(a)). If thereexists a point pV ∈ V , which is incident to infinitely many lines fully contained in V ,then pV is called an exceptional point of V . By Guth and Katz [15], V can contain atmost one exceptional point pV (in which case V is a cone with pV as its apex), and(as also asserted in the theorem) at most two exceptional lines.

The flecnode polynomial in four dimensions Let f ∈ C[x, y, z, w] be a polynomialof degree D ≥ 4. A flecnode of f is a point p ∈ Z( f ) for which there exists a line thatpasses through p and osculates to Z( f ) to order four at p. Therefore, if the direction ofthe line is v = (v0, v1, v2, v3), then it osculates to Z( f ) to order four at p if f (p) = 0and

Fi (p; v) = 0, for i = 1, 2, 3, 4. (10)

The four-dimensional flecnode polynomial of f , denoted FL4f , is the polynomialobtained by eliminating v from the four equations in the system (10). (See Salmon [32],and the relevant applications thereof in [10,15], for details concerning flecnode poly-nomials in three dimensions; see also Ivey and Landsberg [19] for a more moderngeneralization of this concept.) Note that these four polynomials are homogeneous inv (of respective degrees 1, 2, 3, and 4).We thus have a system of four equations in eightvariables, which is homogeneous in the four variables v0, v1, v2, v3. Eliminating thosevariables results in a single polynomial equation in p = (x, y, z, w). Using standardtechniques, as in Cox et al. [4], the resulting polynomial FL4f is the multipolynomialresultant Res4(F1, F2, F3, F4) of F1, F2, F3, F4, regarding these as polynomials in v

(where the coefficients are polynomials in p). By definition, FL4f vanishes at all theflecnodes of f . The following results are immediate consequences of the theory ofmultipolynomial resultants, presented in Cox et al. [4].

Lemma 2.9 Given a polynomial f ∈ C[x, y, z, w] of degree D ≥ 4, its flecnodepolynomial FL4f has degree O(D).

Proof The polynomial Fi , for i = 1, . . . , 4, is a homogeneous polynomial in v ofdegree di = i over C[x, y, z, w]. By [4, Thm. 4.9], putting d := (

∑4i=1 di ) − 3 = 7,

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themultipolynomial resultantFL4f = Res4(F1, F2, F3, F4) is equal toD3D′3, where D3 is

a polynomial of degree(d+3

3

) = (103

) = 120 in the coefficients of the polynomials Fi ,and D′

3 is a polynomial of degree d1d2d3 + d1d2d4 + d1d3d4 + d2d3d4 = 6 + 8+ 12 + 24 = 50 in these coefficients (see Cox et al. [4, Chap. 3.4, exercises1,3,6,12,19]). Since each coefficient of any of the polynomials Fi is of degree atmost D − 1, we deduce that FL4f is of degree at most O(D). Lemma 2.10 Given a polynomial f ∈ C[x, y, z, w] of degree D ≥ 4, every line thatis fully contained in Z( f ) is also fully contained in Z(FL4f ).

Proof Every point on any such line is a flecnode of f , so FL4f vanishes identically onthe line.

Ruled Surfaces in four dimensions Flecnode polynomials are a major tool for charac-terizing ruled surfaces. This is manifested in the following theorem of Landsberg [25],which is a crucial tool for our analysis. It is established in [25] as a considerably moregeneral result, but we formulate here a special instance that suffices for our needs.

Theorem 2.11 (Landsberg [25]) Let f ∈ C[x, y, z, w] be a polynomial of degreeD ≥ 4. Then Z( f ) is ruled by (complex) lines if and only if Z( f ) ⊆ Z(FL4f ).

We note that Theorem 2.11 extends the classical Cayley–Salmon theorem in threedimensions (see Salmon [32]). A quick review of this result is given below. We alsonote that we will use a refined version of this theorem, also due to Landsberg, givenas Theorem 3.8 in Sect. 3. When f is of degree ≤ 3, we have the following simplersituation.

Lemma 2.12 For every polynomial f ∈ C[x, y, z, w] of degree ≤ 3, Z( f ) is ruledby (possibly complex) lines.

Proof Let v = (v0, v1, v2, v3) ∈ C4 be a direction. First notice that for a point

p ∈ C4, the line through p in direction v is contained in Z( f ) if and only if the first

three equations in (10) are satisfied, because all the other terms in the Taylor expan-sion of f (p + tv) always vanish for a polynomial f of degree ≤ 3. This is a systemof three homogeneous polynomials in v0, v1, v2, v3, of degrees 1, 2, 3, respectively.By Bézout’s theorem, as stated in Theorem 2.2 below, the number of solutions (com-plex projective, counted with multiplicities) of this system is either six or infinite, sothere is at least one (possibly complex) line that passes through p and is containedin Z( f ).

Back to three dimensions In three dimensions the analysis is somewhat simpler, andgoes back to the 19th century, in Salmon’s work [32] ond others. The flecnode poly-nomial FL f of f , defined in an analogous manner, is of degree 11 deg( f ) − 24 [32].Theorem 2.11 is replaced by the Cayley–Salmon theorem [32], with the analogousassertion that Z( f ) is ruled by lines if and only if Z( f ) ⊆ Z(FL f ). A simple proofof the Cayley–Salmon theorem can be found in Terry Tao’s blog [44].

We will be using the following result, established by Guth and Katz [14]; see also[10].

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Proposition 2.13 Let f be a trivariate irreducible polynomial of degree D. If Z( f )fully contains more than 11D2 − 24D lines then Z( f ) is ruled by (possibly complex)lines.

Proof Apply Corollary 2.4 to FL f and f , to conclude that FL f and f must havea common factor. Since f is irreducible, this factor must be f itself, and then theCayley–Salmon theorem implies that Z( f ) is ruled.

2.5 Flat Points and the Second Fundamental Form

We continue with the four-dimensional setup. Extending the notation in Guth andKatz [14] (see also [10], and also Pressley [28] and Ivey and Landsberg [19] for morebasic references), we call a non-singular point p of Z( f ) linearly flat, if it is incident toat least three distinct 2-flats that are fully contained in Z( f ) (and thus also in the tangenthyperplane TpZ( f )). (The original definition, in [10,15], for the three-dimensionalcase, is that a non-singular point p ∈ Z( f ) is linearly flat if it is incident to three distinctlines that are fully contained in Z( f )) The condition for a point p to be linearly flat canbe worked out as follows, suitably extending the technique used in three dimensionsin [10,14]. Although this extension is fairly routine, we are not aware of any previousconcrete reference, so we spell out the details for the sake of completeness.

Let p be a non-singular point of Z( f ), and let f (2) denote the second-order Taylorexpansion of f at p. That is, we have, for any direction vector v and t ∈ C,

f (2)(p + tv) = t∇ f (p) · v + 12 t

2vT H f (p)v. (11)

If p is linearly flat, there exist three 2-flats π1, π2, π3, contained in the tangent hyper-plane TpZ( f ), such that vT H f (p)v = 0, for all v ∈ π1, π2, π3 (clearly, the first term∇ f (p) · v also vanishes for any such v). Using a suitable coordinate frame withinTpZ( f ), we can regard vT H f (p)v as a quadratic trivariate homogeneous polyno-mial.

Since vT H f (p)v vanishes on three 2-flats inside TpZ( f ), a (generic) line , fullycontained in TpZ( f ) and not passing through p, intersects these 2-flats at three distinctpoints, at which vT H f v vanishes. Since this is a quadratic polynomial, it must vanishidentically on . Thus, vT H f v is zero for all vectors v ∈ TpZ( f ), and thus f (2)

vanishes identically on TpZ( f ). In this case, we say that p is a flat point of Z( f ).Therefore, every linearly flat point of Z( f ) is also a flat point of Z( f ) (albeit notnecessarily vice versa).7 The same definition applies in three dimensions too.

We next express the set of linearly flat points of Z( f ) as the zero set of a certaincollection of polynomials. To do so, we define three canonical 2-flats, on which wetest the vanishing of the quadratic form vT H f v. (The preceding analysis shows that,for a linearly flat point, it does not matter which triple of 2-flats is used for testing thelinear flatness, as long as they are distinct.) These will be the 2-flats

7 For example, for the surface inR3 defined by the zero set of f = x+ y+ z+ x3, the point 0 = (0, 0, 0) ∈Z( f ) is flat (because the second order Taylor expansion of f near 0 is the plane x + y + z = 0), but is notlinearly flat, since there is no line incident to 0 and contained in Z( f ).

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π xp : = TpZ( f ) ∩ x = xp, π

yp := TpZ( f ) ∩ y = yp,

and π zp := TpZ( f ) ∩ z = z p. (12)

These are indeed distinct 2-flats, unless TpZ( f ) is orthogonal to the x-, y-, or z-axis.Denote by Z( f )axis the subset of non-singular points p ∈ Z( f ), for which TpZ( f ) isorthogonal to one of these axes, and assume in what follows that p ∈ Z( f )\ Z( f )axis.We can ignore points in Z( f )axis by assuming that the coordinate frame of the ambientspace is generic, to ensure that none of our (finitely many) input points has a tangenthyperplane that is orthogonal to any of the axes.

Lemma 2.14 Let p be a non-singular point of Z( f ) \ Z( f )axis . Then p is aflat point of Z( f ) if and only if p is a flat point of each of the varietiesZ( f |x=xp), Z( f |y=yp), Z( f |z=z p).

Proof Note that the three varieties in the lemma are two-dimensional varieties withinthe corresponding three-dimensional cross-sections x = xp, y = yp, and z = z p, of4-space.

If p is a flat point of Z( f )\ Z( f )axis , then the second-order Taylor expansion f (2)

vanishes identically on TpZ( f ). By the assumption on p, we have

TpZ( f |x=xp) = TpZ( f ) ∩ x = xp,TpZ( f |y=yp) = TpZ( f ) ∩ y = yp, and

TpZ( f |z=z p) = TpZ( f ) ∩ z = z p,

and these are three distinct 2-flats. Therefore, f |(2)x=xp vanishes identically onTpZ( f |x=xp), implying that p is a flat point of Z( f |x=xp); similarly p is a flat pointof Z( f |y=yp) and of Z( f |z=z p). For the other direction, notice that if p satisfies theassumptions in the lemma, and is a flat point of each of Z( f |x=xp), Z( f |y=yp), andZ( f |z=z p), then f (2) vanishes on three distinct 2-flats contained in TpZ( f ) (namely,the intersection of TpZ( f ) with x = xp, y = yp and z = z p), which are distinctsince p /∈ Z( f )axis. Since f (2) is quadratic, the argument given above implies that itis identically 0 on TpZ( f ).

Recall from Elekes et al. [10] that p is flat for f |x=xp if and only if 1

j := j ( f |x=xp) vanishes at p, for j = 1, 2, 3, where j (h) = (∇h × e j )T

Hh(∇h × e j ), and where e1, e2, e3 denote the unit vectors in the respective y-, z-, and w-directions, and the symbol × stands for the vector product in x = xp,regarded as a copy of C

3. In fact, when xp is also considered as a variable (call it xthen), we get that, as in the three-dimensional case, each of 1

j , for j = 1, 2, 3, is apolynomial in x, y, z, w of (total) degree 3D − 4. Similarly, the analogously definedpolynomials 2

j := j ( f |y=yp), 3j := j ( f |z=z p), for j = 1, 2, 3, vanish at p

if and only if p is a flat point of f |y=yp and f |z=z p. By Lemma 2.14, we concludethat a non-singular point p ∈ Z( f ) \ Z( f )axis is flat if and only if i

j (p) = 0, for1 ≤ i, j ≤ 3.

We say that a line ⊂ Z( f ) is a singular line of Z( f ), if all of its points aresingular. We say that a line ⊂ Z( f ) is a flat line of Z( f ) if it is not a singular line of

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Z( f ), and all of its non-singular points are flat. An easy observation is that a flat linecan contain at most D− 1 singular points of Z( f ) (these are the points on where allfour first-order partial derivatives of f vanish). Similarly, a non-singular line is flat if(and only if) it is incident to at least 3D − 3 flat points.

The second fundamental form We use the following notations and results from dif-ferential geometry; see Pressley [28] and Ivey and Landsberg [19] for details. For avariety X , the differential dγ of the Gauss mapping γ that maps each point p ∈ X toits tangent space TpX , is called the second fundamental form of X . In four dimensions,for X = Z( f ), and for any non-singular point p ∈ Z( f ), the second fundamentalform, locally near p, can be written as (see [19])

∑1≤i, j≤3

ai j dui du j ,

where x = x(u1, u2, u3) is a parametrization of Z( f ), locally near p, andai j = xui u j · n, where n = n(p) = ∇ f (p)/‖∇ f (p)‖ is the unit normal to Z( f )at p. Since the second fundamental form is the differential of the Gauss mapping,it does not depend on the specific local parametrization of f near p. An importantproperty of the second fundamental form is that it vanishes at every non-singular flatpoint p ∈ Z( f ) (see, e.g., Pressley [28] and Ivey and Landsberg [19]).

Lemma 2.15 If a line ⊂ Z( f ) is flat, then the tangent space TpZ( f ) is fixed for allthe non-singular points p ∈ .

Proof The proof applies a fairly standard argument in differential geometry (see, e.g.,Pressley [28]); see also a proof of a similar claim for the three-dimensional case in[10, Appendix]. Fix a non-singular point p ∈ , and assume that x = x(u1, u2, u3)is a parametrization of Z( f ), locally near p. We assume, as we may, that the relevantneighborhood Np of p consists only of non-singular points. For any point (a, b, c)in the corresponding parameter domain, xu1 , xu2 , xu3 span the tangent space to Z( f )at x(a, b, c). Indeed, since x(u1, u2, u3) is a local parametrization, its differential(dx)(a,b,c) : T(a,b,c)C

3 → Tx(a,b,c)Z( f ) is an isomorphism. Hence, the image of thislatter map is spanned by xu1 , xu2 , xu3 at x(a, b, c). In particular, we have

xui · n = 0, i = 1, 2, 3,

over Np. We now differentiate these equations with respect to u j , for j = 1, 2, 3, andobtain

xui u j · n + xui · nu j ≡ 0 on ∩ Np, for 1 ≤ i, j ≤ 3.

The first term vanishes because is flat, so, as noted above, the second fundamentalform vanishes at each non-singular point of . We therefore have

xui · nu j ≡ 0 on ∩ Np, for i, j = 1, 2, 3.

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Since xu1 , xu2 , xu3 span the tangent space Tq Z( f ), for each q ∈ Np, it follows thatnu j (q) is orthogonal to Tq Z( f ) for each q ∈ ∩ Np, and thus must be parallel ton(q) in this neighborhood. However, since n is of unit length, we have n · n ≡ 1, anddifferentiating this equation yields

nu j · n ≡ 0 on ∩ Np, for j = 1, 2, 3.

Since nu j (q) is both parallel and orthogonal to n(q), it must be identically zero on ∩ Np, for j = 1, 2, 3.

Write = p + tv, t ∈ C, and define h(t) := n(p + tv), for t ∈ C. Then, in asuitable tensor notation,

h′(t) = (nu1(p + tv), nu2(p + tv), nu3(p + tv)) · v ≡ 0,

locally near t = 0. Thus, n(p + tv) is constant locally near t = 0, implying that n isconstant along , locally near p.

It still remains to show that n is constant on the set of all the non-singular points ofZ( f ) contained in . Set

Zs() := t ∈ C | p + tv is a singular point of Z( f ).

As is not singular, |Zs()| ≤ D − 1 (as already observed). The map t → n(p + tv)

is constant in a neighborhood of every point t of Zns() := C \ Zs(). Since Zns()

is a connected set,8 n has a fixed value at all the non-singular points on , as asserted.Since the tangent hyperplanes TpZ( f ) along all contain the line itself, and all havethe same normal, we deduce that TpZ( f ) is fixed for all non-singular points p ∈ .

2.6 Finitely and Infinitely Ruled Surfaces in Four Dimensions, and u-Resultants

Recall again the definition of p, for a polynomial f ∈ C[x, y, z, w] and a pointp ∈ Z( f ), which is the union of all (complex) lines passing through p and fullycontained in Z( f ), and that of p, as the set of directions (considered as points inPTpZ( f )) of these lines.

Fix a line ∈ p, and let v = (v0, v1, v2, v3) ∈ P3 represent its direction. Since

⊂ Z( f ), the four terms Fi (p; v) = ∇ iv f (p), for i = 1, 2, 3, 4, must vanish at

p. These terms, which we denote shortly as Fi (v) at the fixed p, are homogeneouspolynomials of respective degrees 1, 2, 3, and 4 in v = (v0, v1, v2, v3). (Note thatwhen D ≤ 3, some of these polynomials are identically zero.)

In this subsection we provide a (partial) algebraic characterization of pointsp ∈ Z( f ) for which |p| is infinite; that is, points that are incident to infinitelymany lines that are fully contained in Z( f ). We refer to this situation by saying thatZ( f ) is infinitely ruled at p. To be precise, here we only characterize points that are

8 This property holds for C but not for R.

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incident to infinitely many lines that osculate to Z( f ) to order three. The passage fromthis to the full characterization will be done during the analysis in the next section.

u-Resultants The algebraic tool that we use for this purpose are u-resultants. Specifi-cally, following and specializing Cox et al. [4, Chap. 3.5, p. 116], define, for a vectoru = (u0, u1, u2, u3) ∈ P

3,

U (p; u0, u1, u2, u3) = Res4(F1(p; v), F2(p; v), F3(p; v),

u0v0 + u1v1 + u2v2 + u3v3),

where Res4(·) denotes, as earlier, the multipolynomial resultant of the four respective(homogeneous) polynomials, with respect to the variables v0, v1, v2, v3. For fixed p,this is the so-called u-resultant of F1(v), F2(v), F3(v).

Theorem 2.16 The function U (p; u0, u1, u2, u3) is a homogeneous polynomial ofdegree six in the variables u0, u1, u2, u3, and is a polynomial of degree O(D) inp = (x, y, z, w). For fixed p ∈ Z( f ),U (p; u0, u1, u2, u3) is identically zero as apolynomial in u0, u1, u2, u3, if and only if there are infinitely many (complex) direc-tions v = (v0, v1, v2, v3), such that the corresponding lines p+ tv | t ∈ C osculateto Z( f ) to order three at p.

Proof Bydefinition, the osculation property in the theorem, for given p and v, is equiv-alent to F1(p; v) = F2(p; v) = F3(p; v) = 0. Regarding F1, F2, F3 as homogeneouspolynomials in v, the degree ofU in u0, u1, u2, u3 is deg(F1) deg(F2) deg(F3) = 3!=6(see Cox et al. [4, Exe. 3.4.6.b]). Put d = deg(F1) + deg(F2) + deg(F3) + 1 = 7.Then the total degree of U in the coefficients of Fi , each being a polynomial in p ofdegree at most D, is at most

(d3

) = (73

) = 35 (see also the proof of Lemma 2.9 andCox et al. [4, Exes. 3.4.6.c, 3.4.19]), and thus the degree of U as a polynomial in p isO(D).

Put H(u, v) = u0v0 + u1v1 + u2v2 + u3v3, and, for any v ∈ C4, denote by

Hv the hyperplane H(u, v) = 0. Fix p ∈ Z( f ), and regard F1, F2, F3, H(u, ·)as polynomials in v. If the osculation property holds at p (for infinitely manylines) then Z(F1, F2, F3) is infinite, so it is at least 1-dimensional. Thus, for anyu = (u0, u1, u2, u3) ∈ C

4, the variety Z(F1, F2, F3, H(u, v)) is non-empty, so themultipolynomial resultant of these four polynomials (in v) vanishes at u. Since thisholds for all u ∈ C

4, It follows from Cox et al. [4, Prop. 1.1.5] that U ≡ 0.Suppose then that the osculation property does not hold (for infinitely many lines)

at p, so Z(F1, F2, F3) is finite. Pick any u0 /∈ ⋃v∈Z(F1,F2,F3) Hv . Then, for every

v ∈ Z(F1, F2, F3), we have H(u0, v) = 0, implying that

Z(F1, F2, F3, H(u0, ·)) = v ∈ Z(F1, F2, F3) | H(u0, v) = 0 = ∅.

Therefore, by the properties of multipolynomial resultants, U (u0) = 0, and U is notidentically zero.

Remark Theorem 2.16 shows that the subset of Z( f ) consisting of the points incidentto infinitely many lines that osculate to Z( f ) to order three is contained in a subvariety

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of Z( f ), which is the intersection of Z( f )with the common zero set of the coefficientsof U (considered as polynomials in x, y, z, w).

Corollary 2.17 Fix p ∈ Z( f ). The polynomial U (p; u0, u1, u2, u3) is identicallyzero, as a polynomial in u0, u1, u2, u3, if and only if there are more than six (complex)lines osculating to Z( f ) to order 3 at p.

Proof The polynomial Fi is either 0 or of degree i (in v, for a fixed value of p), fori = 1, 2, 3. By Theorem 2.2, the number of their common zeros v = (v0, v1, v2, v3) iseither six (counting complex projective solutions with multiplicity; see also the proofof Theorem 2.16) or infinite. The result then follows from Theorem 2.16.

3 Proof of Theorem 1.3

Let P, L ,m, n, q, and s be as in the theorem.The proof proceeds by induction on m, where we establish the inequality

I (P, L)≤2c√logm(m2/5n4/5 + m)+A(m1/2n1/2q1/4 + m2/3n1/3s1/3 + n), (13)

where c and A are constants that will be fixed later, and where the base cases of theinduction are the ranges m ≤ √

n and m ≤ M0, for a sufficiently large constant M0.In both cases we have I (P, L) ≤ A(m + n), for a suitable choice of A.9 Assume thenthat the bound holds for all m′ < m, and consider an instance involving sets P, L ,with |P| = m >

√|L| = √n, and m > M0.

As already discussed, the bound in (5) is qualitatively different in the two rangesm = O(n4/3) and m = (n4/3), and the analysis will occasionally have to bifurcateaccordingly. Nevertheless, the bifurcation is mainly in the choice of various parame-ters, and inmanipulating them.Most of the technical details that dealwith the algebraicstructure of the problem are identical. We will therefore present the analysis jointly forboth cases, and bifurcate only locally, when the induction itself, or tools that preparefor the induction, get into action, and require different treatments in the two cases.

As promised in the overview, we will use two different partitioning schemes, onewith a polynomial of “large” degree, and one with a polynomial of “small” degree.We start naturally with the first scheme.

An important issue to bear in mind is that, unlike most of the material in thepreceding section, where the underlying field wasC, the analysis in this section is overthe reals. Nevertheless, this is essentially needed only for constructing a polynomialpartitioning, which ismeaningless overC. Once this is done, the analysis of incidencesbetween points and lines on the zero set of the partitioning polynomial can be carriedout over the complex field just as well as over R, and then the machinery reviewedand developed in the previous section can be brought to bear.

9 When m ≤ √n (or when n ≤ √

m), an immediate application of the Szemerédi–Trotter theorem yieldsthe linear bound O(m + n).

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First partitioning scheme Fix a parameter r , given by

r =cm8/5/n4/5 if m ≤ an4/3,

cn4/m2 if m ≥ an4/3,

where a and c are suitable constants. Note that, in both cases, 1 ≤ r ≤ m, for a suitablechoice of the constants of proportionality, unless either m = (n2) or n = (m2),extreme cases that have already been handled.We refer to the cases n1/2 ≤ m ≤ an4/3

and an4/3 < m ≤ n2 as the cases of small m and of large m, respectively.We now apply the polynomial partitioning theorem of Guth and Katz (see [15] and

[22, Thm. 2.6]), to obtain an r -partitioning 4-variate (real) polynomial f of degree

D = O(r1/4) ≤c0m2/5/n1/5 if m ≤ an4/3,

c0n/m1/2 if m ≥ an4/3,(14)

for another suitable constant c0. That is, every connected component of R4 \ Z( f )

contains at most m/r points of P , where, as above, Z( f ) denotes the zero set of f .By Warren’s theorem [46] (see also [22]), the number of components of R

4 \ Z( f ) isO(D4) = O(r).

Set P0 := P ∩ Z( f ) and P ′ := P \ P0. We recall that, although the points of P ′ aremore or less evenly partitioned among the cells of the partition, no nontrivial boundcan be provided for the size of P0; in the worst case, all the points of P could lie inZ( f ). Each line ∈ L is either fully contained in Z( f ) or intersects it in at mostD points (since the restriction of f to is a univariate polynomial of degree at mostD). Let L0 denote the subset of lines of L that are fully contained in Z( f ) and putL ′ = L \ L0.

We have

I (P, L) = I (P0, L0) + I (P0, L′) + I (P ′, L ′). (15)

As can be expected (and noted earlier), the harder part of the analysis is the estimationof I (P0, L0). Indeed, it might happen that Z( f ) is a hyperplane, and then the best(and worst-case tight) bound we can offer is the bound specified by Theorem 1.1. Itmight also happen that Z( f ) contains some 2-flat, in which case we are back in theplanar scenario, for which the best (and worst-case tight) bound we can offer is theSzemerédi–Trotter bound (1). Of course, the assumptions of the theorem come to therescue, and we will see below how exactly they are used.

We first bound the second and third terms of (15). We have

I (P0, L′) ≤ |L ′| · D ≤ nD, (16)

because, as just noted, a line not fully contained in Z( f ) can intersect this set in at mostD points. To estimate I (P ′, L ′), we put, for each cell τ of the partition, Pτ = P ∩ τ ,and let Lτ denote the set of the lines of L ′ that cross τ ; put mτ = |Pτ | ≤ m/r , and

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nτ = |Lτ |. Since every line ∈ L ′ crosses at most 1 + D components of R4 \ Z( f )

(because it has to pass through Z( f ) in between cells), we have

∑τ

nτ ≤ |L ′|(1 + D) ≤ n(1 + D). (17)

Clearly, we haveI (P ′, L ′) =

∑τ

I (Pτ , Lτ ).

We now bifurcate depending on the value of m.

Estimating I (P ′, L ′): The case of small m. Here we use the easy upper bound (whichholds for any pair of sets Pτ , Lτ )

I (Pτ , Lτ ) = O(|Pτ |2 + |Lτ |) = O((m/r)2 + nτ ).

Summing these bounds over the cells, using (17), and recalling the value of r (and ofD), we get

I (P ′, L ′) =∑τ

I (Pτ , Lτ ) = O(m2/r + nr1/4) = O(m2/5n4/5).

Estimating I (P ′, L ′): The case of large m. Here we use the dual (generally applicable)upper bound I (Pτ , Lτ ) = O(|Lτ |2+|Pτ |), which, by splitting Lτ into |Lτ |/|Pτ |1/2subsets of size at most |Pτ |1/2, becomes

I (Pτ , Lτ ) = O(|Pτ |1/2|Lτ | + |Pτ |) = O((m/r)1/2nτ + mτ ).

Summing these bounds over the cells, using (17), and recalling the value of r , we get

I (P ′, L ′)=∑τ

I (Pτ , Lτ ) = O((m/r)1/2nr1/4 + m) = O(m1/2n/D + m) = O(m).

Combining both bounds, we have:

I (P0, L′) + I (P ′, L ′) = O(m2/5n4/5 + m). (18)

Note that in this part of the analysis we do not need the assumptions involving q ands—the large degree trivializes the analysis within the cells of the partition.

Estimating I (P0, L0). We next bound the number of incidences between points andlines that are contained in Z( f ). To simplify the notation, write P for P0 and L forL0, and denote their respective cardinalities as m and n. (The reader should keepthis convention in mind, as we will “undo” it towards the end of the analysis.) To beprecise, we will not be able to account explicitly for all types of these incidences (forthe present choices of D). Our strategy is to obtain an explicit bound for a subset ofthe incidences, which is subsumed by the bound in (5), and then prune away those

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lines and points that participate in these incidences. We will be left with “problematic”subsets of points and lines, and we will then handle them in a second, new, induction-based partitioning step. A major goal for the first stage is to show that, for the set ofsurviving lines, the parameters q and s can be replaced by the respective parametersO(D2) and O(D) that “pass well” through the induction; see below for details.10

By the nature of its construction, f is in general reducible (see [15]). However, toapply successfully certain steps of the forthcoming analysis, we will need to assumethat f is irreducible, so we will apply the analysis separately to each irreducible factorof f , and then sum up the resulting bounds. (The actual problem decomposition issubtler — see below.)

Write the irreducible factors of f , in an arbitrary order, as f1, . . . , fk , for somek ≤ D. The points of P are partitioned among the zero sets of these factors, byassigning each point p ∈ P to the first factor in this order whose zero set contains p.A line ∈ L is similarly assigned to the first factor whose zero set fully contains

(there always exists such a factor). Then I (P, L) is the sum, over i = 1, . . . , k, of thenumber of incidences between the points and the lines that are assigned to the (same)i th factor, plus the number of incidences between points and lines assigned to differentfactors. The latter kind of incidences is easier to handle. Indeed, if (p, ) is an incidentpair in P × L , so that p is assigned to fi and is assigned to f j , for i = j (necessarilyi < j), then the incidence occurs at an intersection of with Z( fi ). By construction, is not fully contained in Z( fi ), so it intersects it in at most deg( fi ) points, so theoverall number of incidences on of this kind is at most

∑i = j deg fi < D, and the

overall number of such incidences is therefore at most nD.For the former kind of incidences, we assume in what follows that we have a single

irreducible polynomial f , and denote by P andm, for short, the set of points assignedto f and its cardinality, and by L and n the set of lines assigned to f (and thus fullycontained in Z( f )) and its cardinality. We continue to denote the degree of f as D.(Again, we will undo these conventions towards the end of the analysis.)

This is not yet the end of the reduction, because, in most of the analysis about tounfold, we need to assume that the points of P are non-singular points of Z( f ). Toreduce the setup to this situation we proceed as follows. We construct a sequence ofpartial derivatives of f that are not identically zero on Z( f ). For this we assume, aswe may, that f , and each of its derivatives, are square-free; whenever this fails, wereplace the corresponding derivative by its square-free counterpart before continuingto differentiate. Without loss of generality, assume that this sequence is f, fx , fxx ,and so on. Denote the j-th element in this sequence as f j , for j = 0, 1, . . . (sof0 = f, f1 = fx , and so on). Assign each point p ∈ P to the first polynomial f j inthe sequence for which p is non-singular; more precisely, we assign p to the first f jfor which f j (p) = 0 but f j+1(p) = 0 (recall that f0(p) is always 0 by assumption.Similarly, assign each line to the first polynomial f j in the sequence for which isfully contained in Z( f j ) but not fully contained in Z( f j+1) (again, by assumption,there always exists such f j ). If is assigned to f j then it can only contain points p

10 Note that in general the bounds O(D2) and O(D) are not necessarily smaller than their respectiveoriginal counterparts q and s. Nevertheless, they uniformly depend on m and n in a way that makes themfit the induction process, whereas the parameters q and s, over which we have no control, do not.

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that were assigned to some fk with k ≥ j . Indeed, if contained a point p assigned tofk with k < j then fk+1(p) = 0 but is fully contained in Z( fk+1), since k + 1 ≤ j ;this is a contradiction that establishes the claim.

Fix a line ∈ L , which is assigned to some f j . An incidence between and apoint p ∈ P , assigned to some fk , for k > j , can be charged to the intersection of with Z( f j+1) at p (by construction, p belongs to Z( f j+1)). The number of suchintersections is at most D− j−1, so the overall number of incidences of this sort, overall lines ∈ L , is O(nD). It therefore suffices to consider only incidences betweenpoints and lines that are assigned to the same zero set Z( fi ).

The reductions so far have produced a finite collection of up to O(D) polynomials,each of degree atmost D, so that the points of P are partitioned among the polynomialsand so are the lines of L , and we only need to bound the number of incidences betweenpoints and lines assigned to the same polynomial. This is not the end yet, because thevarious partial derivatives might be reducible, which we want to avoid. Thus, in a finaldecomposition step, we split each derivative polynomial f j into its irreducible factors,and reassign the points and lines that were assigned to Z( f j ) to the various factors, bythe same “first come first served” rule used above. The overall number of incidencesthat are lost in this process is again O(nD). The overall number of polynomials isO(D2), as can easily be checked. Note also that the last decomposition step preservesnon-singularity of the points in the special sense defined above; that is, as is easilyverified, a point p ∈ Z( f j ) with f j+1(p) = 0, continues to be a non-singular pointof the irreducible component it is reassigned to.

We now fix one such final polynomial, still call it f , denote its degree by D (whichis upper bounded by the original degree D), and denote by P and L the subsets of theoriginal sets of points and lines that are assigned to f , and by m and n their respectivecardinalities. (Again, this simplifying convention will be undone towards the end ofthe analysis.) We now may assume that P consists exclusively of non-singular pointsof the irreducible variety Z( f ).

If D ≤ 3, then, by Lemma 2.12, Z( f ) is ruled by lines. Hypersurfaces ruled bylines will be handled in the later part of the analysis. (Note that the cases D = 1 orD = 2 can be controlled by assumption (i’) of the theorem (see below), whereas thecase D = 3 requires a different treatment.) Suppose then that D ≥ 4. The flecnodepolynomial FL4f of f (see Sect. 2.4) vanishes identically on every line of L (and thus

also on P , assuming that each point of P is incident to at least one line of L). If FL4fdoes not vanish identically on Z( f ), then Z( f,FL4f ) := Z( f ) ∩ Z(FL4f ) is a two-dimensional variety (see, e.g., Hartshorne [18, Exercise I.1.8]). It contains P and allthe lines of L (by Lemma 2.10), and is of degree O(D2) (by Theorem 2.2). The otherpossibility is that FL4f vanishes identically on Z( f ), and then Theorem 2.11 impliesthat Z( f ) is ruled by lines. This latter case, which requires several more refined toolsfrom algebraic geometry, will be analyzed later.

First Case: Z( f,FL4f ) is Two-Dimensional

Put g = FL4f . In the analysis below, we only use the facts that deg(g) = O(D), andthat Z( f, g) is two-dimensional, so the analysis applies for any such g; this comment

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will be useful in later steps of the analysis. Recall that in this part of the analysis f isassumed to be an irreducible polynomial of degree ≥ 4.

We have a set P of m points and a set L of n lines in C4, so that P is contained in

the two-dimensional algebraic variety Z( f, g) ⊂ C4. By pruning away all the lines

containing at most max(D, deg(g)) points of P , we lose O(nD) incidences, and allthe surviving lines are contained in Z( f, g), as is easily checked. For simplicity ofnotation, we continue to denote by L the set of surviving lines.

Let Z( f, g) = ⋃si=1 Vi be the decomposition of Z( f, g) into its irreducible com-

ponents, as described in Section 2.2. By Theorem 2.2, we have∑s

i=1 deg(Vi ) ≤deg( f ) deg(g) = O(D2).

Incidences Within Non-planar Components of Z( f, g). Our next step is to analyzethe number of incidences between points and lines within the components of Z( f, g)that are not 2-flats. For this we first need the following bound on point-line incidenceswithin a two-dimensional surface in three dimensions. This part of the analysis is takenfrom our paper [37]. We also refer to Sect. 2.4 for properties of ruled surfaces.

For a point p on an irreducible singly ruled surface V , which is not the exceptionalpoint of V , we let V (p) denote the number of generator lines passing through p andfully contained in V (so if p is incident to an exceptional line, we do not count that lineinV (p)).We also put∗

V (p) := max0,V (p)−1. Finally, if V is a cone and pV isits exceptional point (that is, apex),we putV (pV ) = ∗

V (pV ) := 0.We also considera variant of this notation, where we are also given a finite set L of lines (where not alllines of L are necessarily contained in V ), which does not contain any of the (at mosttwo) exceptional lines of V . For a point p ∈ V , we let λV (p; L) denote the number oflines in L that pass through p and are fully contained in V , with the same provisions asabove, namely that we do not count incidences with exceptional lines, nor do we countincidences with an exceptional point, and put λ∗

V (p; L) := max0, λV (p; L) − 1. IfV is a cone with apex pV , we put λV (pV ; L) = λ∗

V (pV ; L) = 0. We clearly haveλV (p; L) ≤ V (p) and λ∗

V (p; L) ≤ ∗V (p), for each point p.

Lemma 3.1 Let V be an irreducible singly ruled two-dimensional surface of degreeD > 1 in R

3 or in C3. Then, for any line , except for the (at most) two exceptional

lines of V , we have∑

p∈∩VV (p) ≤ D if is not fully contained in V,

∑p∈∩V

∗V (p) ≤ D if is fully contained in V .

The following lemma provides the needed infrastructure for our analysis, and istaken from Sharir and Solomon [37, Thm. 15].

Lemma 3.2 Let V be a possibly reducible two-dimensional algebraic surface ofdegree D > 1 in R

3 or in C3, with no linear components. Let P be a set of m distinct

points on V and let L be a set of n distinct lines fully contained in V . Then there existsa subset L0 ⊆ L of at most O(D2) lines, such that the number of incidences betweenP and L \ L0 satisfies

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I (P, L \ L0) = O(m1/2n1/2D1/2 + m + n). (19)

Sketch of Proof. We provide the following sketch of the proof; the full details aregiven in the companion paper [37]. Consider the irreducible componentsW1, . . . ,Wu

of V . We first argue that the number of lines that are either contained in the union ofthe non-ruled components, or those contained in more than one ruled component of Vis O(D2), and we place all these lines, as well as the exceptional lines of any singlyruled component, in the exceptional set L0. We may thus assume that each survivingline in L1 := L \ L0 is contained in a unique ruled component of V , and is a generatorof that component.

The strategy of the proof is to consider each line of L1, and to estimate the numberof its incidences with the points of P in an indirect manner, via Lemma 3.1, appliedto and to each of the ruled components Wj of V .

Specifically, we fix some threshold parameter ξ , and dispose of points that areincident to at most ξ lines of L1, losing at most mξ incidences. Let P1 denote the setof surviving points.

Now if a line ∈ L1 is incident to a point p ∈ P1, it meets at least ξ other linesof L1 at p. It follows from Lemma 3.1 that the overall number of such lines, over allpoints in P1 ∩ , is roughly D, so the number of such points on is at most roughlyD/ξ , for a total of nD/ξ incidences of this kind. Choosing ξ = (nD/m)1/2 yieldsthe bound O(m1/2n1/2D1/2), and the lemma follows.

We can now proceed, by deriving two upper bounds for certain types of incidencesbetween P and L . The first bound is relevant for the range m = O(n4/3), and thesecond bound is relevant for the rangem = (n4/3). Nevertheless, both bounds applyto the entire range of m and n.

Proposition 3.3 The number of incidences involving non-singular points of Z( f ) thatare contained in components of Z( f, g) that are not 2-flats is

minO(mD2 + nD), O(m + nD4). (20)

Proof We first establish the bound O(mD2 + nD). Let p ∈ Z( f ) be a non-singularpoint. The irreducible decomposition of Sp := Z( f, g)∩ TpZ( f ) is the union of one-and two-dimensional components. Clearly, Sp contains all the lines that are incidentto p and are fully contained in Z( f, g); it is a variety, embedded in 3-space (namely,in TpZ( f )), of degree O(D2). The union of the one-dimensional components is acurve of degree O(D2), so, by Lemma 2.3, it can contain at most O(D2) lines; whensumming over all p ∈ P , the total number of incidences with those lines is O(mD2).

It remains to bound incidences involving the two-dimensional components of Spthat are not 2-flats. By Sharir and Solomon [36, Lem. 5], the number of lines incident top inside these two-dimensional components of Sp is at most O(D2), except possiblyfor lines that lie in a component that is a cone and has p as its apex. Summing overall p ∈ P , we get a total of O(mD2) incidences for this case too, ignoring lines thatlie only in conic (or flat) components.

Note that each two-dimensional component of Sp is necessarily also a two-dimensional irreducible component of Z( f, g). Hence the analysis performed so far

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takes care of all incidences except for those that occur on conic two-dimensional com-ponents of Z( f, g) (and on 2-flats, which we totally ignore in this proposition). LetV be a conic component of Z( f, g) with apex pV , which is not a 2-flat. We notethat V cannot fully contain a line that is not incident to pV . Indeed, suppose to thecontrary that V contained such a line . Since V is a cone with apex pV , for eachpoint a ∈ , the line connecting a to pV is fully contained in V , and therefore the2-flat containing pV and is fully contained in V . As V is irreducible and is not a2-flat, we obtain a contradiction, showing that no such line exists. We conclude thatany point on V , except for pV , is incident to at most one line that is fully containedin V (a “generator” through pV ), for a total of O(m) incidences. Since Z( f, g) is ofdegree O(D2), the number of conic components of Z( f, g) is O(D2), so, summingthis bound over all components V , we get again the bound O(mD2) on the numberof relevant “non-apex” incidences.

Therefore, it remains to bound the number of incidences between the points of

Pc := pV | pV is an apex of an irreducible conic component V of Z( f, g)

and the lines of L . Since there are at most O(D2) irreducible components of Z( f, g),we have |Pc| ≤ cD2, for some suitable constant c. We next let Lc denote the set oflines in L containing fewer than cD points of Pc, and claim that any point p ∈ Pc isincident to fewer than D lines of L\Lc. Indeed, otherwise,wewould get at least D linesincident to p, each containing at least cD+1 points of Pc, i.e., at least cD points otherthan p. As these points are all distinct, we would get that |Pc| ≥ 1 + D · cD > cD2,a contradiction. On the other hand, by definition of Lc, we have

I (Pc, Lc) = O(nD).

We have thus shown that the number of incidences involving points of Pc is

I (Pc, L) = I (Pc, Lc) + I (Pc, L \ Lc) = O(nD) + O(mD) = O(nD + mD),

well within the bound that we seek to establish.

The second bound We next establish the second bound O(m + nD4). Let V bean irreducible two-dimensional component of Z( f, g). If V is not ruled, then byProposition 2.13, it contains at most 11 deg(V )2 − 24 deg(V ) < 11 deg(V )2 lines.Summing over all irreducible components of Z( f, g) that are not ruled, we get at most11

∑V deg(V )2 = O(D4) lines. Let be one of those lines, and let p ∈ ∩ P . For

any other line λ ∈ L that passes through p, we charge its incidence with p to its inter-section with . This yields a total of O(nD4) incidences, to which we add O(m) forincidences with those points that lie on only one line of L , for a total of O(m + nD4)

incidences.We next analyze the irreducible components of Z( f, g) that are ruled but are not

2-flats. Let V1, . . . , Vk denote these components, for some k = O(D2). Projectall these components onto some generic hyperplane, and regard them as a single(reducible) ruled surface in 3-space, whose degree is

∑ki=1 deg(Vi ) = O(D2).

Lemma 3.2 then yields a subset L0 of L of size O(D4), and shows that

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I (P, L \ L0) = O(m1/2n1/2D + m + n).

The lines of L0 are simply added to the set of O(D4) lines not belonging to ruledcomponents. This does not affect the asymptotic bound O(nD4) derived above. Intotal we get

O(m1/2n1/2D + m + nD4)

incidences. Since

m1/2n1/2D ≤ 12 (m + nD2),

we obtain the second bound asserted in the proposition. Remark The term O(nD4) appears to be too weak, and can probably be improved,using ideas similar to those in the proof of Lemma 3.2. Since such an improvementdoes not have a significant effect on our analysis, we leave it as an interesting problemfor further research.

Restrictedness of hyperplanes and quadrics, and lines on 2-flats The bounds in Propo-sition 3.3 might be too large, for the current choices of D, because of the respectiveterms O(mD2) and O(nD4). (Technically, the m and n in the definition of D arenot necessarily the same as the m and n that denote the size of the current subsetsof the original P and L , but let us assume that they are the same for the present dis-cussion.) For example, when m = O(n4/3) and D = (m2/5/n1/5) (recall that thisis the “large” value of D for this range), we have mD2 = (m9/5/n2/5), and this is m2/5n4/5 when m n6/7. Similarly, when m = (n4/3) and D = (n/m1/2)

(which is the value chosen for this range), we have nD4 = (n5/m2), and this is mwhen m n5/3. These bounds will be used in the second partitioning step, wherewe use a smaller-degree partitioning polynomial, and for m outside the problematicranges, i.e., for m ≤ n6/7 or m ≥ n5/3; see below for details. Otherwise, for thecurrent D, these bounds need to be finessed and replaced by the following alternativeanalysis.11

In the first step of this analysis, we estimate the number of lines contained in ahyperplaneor a quadric (when Z( f, g) is two-dimensional), and establish the followingproperties.

Lemma 3.4 Each hyperplane or quadric H is O(D2)-restricted for the lines of Lthat are contained in non-planar components of Z( f, g).

Proof Fix a hyperplane or quadric H . Recall that all the lines in the current set Lare contained in Z( f, g). Let V be an irreducible component of Z( f, g), which isnot a 2-flat. If V ∩ H is a curve, then (recalling Theorem 2.2) its degree is at most

11 As the calculations worked out above indicate, the bounds in Proposition 3.3 will be within the bound(5) whenm is sufficiently small (below n4/3) or sufficiently large (above n5/3). For such values ofm we canbypass the induction process, and obtain the desired bounds directly, in a single step. See a more detaileddescription towards the end of this section.

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deg(V ) (when H is a hyperplane) or 2 deg(V ) (when H is a quadric), and can there-fore contain at most 2 deg(V ) lines, by Lemma 2.3. Therefore, the union of all theirreducible components V of Z( f, g) which intersect H in a curve, contains at most2

∑V deg(V ) = O(D2) lines. Assume then that V ∩ H is two-dimensional. Since V

is irreducible, we must have V ∩H = V , so V is fully contained in H . Moreover, V isan irreducible two-dimensional surface contained in Z( f )∩ H , and therefore must bean irreducible component of Z( f )∩ H , which is a two-dimensional surface of degree≤ D. By Theorem 2.2,

∑V⊂H deg(V ) ≤ deg(Z( f ) ∩ H) ≤ deg( f ) ≤ D. If V is

not ruled by lines (and, by assumption, is not a 2-flat), then by Proposition 2.13, itcontains at most 11 deg(V )2 lines, and summing over all such components V withinH , we get a total of at most

∑V 11 deg(V )2 = O(D2) lines.

The remaining irreducible (two-dimensional) components V of Z( f, g) that meetH (if such components exist) are fully contained in H , and are ruled by lines. Asalready observed, these components are also irreducible components of Z( f ) ∩ H ,and so, with the exception of O(D2) lines (those contained in the components alreadyanalyzed), all the lines of L that lie in H are contained in components of Z( f ) ∩ Hthat are ruled by lines. Since f restricted to H is a polynomial of degree ≤ D, andsince we are interested in lines of L that are not contained in planar components ofZ( f ) ∩ H we conclude that H is O(D2)-restricted, with respect to the subset of Lmentioned in the lemma.

We next analyze the number of lines contained in a 2-flat.

Lemma 3.5 Let π be a 2-flat that is not fully contained in Z( f, g). Then the numberof lines fully contained in Z( f ) ∩ π is O(D).

Proof The intersection Z( f )∩π is either π itself, or a curve of degree≤ D. The lattercase implies (using Lemma 2.3) thatπ contains at most D lines that are fully containedin Z( f ). In the former case π ⊂ Z( f ). By assumption, π is not contained in Z( f, g),implying that g intersectsπ in a curve of degree O(D) (sinceπ∩Z( f, g) = π∩Z(g)),and can therefore contain at most O(D) lines that are fully contained in Z( f ).

Recap Summing up what has been done so far, we can classify the incidences inI (P, L) into the following types. Recall that the analysis is confined to a single irre-ducible factor f of the original polynomial or of some higher partial derivative of sucha factor.

(a) We treat the cases where f is linear or quadratic separately, using a variant of The-orem 1.1, which takes into account the restrictedness of hyperplanes and quadrics;see Proposition 3.6 below.

(b) We treat the case where Z( f ) is ruled by lines separately (this is the second casein the analysis, when Z( f,FL(4)

f ) is three-dimensional).If f is not ruled by lines and is of degree ≥ 4 (recall that each surface of degree atmost 3 is ruled by lines—see Lemma 2.12), then there are two kinds of incidencesthat need to be considered.

(c) Incidences between points and lines that are contained in irreducible componentsof Z( f,FL(4)

f ) (or, more generally, of Z( f, g), for other suitable polynomials g)

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that are not 2-flats. We have bounded the number of these incidences in Proposi-tion 3.3 in two different ways, but we also ignored these incidences, passing themto the induction in the second partitioning step, to be presented later, where wenow know that each hyperplane and quadric is O(D2)-restricted, and each 2-flatcontains at most O(D) lines of L . For both properties to hold, we first have to getrid of all the lines of L that are contained in 2-flats within Z( f, g), and wewill per-form this pruning after bounding the number of incidences involving lines that arecontained in such 2-flats. This will make the O(D2)-restrictedness in Lemma 3.4hold with respect to the entire (pruned) set L , and will make Lemma 3.5 hold foreach 2-flat.

(d) Incidences between points and lines that are contained in some irreducible com-ponent of Z( f, g) that is a 2-flat. These incidences will be analyzed explicitlybelow, using the properties of flat points and lines, as presented in Sect. 2.5.

This classification of incidences, especially those of types (c) and (d), holds ingeneral, for any polynomial g satisfying the properties assumed in this treatment (thatit has degree O(D) and that Z( f, g) is two-dimensional), and their treatment alsoapplies to these more general scenarios.

Incidences within hyperplanes and quadrics We next derive a bound that we will useseveral times later on, in cases where we can partition P and L (or, more precisely,subsets thereof) among some finite collection of hyperplanes and quadrics, so that allthe relevant incidences occur between points and lines that are assigned to the samesurface. Recall that we have already applied a similar partitioning among the factorsof f and of its derivatives. The prime application of this bound will be to incidencesof type (a) above, but it will also be used in the analysis of type (d) incidences, and inthe analysis of the second case (b), where Z( f,FL(4)

f ) is three-dimensional, i.e., when

Z( f,FL(4)f ) = Z( f ). In particular, we emphasize that the following proposition does

not require that Z( f,FL(4)f ) be two-dimensional.

Proposition 3.6 Let H1, . . . , Ht be a finite collection of hyperplanes and quadrics.Assume that the points of P and the lines of L are partitioned among H1, . . . , Ht , sothat each point p ∈ P (resp., each line ∈ L) is assigned to a unique hyperplaneor quadric that contains p ( resp., fully contains ), and assume further that each Hi

is q-restricted and that each 2-flat contains at most s lines of L. Then the overallnumber of incidences between points and lines that are assigned to the same surface(i.e., hyperplane or quadric) is

O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + n). (21)

Proof For i = 1, . . . , t , let Li (resp., Pi ) denote the set of lines of L (resp., points ofP), that are assigned to Hi , and put ni := |Li |,mi := |Pi |. We have

∑i mi = m and∑

i ni = n. For each i , since Hi is q-restricted, there exists a polynomial gi = gHi ,of degree O(

√q), such that all the lines of Li , with the exception of at most q of

them, are fully contained in ruled components of Hi ∩ Z(gi ) that are not 2-flats. WriteLi = Lnr

i ∪ Lri , where L

ri is the subset of those lines that are fully contained in ruled

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components of Hi ∩ Z(gi ) that are not 2-flats, and Lnri is the complementary subset, of

size at most q. The lines in Lri are contained in the unionW

r of the ruled componentsof Hi ∩ Z(gi ) that are not 2-flats. We also remove from Lr

i the subset Lri0 of O(q)

lines, as provided by Lemma 3.2 (including all the lines that are fully contained inmorethan one component Wi ), and put them in Lnr

i ; we continue to use the same notationsfor these modified sets. To apply Lemma 3.2 to the case where Hi is a quadric, wefirst project the configuration onto some generic 3-space, and note that by Sharir andSolomon [35, Lem. 2.1], the projection of Wr does not contain any 2-flat. Since thesize of Lnr

i is still O(q), we have, by Theorem 1.1,

I (Pi , Lnri ) = O(m1/2

i |Lnri |3/4 + m2/3

i |Lnri |1/3s1/3 + mi + |Lnr

i |)= O(m1/2

i n1/2i q1/4 + m2/3i n1/3i s1/3 + mi + ni ).

(Note that Theorem 1.1 is directly applicable when Hi is a hyperplane, and that it canalso be appliedwhen Hi is a quadric, by projecting the configuration onto some generichyperplane, similar to what we have just noted for the application of Lemma 3.2.)

We next bound I (Pi , Lri ), using Lemma 3.2 (when Hi is a quadric, we apply it to the

generic projection ofWr to three dimensions, as above). Since deg(gi ) = O(√q),Wr

is of degree O(√q), and thus also its projection to three dimensions (see, e.g., [17]),

in case Hi is a quadric. We have already removed from Lri the subset L

ri0 provided by

the lemma, and so the lemma yields the bound

I (Pi , Lri ) = O(m1/2

i n1/2i q1/4 + mi + ni ).

That is, we have:

I (Pi , Li ) = O(m1/2i n1/2i q1/4 + m2/3

i n1/3i s1/3 + mi + ni ).

Summing these bounds for i = 1, . . . , t , and using Hölder’s inequality (twice), weget the bound asserted in (21). The case where f is linear or quadratic (These are the cases D = 1, 2) Let us applyProposition 3.6 right away to bound the number of incidences when our (irreducible) fis linear or quadratic, that is, when Z( f ) is a hyperplane or a quadric. Proposition 3.6(together with assumption (i) of the theorem) then implies the following bound.

I (P, L) = O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + n), (22)

which is subsumed by the main bound (5).

Incidences within 2-flats fully contained in Z( f, g)Assuming generic directions of thecoordinate axes, we may assume that, for every non-singular point p ∈ P, TpZ( f ) isnot orthogonal to any of the axes. This allows us to use the flatness criterion developedin Sect. 2.5 to each point of P .

As in previous steps of the analysis, we simplify the notation by denoting thesubsets of the points and lines that lie in the 2-flat components of Z( f, g) as Pand L , and their respective sizes as m and n. Each point p ∈ P (resp., each line

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734 Discrete Comput Geom (2017) 57:702–756

∈ L) under consideration is contained (resp., fully contained) in at least one of the2-flats π1, . . . , πk that are fully contained in Z( f, g) (these are the linear irreduciblecomponents of Z( f, g), and we have k = O(D2)). Let P(2) (resp., P(3)) denote theset of points p ∈ P that lie in at most two (resp., at least three) of these 2-flats. Assigneach point p ∈ P(2) to the (at most) two 2-flats containing it. Note that if p ∈ P(2),then every line that is incident to p can be contained in at most two of the 2-flats πi ,and we assign to those 2-flats. Let L(2) denote the set of lines ∈ L such that isincident to at least one point in P(2) (and is thus contained in at most two 2-flats πi ),and put L(3) = L\L(2). For i = 1, . . . , k, let L(2)

i (resp., P(2)i ) denote the set of lines of

L(2) (resp., points of P(2)), that are contained in πi , and put ni := |L(2)i |,mi = |P(2)

i |.(Note that we ignore here lines that are not fully contained in one of these 2-flats; theselines are fully contained in other components of Z( f, g) and their contribution to theincidence count has already been taken care of.) By construction,

k∑i=1

mi ≤ 2m, andk∑

i=1

ni ≤ 2n.

Moreover, a point p ∈ P(2) can be incident only to lines of L that are containedin one of the (at most) two 2-flats that contain p, so we have I (P(2), L(2)) ≤∑k

i=1 I (P(2)i , L(2)

i ). The Szemerédi–Trotter bound (1) yields

I (P(2)i , L(2)

i ) = O(m2/3i n2/3i + mi + ni ), i = 1, . . . , k. (23)

By assumption (ii) of the theorem, ni ≤ s for each i = 1, . . . , k, so, summing overi = 1, . . . , k and using Hölder’s inequality, we obtain

I (P(2), L(2)) ≤k∑

i=1

I (P(2)i , L(2)

i ) = O( k∑

i=1

(m2/3

i n2/3i + mi + ni))

= O(( k∑

i=1

m2/3i n1/3i s1/3

)+ m + n

)

= O(( k∑

i=1

mi

)2/3( k∑i=1

ni)1/3

s1/3 + m + n)

= O(m2/3n1/3s1/3 + m + n). (24)

Consider next the points of P(3), each contained in at least three 2-flats that are fullycontained in Z( f ). All the points of P(3) are linearly flat (see Sect. 2.5 for details),and are therefore flat. Notice that each such point can be incident to lines of L(2) andto lines of L(3). We prune away each line ∈ L that contains fewer than 3D points ofP(3), losing at most 3nD incidences in the process.

Each of the surviving lines contains at least 3D−3 flat points, and is therefore flat,because the degrees of the nine polynomials whose vanishing at p captures the flatnessof p, are all at most 3D − 4. In other words, we are left with the task of bounding

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the number of incidences between flat points and flat lines. To simplify this part ofthe presentation, we again rename the sets of these points and lines as P and L , anddenote their sizes by m and n, respectively.

Incidences between flat points and lines By Lemma 2.15, all the (non-singular) pointsof a flat line have the same tangent hyperplane. We assign each flat point p ∈ P (resp.,flat line ∈ L) to TpZ( f ) (resp., to TpZ( f ) for some (any) non-singular pointp ∈ P ∩ ; again we only consider lines incident to at least one such point). We havetherefore partitioned P and L among distinct hyperplanes H1, . . . , Ht , and we onlyneed to count incidences between points and lines assigned to the same hyperplane.By assumptions (i) and (ii) of the theorem, the conditions of Proposition 3.6 hold,implying that the number of these incidences is

O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + n). (25)

As promised, after having bounded the number of incidences within the 2-flats thatare fully contained in Z( f, g), we remove from L the lines that are contained in such2-flats, and continue the analysis with the remaining subset.

In summary, combining the bounds in (24) and (25), Proposition 3.3, and Lemmas 3.4and 3.5, the overall outcome of the analysis for the first case is summarized in thefollowing proposition. (In the proposition, f is one of the irreducible factors of theoriginal polynomial or of one of its derivatives, and P and L refer to the subsetsassigned to that factor.)

Proposition 3.7 Let g be any polynomial of degree O(D) such that Z( f, g) is two-dimensional, let P be a set of m points contained in Z( f, g), and let L be a set of nlines contained in Z( f, g). Then

I (P, L) = I (P∗, L∗) + O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + nD), (26)

where P∗ and L∗ are subsets of P and L, respectively, so that each hyperplane orquadric is O(D2)-restricted with respect to L∗, and each 2-flat contains at most O(D)

lines of L∗. We also have the explicit estimate

I (P∗, L∗) = minO(mD2 + nD), O(m + nD4). (27)

Remark (1) As already noted, lines that are contained in 2-flats that are fully containedin Z( f, g) have already been taken care of, and thus do not belong to L∗, so theapplication of Lemma 3.5 shows that every 2-flat contains only O(D) lines ofL∗, and the application of Lemma 3.4 shows that every hyperplane or quadric isO(D2)-restricted.

(2) When m and n are such that the bound on I (P∗, L∗) in (27) is dominated byO(m2/5n4/5 + m), we use these bounds explicitly, and get the induction-freerefined bound in (6). This remark will be expanded and highlighted later, as wespell out the details of the induction process.

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736 Discrete Comput Geom (2017) 57:702–756

Second Case: Z( f ) is Ruled by Lines

We next consider the case where the four-dimensional flecnode polynomial FL4fvanishes identically on Z( f ). By Theorem 2.11, this implies that Z( f ) is ruled by(possibly complex) lines.

In what follows we assume that D ≥ 3 (the cases D = 1, 2 have already beentreated earlier, using Proposition 3.6). We prune away points p ∈ P , with |p| ≤ 6(the number of incidences involving these points is at most 6m = O(m)). For sim-plicity of notation, we still denote the set of surviving points by P . Thus we now have|p| > 6, for every p ∈ P .

Recalling the properties of the u-resultant of f (that is, the u-resultant associatedwith F1(p; v), F2(p; v), F3(p; v)), as reviewed in Sect. 2.6, we have, by Corol-lary 2.17, that U (p; u0, u1, u2, u3) ≡ 0 (as a polynomial in u0, . . . , u3) for everyp ∈ P .

We will use the following theorem of Landsberg, which generalizes Theorem 2.11.It is stated here in a specialized and slightly revised form, but still for an arbitraryhypersurface in any dimension, and for any choice of the parameter k. Recall that k

is the union of kp over all p ∈ X , namely, it is the set of all lines that osculate to

Z( f ) to order three at some point on Z( f ).The actual application of the theorem will be for X = Z( f ) (and d = 4, k = 3).

We refer the reader to Sect. 2.1 for notations and further details.

Theorem 3.8 (Landsberg [19, Thm. 3.8.7]) Let X ⊂ Pd(C) be a hypersurface, and let

k ≥ 2 be an integer, such that there is an irreducible component k0 ⊂ k satisfying,

for every point p in a Zariski open setO ⊂ Z( f ), dimk0,p > d − k − 1, where k

0,p

is the set of lines in k0 incident to p. Then, for each point p ∈ O, all lines in k

0,pare contained in X.

To appreciate the theorem, we note that, informally, lines through a fixed point phave d − 1 degrees of freedom, and the constraint that such a line osculates to X toorder k removes k degrees of freedom, leaving d − k−1 degrees. The theorem assertsthat if the dimension of this set of lines is larger, for most points on X , then these linesare fully contained in X . Note also that this is a “local-to-global” theorem—the largedimensionality condition has to hold at every point of some Zariski open subset ofZ( f ), for the conclusion to hold.

IfU (p; u0, u1, u2, u3)does not vanish identically (as a polynomial inu0, u1, u2, u3)at every point p ∈ Z( f ), then at least one of its coefficients, call it cU , does not vanishidentically on Z( f ). In this case, as U vanishes identically at every point of P (as apolynomial in u0, u1, u2, u3), it follows that P is contained in the two-dimensionalvariety Z( f, cU ). Since cU has degree O(D) in x, y, z, w (by Theorem 2.16), we canproceed exactly as we did in the case where Z( f,FL4f ) was 2-dimensional. That is,we obtain the bound (26) in Proposition 3.7, namely,

I (P, L) = I (P∗, L∗) + O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + nD), (28)

where P∗ and L∗ are subsets of P and L , respectively, so that each hyperplane orquadric is O(D2)-restricted with respect to L∗, and each 2-flat contains at most O(D)

lines of L∗. We also have the explicit estimate

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Discrete Comput Geom (2017) 57:702–756 737

I (P∗, L∗) = minO(mD2 + nD), O(m + nD4).

Therefore, since this case does not require the following analysis, it suffices toconsider the complementary situation,wherewe assume thatU (p; u0, u1, u2, u3) ≡ 0at every point p ∈ Z( f ) (as a polynomial in u0, u1, u2, u3). By Theorem 2.16, 3

p isinfinite, so its dimension is positive, for each such p.

Informally, the analysis proceeds as follows. Since3p is (at least) one-dimensional

for every point p ∈ Z( f ), the set3, which is the union of3p, over all p ∈ Z( f ), has

(at least) three degrees of freedom—three for specifying p, at least one for specifyingthe line in 3

p, and one removed because the same line may arise at each of its points(if it is fully contained in Z( f )). In what follows we show that we can find a singleirreducible component 3

0 of 3, which is three-dimensional, and such that for anypoint p ∈ Z( f ), the variety 3

0,p is at least one-dimensional. This will facilitate theapplication of Theorem 3.8 in our context.

Theorem 3.9 There exists an irreducible component 30 of 3 of dimension at least

three, such that for each non-singular p ∈ Z( f ), the variety 30,p is at least one-

dimensional.

Proof The proof makes use of the Theorem of the Fibers and related results, asreviewed in Sect. 2.3. Put

W := (p, ) | p ∈ , ∈ 3p ⊂ Z( f ) × 3.

Note thatW is naturally embedded in P3 × P

5, where the second component containsthe Plücker hypersurface of lines in 3-space. W can formally be defined as the zeroset of homogeneous polynomials; one polynomial defines the Plücker quadric, otherpolynomials express the condition p ∈ , and other polynomials are those definingthe projective variety 3

p, whose elements are now represented by their Plücker coor-dinates in the appropriate projective space (see Sect. 2.1 for details). Therefore, W isa projective variety.

Let1 : W → Z( f ), 2 : W → 3

be the (restrictions to W of the) projections to the first and second factors of theproduct. For an irreducible component 3

0 of 3 (which is also a projective variety),put

W0 := −12 (3

0) = (p, ) ∈ W | ∈ 30,p.

Since W and 30 are projective varieties, so is W0. (Indeed, if W = Z( fi (p, )),

and 30 = Z(g j ()), for suitable sets of homogeneous polynomials fi , g j , then

W0 = Z( fi (p, ), g j ()).)Let W0 denote some irreducible component ofW0, and put Y := 1(W0) ⊂ Z( f ).

By the projective extension theorem (see, e.g., Cox et al. [5, Thm. 8.6]), Y is also aprojective variety.

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738 Discrete Comput Geom (2017) 57:702–756

For a point p ∈ Y , the fiber of the map 1|W0: W0 → Y over p is contained in

p × 30,p = (p, ) | ∈ 3

0,p (this is the fiber of 1|W0 over p, which clearly

contains the fiber of 1|W0over p, as W0 ⊆ W0).

We will show that there exists some component 30 , and some irreducible com-

ponent W0 of W0 = −12 (3

0), such that (i) Y = 1(W0) is equal to Z( f ),and (ii) for every point p ∈ Z( f ), the fiber of 1|W0

: W0 → Y over p is (at

least) one-dimensional; in this case we say that 30 and W0 form a one-dimensional

line cover of Z( f ). Suppose that we have found such a pair 30 , W0. As noted

above, the fiber of 1|W0over p is contained in (or equal to) p × 3

0,p, and

dim(p × 30,p) = dim(3

0,p). Therefore, since Y = Z( f ), this would imply that,

for every p ∈ Z( f ), we have dim(30,p) ≥ 1, which is what we want to prove.

We pick some component 30 , and some irreducible component W0 of

W0 = −12 (3

0), and analyzewhendo30 and W0 formaone-dimensional line cover of

Z( f ). Put, as above, Y = 1(W0). For a point p ∈ Y , put λ(p) = dim(1|−1W0

(p)),and let λ = minp∈Y λ(p). As noted above, λ(p) ≤ dim(3

0,p).

By the Theorem of the Fibers (Theorem 2.6), applied to the map 1|W0: W0 →

Y ⊆ Z( f ), we havedim(W0) = dim(Y ) + λ. (29)

Observe that λ ≤ 1. Indeed, if λ = 2, then there exists some non-singular pointp ∈ Y , such that 3

0,p is (at least) two-dimensional, implying that Z( f ) is a three-dimensional cone; since p is non-singular, Z( f ) is thus a hyperplane, contrary to ourassumptions.

Assume first that Y = 1(W0) is equal to Z( f ) (this is part (i) of the definitionof a one-dimensional line cover). If λ = 1, part (ii) of this property also holds, andwe are done. Assume then that λ = 0. By the first part of the Theorem of the Fibers(Theorem 2.6), the subset Y1 = p ∈ Y | λ(p) ≥ 1 is Zariski closed in Y , so it isa subvariety of Z( f ), of dimension at most 2. Hence, for each p in the Zariski opencomplement Y \ Y1, the fiber 1|−1

W0(p) is finite.

The remaining case is when Y = 1(W0) is properly contained in Z( f ). SinceZ( f ) is irreducible, Y is of dimension at most two.

To recap, we have proved that for each component 30 of 3, and each component

W0 of W0, if the associated Y is properly contained in Z( f ), then the image of W0under 1 (that is, Y ) is at most two-dimensional; we refer to this situation as being ofthe first kind. If Y = Z( f ) but λ = 0 (these are refered to as situations of the secondkind), then, except for a two-dimensional subvariety Y1 of Z( f ), the fibers of the map1|W0

are finite.However, in the case under consideration, we have argued that, for any non-singular

point p ∈ Z( f ), the fiber −11 (p) = p × 3

p is (at least) one-dimensional.We apply this analysis to all the irreducible components 3

0 of 3, and to all theirreducible components of the correspondingW0 = −1

2 (30). LetY

∗ denote the unionof all the images Y of the first kind, and of all the excluded subvarieties Y1 of the secondkind. Being a finite union of two-dimensional varieties, Y ∗ is two-dimensional.

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Discrete Comput Geom (2017) 57:702–756 739

The union, over the irreducible components 30 of 3, of all the corresponding

components W0, covers W , and therefore, for any non-singular point p ∈ Z( f ), theunion over all the components W0 of the fibers of 1|W0

over p is equal to the fiberof 1 over p, which is one-dimensional (and thus infinite).

We claim that there must exist some irreducible component 30 of 3, and a corre-

sponding irreducible component W0 of W0, such that Y = 1(W0) is equal to Z( f ),and the corresponding λ is equal to 1. Indeed, if this were not the case, take any non-singular point p in Z( f ) \ Y ∗. Since p is not in the image 1(W0), for any W0 of thefirst kind, the fiber of 1|W0

at p is empty. Similarly, since p is not in the excluded set

Y1 for any W0 of the second kind, the fiber of 1|W0at p is finite. But then the fiber

of 1 at p, being a finite union of (empty or) finite sets, must be finite, a contradictionthat establishes the claim.

Since for every p ∈ Y = Z( f ), λ ≤ λ(p) ≤ dim(30,p), it follows that all the

fibers 30,p are (at least) one-dimensional, completing the proof.

Remark One interesting corollary of the Theorem of the Fibers is that if we know thatfor any point p in a Zariski open subset O of Z( f ), the fiber of 1 over p (which isequal to p×3

p) is one-dimensional, then this is true for the entire Z( f ). Indeed, by

the Theorem of the Fibers (Theorem 3.9), the set p ∈ Z( f ) | dim(−11 (p)) ≥ 1 is

Zariski closed, and, since it contains the Zariski open setO, it must be equal to Z( f ).

By the preceding remark, Theorem 3.8 (with d = 4, k = 3,O = Z( f ), and 30 as

specified by Theorem 3.9) then implies that Z( f ) is infinitely ruled by lines, in thesense defined in Sect. 2.6; that is, each point p ∈ Z( f ) is incident to infinitely manylines that are fully contained in Z( f ), and, moreover, 3

0,p = 0,p (which is the set

of lines in 0 incident to p). That is, we have shown that 30 = 0. In other words,

for each p ∈ Z( f ),0,p is of dimension at least 1, or, equivalently, the cone 0,p(which is the union of the lines in 0,p) is at least two-dimensional. If, for some non-singular p ∈ Z( f ), the cone 0,p were three-dimensional, then, as already noted,Z( f ) would be a hyperplane, contrary to assumption. Thus, for each non-singularpoint p ∈ Z( f ), the cone 0,p is two-dimensional, and 0,p is one-dimensional. Wealso have dim(0) = dim(3

0) ≥ 3. We thus have

Corollary 3.10 The union of lines in 30 = 0 is equal to Z( f ), and dim(0) =

dim(30) ≥ 3.

Severi’s theorem The following theorem is a major ingredient in the present part ofour analysis. It has been obtained by Severi [34] in 1901, and a variant of it is alsoattributed to Segre [33]; it is mentioned in a recent work of Rogora [31], in anotherwork of Mezzetti and Portelli [26], and also appears in the unpublished thesis ofRichelson [30]. Severi’s paper is not easily accessible (and is written in Italian). As asmall service to the community, we sketch in Appendix A a proof of this theorem (orrather of a special case of the theorem that arises in our context), suggested to us byA.J. de Jong.

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740 Discrete Comput Geom (2017) 57:702–756

Theorem 3.11 (Severi’s Theorem [34]) Let X ⊂ Pd(C) be a k-dimensional irre-

ducible variety, and let 0 be an irreducible component of maximal dimension ofF(X), such that the lines of 0 cover X. Then the following holds.

1. If dim(0) = 2k − 2, then X is a copy of Pk(C) (that is, a complex projective

k-flat).2. If dim(0) = 2k − 3, then either X is a quadric, or X is ruled by copies of

Pk−1(C), i.e., every point12 p ∈ X is incident to a copy of P

k−1(C) that is fullycontained in X.

As is easily checked, the maximum dimension of 0 is 2k − 2. Note also that thecases where dim0 < 2k − 3 are not treated by the theorem (although they mightoccur); see Rogora [31] for a (partial) treatment of these cases.

We apply Severi’s theorem to Z( f ) and to the component 0 obtained in Theo-rem 3.9 and Corollary 3.10, with k = 3 and with dim(0) = 3 = 2k − 3. We thusconclude that either Z( f ) is a quadric, a case ruled out in the present part of theanalysis (which assumed that deg( f ) ≥ 3), or it is ruled by 2-flats.

The case where Z( f ) is ruled by 2-flats. In the remaining case, every point p ∈ Z( f )(see the footnote in Theorem 3.11) is incident to at least one 2-flat τp ⊂ Z( f ). LetDp denote the set of 2-flats that pass through p and are contained in Z( f ).

For a non-singular point p ∈ Z( f ), if |Dp| > 2, then p is a (linearly flat and thus)flat point of Z( f ). Recall that we have bounded the number of incidences involvingflat points (and lines) by partitioning them among a finite number of hyperplanes,and by bounding the incidences within each hyperplane. (Recall that lines incident tofewer than 3D−3 points of P have been pruned away, losing only O(nD) incidences,and that the remaining lines are all flat.) Repeating this argument here, we obtain thebound

O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + n).

In what follows we therefore assume that all points of P are non-singular and non-flat(call these points ordinary for short), and therefore |Dp| = 1 or 2, for each such p. PutH1(p) (resp., H1(p), H2(p)) for the 2-flat (resp., two 2-flats) in Dp, when |Dp| = 1(resp., |Dp| = 2).

Clearly, each line in L , containing at least one ordinary point p ∈ Z( f ), is fullycontained in at most two 2-flats fully contained in Z( f ) (namely, the 2-flats of Dp).

Assign each ordinary point p ∈ P to each of the at most two 2-flats in Dp, andassign each line ∈ L that is incident to at least one ordinary point to the at most two2-flats that fully contain and are fully contained in Z( f ) (it is possible that is notassigned to any 2-flat—see below). Changing the notation, enumerate these 2-flats,over all ordinary points p ∈ P , as U1, . . . ,Uk , and, for each i = 1, . . . , k, let Pi andLi denote the respective subsets of points and lines assigned to Ui , and let mi and nidenote their cardinalities. We then have

∑i mi ≤ 2m and

∑i ni ≤ 2n, and the total

12 Similar to the definition in Sect. 2.4 for the case of lines, it suffices to require this property for everypoint in some Zariski-open subset of X . Here too one can show that the two definitions are equivalent. Seealso the companion paper [37, Lem. 11].

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Discrete Comput Geom (2017) 57:702–756 741

number of incidences within the 2-flatsUi (excluding lines not assigned to any 2-flat)is at most

∑ki=1 I (Pi , Li ). This incidence count can be obtained exactly as in the first

case of the analysis, using the bound in (24). That is, we have

k∑i=1

I (Pi , Li ) = O(m2/3n1/3s1/3 + m + n).

As noted, this bound does not take into account incidences involving lines which arenot contained in any of the 2-flatsUi (and are therefore not assigned to any such 2-flat).It suffices to consider only lines of this sort that are non-singular and non-flat, sincesingular or flat lines are only incident to singular or flat points, and we assumed abovethat all the points of P are ordinary points. If is a non-singular and non-flat line, andis not fully contained in any of the Ui , we call it a piercing line of Z( f ).

Lemma 3.12 If is a piercing line of Z( f ), then the union of lines fully contained inZ( f ) and intersecting is equal to Z( f ).

Proof Let V denote this union. By a suitable extension to four dimensions of a similarresult of Sharir and Solomon [36, Lem. 5], V is a variety in the complex projectivesetting, which we assume throughout this part of the analysis. Clearly V ⊆ Z( f ). If Vis strictly contained in Z( f ), then, since Z( f ) is irreducible, V must be a finite unionof irreducible components V1, . . . , Vk , each of dimension at most two. Let p ∈ bean ordinary point of Z( f ) (since is non-singular and non-flat, such a point exists),and let H1(p) be one of the at most two 2-flats in Dp. Note that H1(p) is containedin V (because it is a union of lines fully contained in Z( f ) and intersecting at p).We claim that there exists some Vj such that H1(p) ⊆ Vj . Indeed, otherwise, theintersection H1(p) ∩ Vj would be (at most) one-dimensional for each j = 1, . . . , k(a variety strictly contained in a 2-flat is of dimension at most one), and therefore

V ∩ H1(p) =( k⋃

j=1

Vj

)∩ H1(p) =

k⋃j=1

(Vj ∩ H1(p))

is a finite union of varieties of dimension at most one, contradicting the fact that H1(p)is contained in V (and is of dimension two). This contradiction establishes the claim.Since H1(p) and Vj are two-dimensional irreducible varieties and H1(p) ⊆ Vj , itfollows that H1(p) = Vj .

In other words, for each ordinary point p ∈ there exists a 2-flat H1(p) ∈ Dp

which is equal to some component Vj . Consider only the components Vj that coincidewith such a 2-flat. Since there are only finitely many components Vj of this kind, oneof them, call it Vj0 , has to intersect in infinitely many points, and therefore ⊆ Vj0 .That is, is contained in the 2-flat Vj0 that is fully contained in Z( f ).

Now pick any ordinary point p ∈ P ∩ . By definition, since p ∈ Vj0 , Vj0 must beone of the (at most) two 2-flats in Dp. But then is fully contained in that 2-flat, whichis one of the Ui ’s, and therefore is not a piercing line. This contradiction completesthe proof.

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742 Discrete Comput Geom (2017) 57:702–756

Remark The last step of the proof shows that if a non-singular and non-flat line

contains a point of P then it is piercing (if and) only if it is not contained in any 2-flatfully contained in Z( f ).

Lemma 3.13 Let p ∈ Z( f ) be an ordinary point. Then p is incident to at most onepiercing line.

Proof Assume to the contrary that p is incident to two piercing lines 1, 2 ∈ L .We claim that the 2-flat π12 that is spanned by 1 and 2 is fully contained in Z( f )(and thus, by the preceding remark, 1 and 2 are not piercing lines). Indeed, for anypoint q ∈ 1, Lemma 3.12 implies that there exists some line q = 1, incident toq, that intersects 2 and is fully contained in Z( f ). When q varies along the non-singular points of 1, we get an infinite collection of lines, fully contained in bothZ( f ) and π12, i.e., in their intersection Z( f ) ∩ π12. If π12 is not contained in Z( f )then Z( f ) ∩ π12 = π12 is a degree-D plane curve, so by Lemma 2.3, it contains atmost D lines, and therefore cannot contain the infinite union of lines

⋃p p.

Therefore, each ordinary point p ∈ P is incident to at most one piercing line, andthe total contribution of incidences involving ordinary points and piercing lines is atmost m.

In summary, combining the bounds that we have obtained for the various subcases ofthe second case, we get the following proposition. As in the first case, here f refersto a single irreducible factor (of the original polynomial or one of its derivatives), Dto its degree, and P and L refer to the subsets of the original respective sets of pointsand lines, that are assigned to f .

Proposition 3.14 Let P be a set of m points contained in Z( f ), and let L be a setof n lines contained in Z( f ), and assume that Z( f ) is ruled by lines and that f is ofdegree ≥ 3. Then

I (P, L) = I (P∗, L∗) + O(m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + nD

), (30)

where P∗ and L∗ are subsets of P and L, respectively, so that each hyperplane orquadric is O(D2)-restricted with respect to L∗, and each 2-flat contains at most O(D)

lines of L∗. We also have the explicit estimate

I (P∗, L∗) = minO(mD2 + nD), O(m + nD4). (31)

The Induction

In summary, after having exhausted all possible cases,we are in the following situation;we finally undo the shorthand notations that we have used, and re-express the variousbounds in terms of the original parameters.

The first partitioning step has resulted in a collection of irreducible polynomials,which we write as f1, . . . , fk , with respective degrees D1, . . . , Dk , all upper bounded

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Discrete Comput Geom (2017) 57:702–756 743

by the degree D chosen in (27) for the original values of m and n. The points of Phave been partitioned among the zero sets Z( f1), . . . , Z( fk), into respective pairwisedisjoint subsets P1, . . . , Pk , including a leftover subset P ′ of points outside all thezero sets, and the lines of L have been partitioned among the zero sets, into respectivepairwise disjoint subsets L1, . . . , Lk , so that the zero set to which a line is assignedfully contains it, and including a leftover subset L ′ of lines not fully contained in anyzero set. Put mi = |Pi |, ni = |Li |, for i = 1, . . . , k, and m′ = |P ′|, n′ = |L ′|. Thenm′ + ∑k

i=1 mi = m, and n′ + ∑ki=1 ni = n.

Then I (P, L) is I (P ′, L ′)+∑ki=1 I (Pi , Li ) plus the number of incidences between

points assigned to some Z( fi ) and lines not fully contained in Z( fi ). (Note thatI (P \P ′, L ′) also counts incidences of this kind.) As we have argued, the total numberof these additional incidences is O(nD). That is, we have, for any choice of thedegree D,

I (P, L) ≤ I (P ′, L ′) + O(nD) +k∑

i=1

I (Pi , Li ). (32)

For each i , the preceding analysis culminates in the following bound.

I (Pi , Li ) = I (P∗i , L∗

i ) + O(m1/2i n1/2i q1/4 + m2/3

i n1/3i s1/3 + mi + ni D), (33)

where, for each i, P∗i and L∗

i are respective subsets of Pi and Li , so that eachhyperplaneor quadric is O(D2)-restricted with respect to L∗

i , and each 2-flat contains at mostO(D) lines of L∗

i . We also have the explicit estimate

I (P∗i , L∗

i ) = minO(mi D2 + ni D), O(mi + ni D

4), for each i. (34)

In addition, for the large values of D in (14), we have

I (P ′, L ′) = O(m2/5n4/5 + m). (35)

Induction-free derivation of the bound To proceedwith the analysis, for general valuesof m and n, we bound the various quantities I (P∗

i , L∗i ) using induction. However, as

asserted in the theorems, the cases where m ≤ n6/7 or m ≥ n5/3 admit an induction-free argument that yields the improved bound in (6), and we first dispose of thesecases. (Recall that these are the original values of m and n, the respective sizes of theentire input sets P and L .)

Assume first thatm ≤ n6/7.We substitute (33), the first bounds in (34), and (35) into(32). Using the Cauchy-Schwarz and Hölder’s inequalities, we have

∑i m

1/2i n1/2i ≤

m1/2n1/2 and∑

i m2/3i n1/3i ≤ m2/3n1/3. We also have

∑i mi ≤ m and

∑i ni ≤ n. In

total we thus get

I (P, L) = O(m2/5n4/5 + m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + mD2 + nD)

= O(m2/5n4/5 + m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + n),

where we have used the fact that mD2 + nD = O(m2/5n4/5 + n) for the choiceD = O(m2/5/n1/5) in (27). This establishes (6) for this case. The case m ≥ n5/3 is

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744 Discrete Comput Geom (2017) 57:702–756

handled in the same manner, using the second bounds O(mi + ni D4) in (34) instead,and the fact that the sum of these bounds is O(m) when m ≥ n5/3.

The induction via a new partitioning We now proceed with the general case, whereinduction is needed. To simplify the notation, we (again, but only temporarily) drop theindices, and consider one of many (possibly a nonconstant number of) subproblems,involving a set P (=P∗

i ) of m (≤ mi ) points and a set L (= L∗i ) of n (≤ ni ) lines, so

that each hyperplane or quadric is O(D2)-restricted for L , and each 2-flat containsat most O(D) lines of L; here D (=Di ) is the degree of the corresponding factor f(= fi ), which is upper bounded by the value in (14). In what follows we will use thislatter bound for (an upper bound on) the Di ’s.

To make the induction work, we choose a degree E , typically much smaller thanD (see below for the actual value), and construct a new partitioning polynomial hof degree E for P . (Although P ⊂ Z( f ) and each line of L is fully contained inZ( f ), we ignore here f completely, possibly losing some structural properties of Pand L , and consider only the partitioning induced by h.) With an appropriate value ofr = (E4), we obtain O(r) cells, each containing at most m/r points of P , and eachline of L either crosses at most E + 1 cells, or is fully contained in Z(h).

Set P0 := P ∩ Z(h) and P ′ := P \ P0. Similarly, denote by L0 the set of lines of Lthat are fully contained in Z(h), and put L ′ := L \ L0. We repeat the whole analysisdone so far, but with h and its degree E instead of f and D, for the points of P andthe lines of L . That is, we apply, to our P and L , the bounds given in (32), (33), and(34) [but not the one in (35)], with E instead of D. Moreover, in this application weexploit the property that each hyperplane or quadric is O(D2)-restricted with respectto L , and each 2-flat contains at most O(D) lines of L . We thus get the followingrecurrence (where the parameters k, Pi , Li , etc., are new and depend on h, but werecycle the notation in the interest of simplicity).

I (P, L) ≤ I (P ′, L ′) + O(nE) +k∑

i=1

I (Pi , Li )

= I (P ′, L ′) + O(nE) +k∑

i=1

I (P∗i , L∗

i )

+k∑

i=1

O(m1/2i n1/2i D1/2 + m2/3

i n1/3i D1/3 + mi + ni E).

Concretely, P ′ is the subset of the points of P contained in the cells of theh-partition,L ′ is the subset of lines of L not fully contained in Z(h), Pi and Li are the subsets of thepoints and lines assigned to the various irreducible factors hi of h and of its derivatives,and P∗

i , L∗i are the excluded subsets, as provided in Propositions 3.7 and 3.14.

Using the Cauchy-Schwarz and Hölder’s inequalities in the second sum, we get,for a suitable absolute constant a,

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Discrete Comput Geom (2017) 57:702–756 745

I (P, L) ≤ I (P ′, L ′) + a(m1/2n1/2D1/2 + m2/3n1/3D1/3 + m + nE)

+k∑

i=1

I (P∗i , L∗

i ).

We have

k∑i=1

I (P∗i , L∗

i ) ≤ a′( k∑i=1

minmi E2 + ni E, mi + ni E

4)

≤ mina′(mE2 + nE), a′(m + nE4),

for a suitable absolute constant a′. That is, slightly increasing the coefficient a, wehave

I (P, L) ≤ I (P ′, L ′) + a(m1/2n1/2D1/2 + m2/3n1/3D1/3 + m + nE)

+ minamE2, anE4. (36)

We next turn to bound I (P ′, L ′). For each cell τ of R4 \ Z(h), put Pτ := P ′ ∩ τ ,

and let Lτ denote the set of the lines of L ′ that cross τ ; put mτ = |Pτ | ≤ m/r (wherer = (E4)), and nτ = |Lτ |. Since every line ∈ L ′ crosses atmost E+1 componentsof R

4 \ Z(h), we have∑

τ nτ ≤ n(1 + E).To simplify the application of the induction hypothesis within the cells of the

partition, we want to make the subproblems be of uniform size, so that mτ = m/E4

and nτ = n/E3 for each τ (the latter quantity, up to some constant, is the averagenumber of lines crossing a cell). This is easy to enforce: To achieve mτ = m/E4, wesimply partition Pτ into mτ /(m/E4) = O(1) subsets, each consisting of at mostm/E4 points, and analyze each subset separately. Similarly, if τ is crossed by ξn/E3

lines, for ξ > 1, we treat τ as if it occurs ξ times, where each incarnation involves allthe points of (each of the constantly many corresponding subsets of) Pτ , and at mostn/E3 lines of Lτ . As is easily verified, the number of subproblems remains O(E4),with a larger constant of proportionality.

We apply the induction hypothesis [i.e., the inequality (13)] for each cell τ . It ishere that the extra factor 2c

√logm in the bound in the theorem comes into play; as

noted earlier, its role is to make the induction step work. We obtain

I (Pτ , Lτ ) ≤ 2c√logmτ (m2/5

τ n4/5τ + mτ )+βA(m1/2τ n1/2τ D1/2 + m2/3

τ n1/3τ D1/3 + nτ )

= 2c√

log(m/E4)((m/E4)2/5(n/E3)4/5 + m/E4)

+ βA((m/E4)1/2(n/E3)1/2D1/2+(m/E4)2/3(n/E3)1/3D1/3 + n/E3),

for a suitable absolute constant β. Summing this bound over all cells τ , that is, multi-plying it by O(E4), we get, for a suitable absolute constant b,

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746 Discrete Comput Geom (2017) 57:702–756

∑τ

I (Pτ , Lτ ) ≤ b · 2c√

log(m/E4)(m2/5n4/5 + m)

+ bA(m1/2n1/2D1/2E1/2 + m2/3n1/3D1/3E1/3 + nE). (37)

We have

2c√

log(m/E4) = 2c√logm−4 log E = 2c

√logm

(1− 4 log E

logm

)1/2

< 2c√logm

(1− 2 log E

logm

)= 2c

√logm

22c log E/√logm

.

We choose E to ensure that

22c log E/√logm > 2b, or

2c log E√logm

> log(2b),

or log E >log(2b)

2c

√logm.

That is, we choose

E > 2c∗√logm, for c∗ = log(2b)

2c< c/3, (38)

where the last constraint can be enforced if c is chosen sufficiently large. With thisconstraint on the choice of E , (37) becomes

∑τ

I (Pτ , Lτ ) ≤ 122

c√logm(m2/5n4/5 + m)

+ bA(m1/2n1/2D1/2E1/2 + m2/3n1/3D1/3E1/3 + nE). (39)

Adding this bound to the one in (36), we get

I (P, L) ≤ 122

c√logm(m2/5n4/5 + m)

+ (bA + a)(m1/2n1/2D1/2E1/2 + m2/3n1/3D1/3E1/3 + nE) + am

+ minamE2, anE4. (40)

Returning to the original notations, we have just bounded I (P∗i , L∗

i ), for anyi = 1, . . . , k. Concretely, we have shown that, for each i ,

I (P∗i , L∗

i ) ≤ 122

c√logmi (m2/5

i n4/5i + mi )

+ (bA + a)(m1/2i n1/2i D1/2E1/2

i + m2/3i n1/3i D1/3E1/3

i + ni Ei ) + ami

+ minami E2i , ani E

4i , (41)

where Ei is the degree of the new partitioning polynomial that is constructed for P∗i

and L∗i .

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Discrete Comput Geom (2017) 57:702–756 747

We now add up these bounds, using (32), (33), and (35), and replacing the Ei ’sby a common upper bound E that we will choose shortly. We thus get the followingbound, where now P and L stand, respectively, for the original, entire input sets ofpoints and lines.

I (P, L) ≤ γ (m2/5n4/5 + m) + γ nD

+ γ

k∑i=1

(m1/2i n1/2i q1/4 + m2/3

i n1/3i s1/3 + mi + ni D)

+ 12

k∑i=1

2c√logmi (m2/5

i n4/5i + mi )

+ γ

k∑i=1

(m1/2i n1/2i D1/2E1/2 + m2/3

i n1/3i D1/3E1/3 + ni E + mi ) (42)

+k∑

i=1

minami E2, ani E

4, (43)

for a suitable absolute constant γ . With several applications of the Cauchy–Schwarzand Hölder’s inequalities we get

I (P, L) ≤ (γ + 122

c√logm)(m2/5n4/5 + m)

+ γ (m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + nD)

+ γ (m1/2n1/2D1/2E1/2 + m2/3n1/3D1/3E1/3 + nE + m)

+ minamE2, anE4. (44)

We now bifurcate depending on the relation betweenm and n, where now, as in therecurrence just derived, m and n refer to the original values of these parameters.

The case m = O(n4/3). Recall that here we take D = O(m2/5/n1/5). It is easilychecked that, for this choice of D, each of the termsm1/2n1/2D1/2,m2/3n1/3D1/3,m,and nD ≥ n, is O(m2/5n4/5), because n1/2 ≤ m = O(n4/3).

We choose13 E = 2c∗√logm . This turns (44) into the bound

I (P, L) ≤ (γ + 122

c√logm + μ22c

∗√logm)(m2/5n4/5 + m)

γ (m1/2n1/2q1/4 + m2/3n1/3s1/3),

for suitable absolute constants μ and γ . The choice of c∗, and the assumption thatm ≥ M0 and that M0 is sufficiently large, ensure that

γ + μ22c∗√logm < 1

22c√logm,

13 This rather minuscule value of E is only needed when m ≈ n4/3; for smaller values of m, much largervalues of E can be chosen.

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748 Discrete Comput Geom (2017) 57:702–756

and thus we get

I (P, L) ≤ 2c√logm(m2/5n4/5 + m) + γ (m1/2n1/2q1/4 + m2/3n1/3s1/3),

which is the bound asserted in (5).

The case m = (n4/3). Here we take D = O(n/m1/2). It is easily checked that,for this choice of D, each of the terms m1/2n1/2D1/2,m2/3n1/3D1/3,m2/5n4/5, andnD ≥ n, is O(m), because m = (n4/3).

We choose, as before, E = 2c∗√logm (or a larger value when applicable), and note

that, for m ≥ M0 sufficiently large, the term nE4 is also O(m). This turns (44) intothe bound

I (P, L) ≤ (γ + 122

c√logm + μ22c

∗√logm)(m2/5n4/5 + m)

+ γ (m1/2n1/2q1/4 + m2/3n1/3s1/3),

for suitable absolute constantsμ and γ . As above, the choice of c∗, and the assumptionthat m ≥ M0 and that M0 is sufficiently large, ensure that

γ + μ22c∗√logm < 1

22c√logm,

and thus we get

I (P, L) ≤ 2c√logm(m2/5n4/5 + m) + γ (m1/2n1/2q1/4 + m2/3n1/3s1/3),

again establishing the bound in (5). Therefore, in both cases, we completed, at last, theinduction step and thus establishing the general upper bound (5) in the theorem. Theimproved bound in (6), for m ≤ n6/7 or for m ≥ n5/3, has already been established.With the lower bound construction, given in the following section, the proof of thetheorem is completed.

4 The Lower Bound

In this section we present a construction that shows that the bound asserted in thetheorem is worst-case tight (except for the factor 2c

√logm), for each m and n, and

for q and s in suitable corresponding ranges, made precise below. The constructionis a generalization to four dimensions of a construction due to Elekes; see [9]. (Athree-dimensional generalization has been used in Guth and Katz [15] for their lowerbound construction.)

We have already remarked that the “lower order” terms m1/2n1/2q1/4 andm2/3n1/3s1/3 are both worst-case tight, as they can be attained by a suitable pack-ing of points and lines into hyperplanes (for the first term) or planes (for the secondterm). Specifically, assume that s ≥ √

q , and create n/q parallel hyperplanes, andplace on each of them q lines and mq/n points in a configuration that attains the

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three-dimensional lower bound as in Guth and Katz [15]. Note that in this construc-tion no plane contains more than

√q ≤ s lines, as desired. Overall, we get

(n/q) · ((mq/n)1/2q3/4) = (m1/2n1/2q1/4)

incidences. A similar (and simpler) construction can be carried out for the second termm2/3n1/3s1/3.

We therefore focus on the term m2/5n4/5 (the remaining terms m and n are trivialto attain).

We fix two integer parameters k and , with concrete values that will be set later,and take P to be the set of vertices of the integer grid

(x, y, z, w) | 1 ≤ x ≤ k, 1 ≤ y, z, w ≤ 2k.

We have |P| = 8k43.We then take L to be the set of all lines of the form

y = ax + b, z = cx + d, w = ex + f, (45)

where 1 ≤ a, c, e ≤ and 1 ≤ b, d, f ≤ k. We have |L| = k36. Note that each linein L has k incidences with the points of P , one for each x = 1, 2, . . . , k, so

I (P, L) = k46 = (|P|2/5|L|4/5),

as is easily checked. Note that |L|1/2 ≤ |P| ≤ 8|L|4/3, which is (asymptotically)the range of interest for this bound to be significant: when |P| < |L|1/2 we have thetrivial bound I (P, L) = O(|L|), andwhen |P| > |L|4/3, the leading term in the boundchanges qualitatively to O(m), which is trivial for a lower bound. Moreover, for anypair of integersm, n, with n1/2 ≤ m ≤ n4/3, we can find k and forwhich |P| = (m)

and |L| = (n). Specifically, choose k = (m2/5/n1/5) and = (n4/5/m3/5); bothare ≥ 1 for the range of m and n under consideration.

To complete the construction, we show that no hyperplane or quadric containsmore than q0 := O

(|L|6/5/|P|2/5) = O(k26) lines of L , and no plane containsmore than s0 := O

(|L|7/5/|P|4/5) = O(kl6) lines of L . As an easy calculationshows, these threshold values of q and s are such that, for q > q0 or s > s0, thecorresponding “lower-dimensional” term m1/2n1/2q1/4 or m2/3n1/3s1/3 dominatesthe “leading” term m2/5n4/5 (for the former domination to arise, we need to assume,as above, that

√q ≤ s), making the above construction pointless (see below for more

details). The actual values of q and s that wewill now derive are actuallymuch smaller.To estimate our q and s, let h be an arbitrary hyperplane. If h is orthogonal to the

x-axis then it does not contain any line of L , as is easily checked, so we may assumethat h intersects any hyperplane of the form x = i in a 2-plane πi . The intersectionof P with x = i is a 2k × 2k × 2k lattice, that we denote as Qi . Every line λ ∈ Lin h meets πi at a single point (as noted, it cannot be fully contained in πi ), which isnecessarily a point in Qi (every line of L contains a point of every Qi ). The size ofπi ∩Qi is easily seen to be O((k)2), and each point is incident to at most 2 lines that

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lie in h. To see this latter property, substitute the equations (45) of a line of L into thelinear equation defining h, say Ax + By +Cz + Dw − 1 = 0 (where B,C and D arenot all 0). This yields a linear equation in x , whose x-coefficient has to vanish. This inturn yields a linear equation in a, c, and e, which can have at most 2 solutions over[1, . . . , ]3 (it is easily checked that the x-coefficient cannot be identically zero forall choices of a, c, e). The number of lines of point-line incidences of P and L withinh is thus O(2(k)2) = O(k24). Since each line is incident to k points, necessarilyall lying in h, it follows that the number of lines of L in h is O(k24/k) = O(k4),which is always smaller than q0.

This analysis easily extends to show that no quadric contains more than O(k4)lines of L; we omit the routine details.

Finally, let π be a 2-plane, where again we may assume that π is not orthogonal tothe x-axis. Then π meets a hyperplane x = i in a line μ, and μ ∩ Qi contains at mostk points. Every line λ in π meets μ at one of these points and, arguing as above, eachsuch point can be incident to at most lines that lie in π (now instead of one linearequation in a, c, e, we get two). Hence, π contains at most k2/k = 2 lines of L ,which is always smaller than s0.

We have thus shown that the bound in Theorem 1.3 is (almost) tight in the worstcase. The bound will be tight when |P| ≤ |L|6/7, which occurs when k ≤ 3/2, as aneasy calculation shows.

Remark As the analysis shows, the various constructions impose certain constraintson the values of q and s, and are therefore not as general (in terms of these parameters)as onemight hope. It would be interesting to extend the constructions so that they applyto more general values of q and s.

5 Conclusion

The results of this paper (almost) settle the problem of point-line incidences in fourdimensions, but they raise several interesting and challenging open problems. Amongthem are:

(a) Get rid of the factor 2c√logm in the bound. We have achieved this improvement

when m is not too close to n4/3, so to speak, allowing us to use the weak but non-inductive bounds and complete the analysis in one step.We believe that the rangesofm where this can be done can be enlarged, e.g. by improving theweak bounds. Aconcrete step in this direction would be to improve the term O(nD4) in the secondbound in Proposition 3.3, which, as already remarked, appears to be too weak. Itwould also be interesting to improve the bound using the strategy in [36,38], whichgenerates a sequence of ranges of m, converging to m = (n4/3), where in eachrange the improved bound (6) holds, with a different constant of proportionalityA. (For readers familiar with the approaches in [35,38], we note that the reasonthis technique does not appear to apply here is the multitude of subproblems, eachwith its ownmi , ni . The induction in [35,38] generates subproblems in which therelation between m and n falls into a range already handled. Here though we donot know how to enforce this property, as we have little control over the values ofm, n in the resulting subproblems.

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(b) Extend (and sharpen) the bound of Corollary 1.4 for any value of k. In particular,is it true that the number of intersection points of the lines (this is the case k = 2;the intersection points are also known as 2-rich points) is O(n4/3 + nq1/2 + ns)?We conjecture that this is indeed the case. (In this conjecture we assume that wehave already managed to get rid of the factor 2c

√logm , as in (a) above.) A deeper

question, extending a similar open problem in three dimensions that has beenposed by Guth and others (see, e.g., Katz’s expository note [23]), is whether theabove conjectured bound can be improved when q = o(n2/3) and s = o(n1/3),that is, when the second and third terms in the conjectured bound become muchsmaller than the term n4/3. We also note that if we could establish such a boundfor the number of k-rich points, for any constant k (when q and s are not too large),then the case of large m (that is, m = (n4/3)) would become vacuous, as onlyO(n4/3) points could be incident to more than k lines.

(c) Extend the study to five and higher dimensions. In a preliminary ongoing study,joint with Adam Sheffer, we can do it using a constant-degree partitioningpolynomial, with the disadvantages discussed above (slightly weaker bounds, sig-nificantly more restrictive assumptions, and inferior “lower-dimensional” terms).The leading terms in the resulting bounds, for points and curves in R

d , areO(m2/(d+1)+εnd/(d+1) + m1+ε), for any ε > 0. See also Dvir and Gopi [7]and Hablicsek and Scherr [16] for recent related studies.Obtaining sharper results in such general settings, like the ones obtained in thispaper, is quite challenging algebraically, although some of the tools developed inthis work seem promising for higher dimensions too.

(d) If we are given in advance that the points and lines lie in some algebraic surface ofa given degree D > 2, can we improve the bound and/or simplify the analysis? Inour companion work [37] we achieve these goals for the three-dimensional case,improving the bound of Guth and Katz [15] in such special cases.

(e) Elaborating on item (a) above, we note that the “culprit” Proposition 3.3, whichproduces the weak bounds that force us to go into the induction, is only used in thecase where Z( f, g) is two-dimensional, and the difficulty there lies in boundingthe number of incidences within a two-dimensional ruled surface (be it eitherone irreducible ruled surface of large degree, or the union of many irreducibleruled surfaces of small degree). The analysis of the three-dimensional analogoussituation (addressed in Guth and Katz [15]), cannot be applied here, since thedegree of the underlying surface in four dimensions is O(D2) instead of D in [15].In a recent study of Szermerédi-Trotter type theorems in three dimensions [24],Kollár uses the arithmetic genus of curves to prove effective bounds on the numberof point-line incidences in three dimensions. In four dimensions, the situation ismore involved, but we hope that the arithmetic genus of the surface Z( f, g) mayyield effective bounds for the number of incidences within this surface.

Acknowledgements Work on this paper by Noam Solomon and Micha Sharir was supported by Grant892/13 from the Israel Science Foundation. Work by Micha Sharir was also supported by Grant 2012/229from theU.S.–IsraelBinational ScienceFoundation, by the Israeli Centers ofResearchExcellence (I-CORE)program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel AvivUniversity. We would like to thank several people whose advice, comments and guidance have helped usa lot in our work on the paper. They are János Kollár, Martin Sombra, Aise J. de Jong, and Saugata Basu.

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We also thank the anonymous referees for their helpful comments on the paper. In addition, as noted,part of the work on the paper was carried out during the special semester on Algebraic Techniques forCombinatorial and Computational Geometry, held at the Institute for Pure and Applied Mathematics atUCLA, in the Spring of 2014. We are grateful for the pleasant working environment provided by IPAM,and for the helpful interaction with additional colleagues, including Larry Guth, Nets Hawk Katz, TerryTao, Jordan Ellenberg, and many others.

Appendix: Severi’s Theorem

In this appendix we sketch a proof of Severi’s theorem (Theorem 3.11).First, recall from Sect. 2.1 that a real (resp., complex) surface X is ruled by real

(resp., complex) lines if every point p ∈ X in a Zariski open dense set is incident toa real (resp., complex) line that is fully contained in X . This definition has been usedin several recent works (see, e.g., [15]); this is a slightly weaker condition than theclassical condition that requires that every point of X be incident to a line containedin X . Nevertheless, as we show next, the two are equivalent.

Lemma 6.1 Let f ∈ R[x, y, z] (resp., f ∈ R[x, y, z, w]) be an irreducible polyno-mial such that there exists a Zariski open dense set U ⊆ Z( f ), so that each point inthe set is incident to a line, fully contained in Z( f ). Then FL f (resp., FL4f ) vanishesidentically on Z( f ), and Z( f ) is ruled by lines.

Proof By assumption and definition, FL f (resp., FL4f ) vanishes on U . If it vanisheson Z( f ), Theorem 2.11 implies that Z( f ) is ruled. Otherwise, Z( f,FL f ) (resp.,Z( f,FL4f )) is properly contained in Z( f ) and contains U . Since Z( f ) is irreducible,this latter variety must be of dimension at most 1 (resp., 2). On the other hand,Z( f,FL f ) (resp., Z( f,FL4f )) is Zariski closed set (by definition of the Zariski topol-ogy) and therefore contains its Zariski closure. As U is Zariski dense, its Zariskiclosure is Z( f ). Remark In Sharir and Solomon [37], we have proved the same statement withoutusing the Flecnode polynomial.

This phenomenon generalizes to k-flats instead of lines (and the proof translatesverbatim).

Lemma 6.2 Let V be an irreducible variety for which there exists a Zariski opensubset U ⊆ V with the property that each point p ∈ U is incident to a k-flat that isfully contained in V . Then this property holds for every point of V .

We now proceed to sketch a proof of Severi’s theorem. For convenience, we repeatits statement.

Theorem 3.11 (Severi’s Theorem [34]) Let X ⊂ Pd(C) be a k-dimensional irre-

ducible variety, and let 0 be a component of maximal dimension of F(X), such thatthe lines of 0 cover X. Then the following holds.

1. If dim(0) = 2k − 2, then X is a copy of Pk(C) (that is, a complex projective

k-flat).

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2. If dim(0) = 2k − 3, then either X is a quadric, or X is ruled by copies ofPk−1(C), i.e., every point p ∈ X is incident to a copy of P

k−1(C) that is fullycontained in X.

We sketch a proof in the case k = 3, d = 4, under the simplifying assumption thatfor any non-singular x ∈ X, 0,x is infinite; this assumption holds in our applicationof the theorem (by the informal dimensionality argument mentioned in the paper, itholds “on average” in general for these parameters). Our proof is based on a sketchprovided by A. J. de Jong, via private communication, and we are very grateful for hisassistance.

Sketch of Proof For x ∈ X , we recall that 0,x denotes the cone of lines (i.e., unionof lines) of 0,x The proof consists of the following steps.

(1) Assume first that dim(0) = 2k − 2 = 4. Then there exists some non-singularpoint x0 ∈ X with dim(0,x0) = 2. Indeed, if, for all non-singular points x ∈X, dim(0,x ) ≤ 1, then dim(0) < 4 (see the analysis in Theorem 3.9, and thepreceding analysis), contradicting the assumption in this case. By an argumentthat has already been sketched earlier, this implies that dim(0,x0) = 3, i.e., thecone of lines in 0,x0 through x0 is three-dimensional, and therefore X = 0,x0 .As x0 is non-singular, it follows that X must be a hyperplane, as claimed.

(2) Consider next the case where dim(0) = 2k − 3 = 3, and for any non-singularpoint x ∈ X, 0,x is 1-dimensional (as just argued, if 0,x is two-dimensionalfor some non-singular x ∈ X , then X is a hyperplane). In other words, 0,x ,parameterized by the direction of its lines, is a curve in PTx X ∼= P

2(C); put exfor its degree. If ex = 1, then 0,x contains a 2-flat.

We next define a “plane-flecnode polynomial system” associated with X , thatexpresses, for a point x ∈ X , the existence of a 2-flat H , such that H osculatesto X to order 3 at x . Since X is a hypersurface, we can write X = Z( f ), for a suit-able 4-variate polynomial f (see Sect. 2), and assume that f is irreducible (as X isirreducible).

We represent a 2-flat through the origin in C4 (ignoring the lower-dimensional

family of 2-flats that cannot be represented in this manner) as

Hv0,v1,v2,v3 := (x, y, z, w) | z = v0x + v1y, w = v2x + v3y, (46)

for v0, v1, v2, v3 ∈ C. The 2-flat Hv0,v1,v2,v3 is said to osculate to X = Z( f ) to orderk at p, if the Taylor expansion of f at p along H satisfies

f (p + (x, y, v0x + v1y, v2x + v3y)) = O(xk+1 + yk+1). (47)

This translates into a system of homogeneous polynomial equations in v0, v1, v2, v3,involving the partial derivatives of f up to order k. Specializing to the casek = 3, the plane-flecnode polynomial system, PFL f , associated with f , is obtainedby eliminating v0, v1, v2, v3 from these equations (for osculation up to order 3). Thisis the multipolynomial resultant system of the polynomials defining these equations

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up to order 3, with respect to v0, v1, v2, v3 (see Van der Waerden [45, Chap. XI] fordetails).

Another theorem of Landsberg [25, Thm. 1] states that, if, for every g ∈ PFL f , gvanishes identically on X , then X is ruled by 2-flats, which finishes the proof in thiscase.

Therefore, wemay assume that X∩Z(PFL f ) is a Zariski closed proper subset of X .By definition of PFL f , it follows that for every non-singular point x ∈ X \ Z(PFL f )

(namely, outside the Zariski closed set Z(PFL f )), we have ex > 1. Indeed, if ex = 1,then, as observed above, there is a 2-flat incident to x , and fully contained in X ,implying that for every g ∈ PFL f , g(x) = 0, contradicting the assumption thatx ∈ X \ Z(PFL f ).

For a generic hyperplane H in P4(C), which is not contained in Z(PFL f ), put

SH := X ∩ H . As observed above, X ∩ Z(PFL f ) is properly contained in X , whichin turn implies that, for a generic hyperplane H in P

4(C), SH is not fully containedin Z(PFL f ). Indeed, let g be a polynomial in PFL f that does not vanish identicallyon X . Then X ∩ Z(g) = Z( f, g) is strictly contained in X = Z( f ), and since Z( f )is irreducible, it follows that Z( f, g) is two-dimensional. Therefore, for a generichyperplane H, X∩Z(PFL f )∩H is contained in the one-dimensional variety Z( f, g)∩H , and thus cannot contain the two-dimensional variety SH .

Let x ∈ X be a non-singular point, and let H be a hyperplane in P4(C), which is

incident to X and not contained in Z(PFL f ). We claim that for a generic H , thereare ex distinct lines that are incident to x and fully contained in SH . Indeed, theintersection of the hyperplane H with Tx X is a 2-flat in Tx X containing x . Takingits projectivization (where the point x is regarded as 0), namely, PTx X ∼= P

2, the(generic) 2-flat Tx X ∩ H becomes a (generic) line. The degree of 0,x ⊂ PTx X is ex .Therefore, the intersection ofx with a line inPTx X ∼= P

2 consists of ex points, whichare distinct since the line is generic. Therefore, its intersection with0,x consists of exdistinct points. These ex distinct (projective) points represent ex distinct lines, incidentto x and fully contained in X ∩ H = SH , as claimed.

We say that a pair (x, H), where H is a hyperplane inP4(C) and x ∈ SH , isadequate

if there are ex distinct lines incident to x that are fully contained in SH . Since a genericpoint x is non-singular, the previous paragraph implies that a generic pair (x, H) isadequate. Therefore, by changing the order of quantifiers, fixing a generic hyperplaneH , a generic point x ∈ SH is such that the pair (x, H) is adequate.

By Bertini’s Theorem (see, e.g., Harris [17, Thm. 17.16]), the irreducibility of Ximplies that for a generic hyperplane H , the surface SH is an irreducible surface inH ∼= P

3(C). For a generic point x ∈ SH , that is, outside an algebraic curve CH in SH ,the pair (x, H) is adequate. Therefore, there are ex distinct lines that are incident to xand fully contained in SH , which, by Lemma 6.1, implies that SH is a ruled surface.Moreover, for any x ∈ SH \ Z(PFL f ), we have ex > 1. As observed above, PFL f

does not vanish identically on SH , implying that Z(PFL f ) ∩ SH is a Zariski closedproper subset of SH , i.e., an algebraic curve contained in SH . Adding this curve toCH , it follows that outside this algebraic curve, each point of SH is incident to at leasttwo lines fully contained in SH . By Sharir and Solomon [37, Lem. 9], this impliesthat SH is either a 2-flat or a regulus. If X is of degree greater than two, then, for ageneric hyperplane H, SH is a (two-dimensional) surface of degree greater than two.

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Therefore, X must be of degree at most two, namely, X is either a hyperplane or aquadric. If X is a hyperplane, then 0 is four-dimensional, contrary to the presentassumption, so finally, we deduce that X is a quadric, and the proof is complete.

References

1. Basu, S., Sombra, M.: Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions. Discrete Comput. Geom. 55(1), 158–184 (2016). Also inarXiv:1406.2144

2. Beauville, A.: Complex Algebraic Surfaces, vol. 34. Cambridge University Press, Cambridge (1996)3. Clarkson, K., Edelsbrunner, H., Guibas, L., Sharir, M., Welzl, E.: Combinatorial complexity bounds

for arrangements of curves and spheres. Discrete Comput. Geom. 5, 99–160 (1990)4. Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Heidelberg (2005)5. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational

Algebraic Geometry and Commutative Algebra. Springer, Heidelberg (2007)6. Dvir, Z.: On the size of Kakeya sets in finite fields. J. Am. Math. Soc. 22, 1093–1097 (2009)7. Dvir, Z., Gopi, S.: On the number of rich lines in truly high dimensional sets. In: Proceedings of 30th

Annual ACM Symposium on Computational Geometry, pp. 584–598 (2015)8. Edge, W.L.: The Theory of Ruled Surfaces. Cambridge University Press, Cambridge (2011)9. Elekes, G.: Sums versus products in number theory, algebra and Erdos geometry—a survey. Paul

Erdos and His Mathematics II. Bolyai Mathematical Society Studies, vol. 11, pp. 241–290. BolyaiMathematical Society, Budapest (2002)

10. Elekes, G., Kaplan, H., Sharir, M.: On lines, joints, and incidences in three dimensions. J. Comb.Theory, Ser. A 118, 962–977 (2011). Also in arXiv:0905.1583

11. Fuchs,D.,Tabachnikov, S.:MathematicalOmnibus:ThirtyLectures onClassicMathematics.AmericanMathematical Society, Providence, RI (2007)

12. Fulton, W.: Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from theCBMS Regional Conference Held at George Mason University, June 27–July 1, 1983, Vol. 54. AMSBookstore (1984)

13. Guth, L.: Distinct distance estimates and low-degree polynomial partitioning. Discrete Comput. Geom.48, 1–17 (2014). Also in arXiv:1404.2321

14. Guth, L., Katz, N.H.: Algebraic methods in discrete analogs of the Kakeya problem. Adv. Math. 225,2828–2839 (2010). Also in arXiv:0812.1043v1

15. Guth, L., Katz, N.H.: On the Erdos distinct distances problem in the plane. Ann. Math. 181, 155–190(2015). Also in arXiv:1011.4105

16. Hablicsek, M., Scherr, Z.: On the number of rich lines in high dimensional real vector spaces. DiscreteComput. Geom. 55(1), 1–8 (2014). Also in arXiv:1412.7025

17. Harris, J.: Algebraic Geometry: A First Course, vol. 133. Springer, New York (1992)18. Hartshorne, R.: Algebraic Geometry. Springer, New York (1983)19. Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and

Exterior Differential Systems, Graduate Studies in Mathematics, vol. 61. American MathematicalSociety, Providence, RI (2003)

20. Kaplan, H., Sharir, M., Shustin, E.: On lines and joints. Discrete Comput. Geom. 44, 838–843 (2010)21. Kaplan, H., Matoušek, J., Safernová, Z., Sharir, M.: Unit distances in three dimensions. Comb. Prob.

Comput. 21, 597–610 (2012). Also in arXiv:1107.107722. Kaplan, H., Matoušek, J., Sharir, M.: Simple proofs of classical theorems in discrete geometry via the

Guth–Katz polynomial partitioning technique. Discrete Comput. Geom. 48, 499–517 (2012). Also inarXiv:1102.5391

23. Katz, N.H.: The flecnode polynomial: a central object in incidence geometry, in arXiv:1404.341224. Kollár, J.: Szemerédi–Trotter-type theorems in dimension 3. Adv. Math. 271, 30–61 (2015). Also in

arXiv:1405.224325. Landsberg, J.M.: Is a linear space contained in a submanifold? On the number of derivatives needed

to tell. J. Reine Angew. Math. 508, 53–60 (1999)26. Mezzetti, E., Portelli, D.: On Threefolds Covered by Lines. Abhandlungen aus dem Mathematischen

Seminar der Universität Hamburg, vol. 70, no. 1. Springer, Heidelberg (2000)

123

756 Discrete Comput Geom (2017) 57:702–756

27. Pach, J., Sharir, M.: Geometric incidences. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs.Contemporary Mathematics, vol. 342, pp. 185–223. American Mathematical Society, Providence, RI(2004)

28. Pressley,A.: ElementaryDifferentialGeometry. SpringerUndergraduateMathematics Series. Springer,London (2001)

29. Quilodrán, R.: The joints problem in Rn . SIAM J. Discrete Math. 23(4), 2211–2213 (2010)30. Richelson, S.: ClassifyingVarieties withMany Lines, a senior thesis, HarvardUniversity (2008). http://

www.math.harvard.edu/theses/senior/richelson/richelson.pdf31. Rogora, E.: Varieties with many lines. Manuscr. Math. 82(1), 207–226 (1994)32. Salmon, G.: A Treatise on the Analytic Geometry of Three Dimensions, vol. 2, 5th edn. Hodges, Figgis

and co. Ltd., Dublin (1915)33. Segre, B.: Sulle Vn contenenti più di ∞n−k Sk , Nota I e II. Rend. Accad. Naz. Lincei 5(193–197),

275–280 (1948)34. Severi, F.: Intorno ai punti doppi impropri etc. Rend. Cir. Math. Palermo 15(10), 33–51 (1901)35. Sharir, M., Solomon, N.: Incidences between points and lines in R

4. In: Proceedings of 30th AnnualSymposium on Computational Geometry, pp. 189–197 (2014)

36. Sharir, M., Solomon, N.: Incidences between points and lines in three dimensions, Intuitive Geometry.Also In: Pach, J. (ed.) Proceedings of 31st Annual Symposium on Computational Geometry. Also inarXiv:1501.02544 (2015)

37. Sharir, M., Solomon, N.: Incidences between points and lines on a two-dimensional variety, inarXiv:1502.01670

38. Sharir, M., Sheffer, A., Zahl, J.: Improved bounds for incidences between points and circles. Comb.Prob. Comput. 24(03), 490–520 (2015). Also in arXiv:1208.0053

39. Solomon, N., Zhang, R., Solomon,: Highly incidental patterns on a quadratic hypersurface in R4, in

arXiv:1601.0181740. Solymosi, J., Tao, T.: An incidence theorem in higher dimensions. Discrete Comput. Geom. 48, 255–

280 (2012)41. Székely, L.: Crossing numbers and hard Erdos problems in discrete geometry. Combt. Prob. Comput.

6, 353–358 (1997)42. Szemerédi, E., Trotter, W.T.: Extremal problems in discrete geometry. Combinatorica 3, 381–392

(1983)43. Tao, T.: From rotating needles to stability of waves: emerging connections between combinatorics,

analysis, and PDE. Notices Am. Math. Soc. 48(3), 294–303 (2001)44. Tao, T.: The Cayley–Salmon theorem via classical differential. https://terrytao.wordpress.com/2014/

03/28/the-cayley-salmon-theorem-via-classical-differential-geometry/45. van der Waerden, B.L., Artin, E., Noether, E.: Modern Algebra, vol. 2. Springer, Heidelberg (1966)46. Warren, H.E.: Lower bound for approximation by nonlinear manifolds. Trans. Am. Math. Soc. 133,

167–178 (1968)47. Zahl, J.: An improved bound on the number of point-surface incidences in three dimensions. Contrib.

Discrete Math. 8(1), 100–121 (2013). Also in arXiv:1104.498748. Zahl, J.: A Szemerédi–Trotter type theorem in R

4. Discrete Comput. Geom. 54(3), 513–572 (2015).Also in arXiv:1203.4600

123

3 Highly incidental patterns on a quadratichypersurface in R4

97

Discrete Mathematics 340 (2017) 585–590

Contents lists available at ScienceDirect

Discrete Mathematicsjournal homepage: www.elsevier.com/locate/disc

Note

Highly incidental patterns on a quadratic hypersurface in R4

Noam Solomon a,*, Ruixiang Zhangb

a School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israelb Department of Mathematics, Princeton University, Princeton, NJ 08540, United States

a r t i c l e i n f o

Article history:Received 14 January 2016Received in revised form 29 November 2016Accepted 8 December 2016Available online 3 January 2017

Keywords:Combinatorial geometryIncidences

a b s t r a c t

In Sharir and Solomon (2015), Sharir and Solomon showed that the number of incidencesbetween m distinct points and n distinct lines in R4 is

O∗(m2/5n4/5

+ m1/2n1/2q1/4 + m2/3n1/3s1/3 + m + n), (1)

provided that no 2-flat contains more than s lines, and no hyperplane or quadric containsmore than q lines, where the O∗ hides a multiplicative factor of 2c

√logm for some absolute

constant c .In this paper we prove that, for integers m, n satisfying n9/8 < m < n3/2, there exist m

points and n lines on the quadratic hypersurface in R4

(x1, x2, x3, x4) ∈ R4| x1 = x22 + x23 − x24,

such that (i) at most s = O(1) lines lie on any 2-flat, (ii) at most q = O(n/m1/3) lineslie on any hyperplane, and (iii) the number of incidences between the points and thelines is Θ(m2/3n1/2), which is asymptotically larger than the upper bound in (1), whenn9/8 < m < n3/2. This shows that the assumption that no quadric contains more thanq lines (in the abovementioned theorem of Sharir and Solomon (2015)) is necessary in thisregime ofm and n.

By a suitable projection from this quadratic hypersurface onto R3, we obtain m pointsand n lines in R3, with at most s = O(1) lines on a common plane, such that the number ofincidences between the m points and the n lines is Θ(m2/3n1/2). It remains an interestingquestion to determine if this bound is also tight in general.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

Let P be a set ofm distinct points inR2 and let L be a set of n distinct lines inR2. Let I(P, L) denote the number of incidencesbetween the points of P and the lines of L; that is, the number of pairs (p, ℓ), such that p ∈ P , ℓ ∈ L and p ∈ ℓ. The classicalSzemerédi–Trotter theorem [12] yields the worst-case tight bound

I(P, L) = O(m2/3n2/3

+ m + n). (2)

Work on this paper by Noam Solomonwas supported by Grant 892/13 from the Israel Science Foundation. Ruixiang Zhangwas supported by PrincetonUniversity. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supportedby the National Science Foundation.

* Corresponding author.E-mail addresses: noam.solom@gmail.com (N. Solomon), ruixiang@math.princeton.edu (R. Zhang).

http://dx.doi.org/10.1016/j.disc.2016.12.0040012-365X/© 2016 Elsevier B.V. All rights reserved.

586 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) 585–590

This bound clearly also holds in three, four, or any higher dimensions which can be easily proved by projecting the givenlines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by placing all the pointsand lines in a common plane, in a configuration that yields the planar lower bound.

In the groundbreaking paper of Guth and Katz [4], an improved bound has been derived for I(P, L), for a set P ofm pointsand a set L of n lines in R3, provided that not too many lines of L lie in a common plane.1 Specifically, they showed:

Theorem 1.1 (Guth and Katz [4]). Let P be a set of m distinct points and L a set of n distinct lines in R3, and let s ≤ n be aparameter, such that no plane contains more than s lines of L. Then

I(P, L) = O(m1/2n3/4

+ m2/3n1/3s1/3 + m + n).

Remark. When s = Θ(√n), this bound is known to be tight, by a generalization to three dimensions of Elekes’ planar

construction of points and lines on an integer grid (see Guth and Katz [4] for the details). For smaller values of s, it is an openproblem to give lower bounds or improve the upper bound, and the case s = O(1) is of particular interest. In Theorem 1.5we give an improved upper bound, and it remains a question (see Question 4.1) whether it is tight.

In a recent paper of Sharir and Solomon [7], the following analogous and sharper result in four dimensions wasestablished.

Theorem 1.2. Let P be a set of m distinct points and L a set of n distinct lines in R4, and let q, s ≤ n be parameters, such that (i)each hyperplane or quadric contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Then

I(P, L) ≤ 2c√logm (

m2/5n4/5+ m

)+ A

(m1/2n1/2q1/4 + m2/3n1/3s1/3 + n

), (3)

where A and c are suitable absolute constants. When m ≤ n6/7 or m ≥ n5/3, there is the sharper bound

I(P, L) ≤ A(m2/5n4/5

+ m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + n). (4)

In general, except for the factor 2c√logm, the bound is tight in the worst case, for any values of m, n, and for corresponding suitable

ranges of q and s.

The termm2/3n1/3s1/3 comes from the planar Szemerédi–Trotter bound (2), and is unavoidable, as it can be attained if wedensely pack points and lines into 2-flats, in patterns that realize the bound in (2).

Likewise, the termm1/2n1/2q1/4 comes from the bound of Guth and Katz [4] in three dimensions (as in Theorem 1.1), andis again unavoidable, as it can be attained if we densely ‘‘pack’’ points and lines into hyperplanes, in patterns that realize thebound in three dimensions.

In this paper we show that the condition in assumption (i) of Theorem 1.2 that quadrics also do not contain too manylines, cannot be dropped, by proving the following theorem.

Theorem 1.3. For each positive integer k and each α > 0, there exists m = Θ(k3+3α) points and n = Θ(k2+4α) lines on thequadratic hypersurface

S := (x1, x2, x3, x4) ∈ R4| x1 = x22 + x23 − x24

inR4, such that there are atmost O(1) lines lying on any 2-flat andO(k1+3α) lines lying on any hyperplane, and I(P, L) = Θ(k3+4α).

Given integers m and n, there are k, α such that m = Θ(k3+3α) points and n = Θ(k2+4α). Substituting these values inTheorem 1.3, we obtain the following corollary.

Corollary 1.4. For integers m, n, there is a configuration of m points and n lines in R4, such that all the points (resp., lines) arecontained (resp., fully contained) in S, and (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in a commonhyperplane is O(n/m1/3), and (iii) the number of incidences between the points and lines isΩ(m2/3n1/2

+ m + n).

Remarks. (1) For integers m, n, satisfying n9/8 < m < n3/2, the number incidences Ω(m2/3n1/2) in Corollary 1.4 isasymptotically larger than the bound of Eq. (3) for the number of incidencesO(m2/5n4/5

+m1/2n1/2q1/4+m2/3n1/3+m+n) =

O(m2/5n4/5+m5/12n3/4

+m+n) (as q = O(n/m1/3)). This implies that the condition in assumption (i) of Theorem 1.2 cannotbe dropped, in this regime ofm and n.(2)We note that the number of 2-rich points determined by n lines in R4 is O(n3/2), provided that at most O(

√n) of the lines

lie on a common plane or regulus.2 To see this, project the lines onto some (generic) hyperplane H , such that no two lines areprojected onto the same line, and similarly, no two 2-rich points are projected onto the same 2-rich point, and such that atmost O(

√n) lines lie on a common plane or regulus. Then, the number of 2-rich points in the configuration of n lines in R4 is

1 The additional requirement in [4], that no regulus contains too many lines, is not needed for the bound given below.2 A regulus is a quadratic surface that is doubly ruled by lines. For more details about reguli, see e.g., Sharir and Solomon [6].

N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) 585–590 587

equal to the number of 2-rich points in the configuration of the projected lines onto H . By Guth and Katz [4], the number of2-rich points determined by the projected lines is O(n3/2), and therefore the same holds for the number of 2-rich points inthe original configuration of lines inR4. We also notice that in a configuration ofm points and n lines inR4, the 1-rich points(i.e., points that are incident to exactly one line) contribute at mostm incidences. Therefore, in Corollary 1.4, as s = O(1), theassumption thatm ≤ n3/2 causes no loss of generality.Proof techniques. It is a common practice to take geometric objects to be integer points on certain hypersurfaces (especiallyquadratic ones) and varieties passing through a lot of such points, in order to obtain lower bounds for their incidences. Forsome most recent applications of this method, see [9,14,15]. In this paper we obtain our incidence lower incidence boundby taking integer points and ‘‘low height’’ lines on the above hypersurface S.Projection to R3. As remarked above, Guth and Katz [4] proved that the number of incidences between m points and nlines in R3 is I(P, L) = O

(m1/2n3/4

+ m2/3n1/3s1/3 + m + n), provided that no plane contains more than s lines of L. When

s = Θ(√n), this bound is tight, by a generalization to three dimensions of Elekes’ construction of points and lines on an

integer grid in the plane (see Guth and Katz [4] for the details). For smaller values of s, it is an open problem to give lowerbounds or improve the upper bound, where the case s = O(1), is of particular interest.

By choosing a generic projection from R4 to R3, we show that Corollary 1.4 directly implies the following Theorem.

Theorem 1.5. For integers m, n, there is a configuration of m points and n lines in R3, such that (i) the number of lines in anycommon plane is s = O(1), and (ii) the number of incidences between the points and lines isΩ(m2/3n1/2

+ m + n).

Remark. When n3/4≪ m ≪ n3/2, the term m2/3n1/2 dominates over m and n, showing that in this regime of m and n, the

construction in Theorem 1.5, of m points and n lines with O(1) lines in a common plane, yields a super-linear number ofincidences. As observed above, the bound of Guth and Katz [4] implies that the number of 2-rich points determined by then lines is O(n3/2), so the assumption that m ≤ n3/2 causes no loss of generality.

Background. Incidence problems have been a major topic in combinatorial and computational geometry for the past thirtyyears, starting with the Szemerédi–Trotter bound [12] back in 1983. Several techniques, interesting in their own right, havebeen developed, or adapted, for the analysis of incidences, including the crossing-lemma technique of Székely [11], and theuse of cuttings as a divide-and-conquer mechanism (e.g., see [2]). Connections with range searching and related problemsin computational geometry have also been noted, and studies of the Kakeya problem (see, e.g., [13]) indicate the connectionbetween this problem and incidence problems. See Pach and Sharir [5] for a comprehensive survey of the topic.

The simplest instances of incidence problems involve points and lines. Szemerédi and Trotter solved completely thisspecial case in the plane [12]. Guth and Katz’s second paper [4] provides a worst-case tight bound in three dimensions,under the assumption that no plane contains too many lines; see Theorem 1.1. Under this assumption, the bound in threedimensions is significantly smaller than the planar bound (unless one of m, n is significantly smaller than the other), andthe intuition is that this phenomenon should also show up as we move to higher dimensions. The first attempt in higherdimensions was made by Sharir and Solomon in [8]. In a recent work, Sharir and Solomon [7] gave a tight bound in four-dimensions provided that the number of lines fully contained in a common hyperplane or quadric is bounded by a parameterq, and the number of lines fully contained in a common 2-flat is bounded by a parameter s. Whereas the condition thatno common hyperplane contains more than a bounded number of lines was known to be necessary, it remained an openquestion whether the condition that the number of lines in a common quadric is bounded is necessary. In this paper, weshow that when n9/8 < m < n3/2, this condition is indeed necessary, by describing an explicit quadratic hypersurface in R4

containing more incidences than the bound prescribed by the main theorem of [7]. This is the content of Theorem 1.3, andCorollary 1.4.

We remark that in [14], another example of points on a quadratic hypersurface in F4 with highly incidental pattern wasnoticed. There F is a finite field. Our current quadratic hypersurface and our counting techniques in R4 are slightly different.The reader may find it interesting to compare the results here to the results in [14].

Another interesting remark is that in three dimensions, there are certain quadratic surfaces, called reguli, such that ifone allows too many lines to lie on such a regulus, the number of 2-rich points determined by them can be larger than theGuth–Katz bound [4] ofO(n3/2). The quadratic hypersurface inR4 presented in this paper can be thought of as a higher degreeanalogs of regulus. However, If one only cares about incidences between points and lines (instead of the number of 2-richpoints determined by the lines), the existence of many lines on a regulus (or any quadratic surface in R3) do not yield morethan a linear number of incidences.

2. Proof of Theorem 1.3

Proof. We start by recalling the quadric

S = (x1, x2, x3, x4) ∈ R4| x1 = x22 + x23 − x24, (5)

on which the construction takes place, and define the set of points by

P = (x1, x2, x3, x4) ∈ S | xi ∈ Z, i = 1, . . . , 4, |x1| ≤ 200k2+2α, |x2|, |x3|, |x4| ≤ 100k1+α, (6)

588 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) 585–590

and the set of lines

L =

x + tv | t ∈ R ⊆ S | x = (x1, . . . , x4), v = (v1, . . . , v4), xi, vi ∈ Z, i = 1, . . . , 4,

|x1| ≤ k2+2α, |x2|, |x3|, |x4| ≤ k1+α,k1+2α

4≤ |v1| ≤ 8k1+2α, |v2|, |v3| ≤ kα, v24 = v22 + v23,

v1 = 2x2v2 + 2x3v3 − 2x4v4,

gcd(v2, v3, v4) = 1, and |v4| ≥kα

2

,

for any positive integer k and any α > 0.Since a point on S is uniquely determined by its last three coordinates, we have

|P| = |(x2, x3, x4) ∈ Z3| |x2|, |x3|, |x4| ≤ 100k1+α| = Θ(k3+3α).

The analysis of (an asymptotically tight bound on) the number of lines of L is a bit more involved. A line x + tv | t ∈ R

in L (assuming x ∈ S, |x1| ≤ k2+2α, |x2|, |x3|, |x4| ≤ k1+α) is fully contained in S if and only if

v1 = 2x2v2 + 2x3v3 − 2x4v4 and v24 = v22 + v23 .

It follows by Benito and Varona [1] that the number of primitive integer triples (v2, v3, v4) (i.e., without a common divisor)satisfying v24 = v22 + v23, |v2|, |v3| ≤ kα , and |v4| ≥

kα2 isΘ(kα). For each such (v2, v3, v4), we claim that there areΩ(k3+3α)

(and trivially also O(|P|) = O(k3+3α)) points x ∈ P , such that v1 = 2x2v2 + 2x3v3 − 2x4v4 satisfying k1+2α

4 ≤ |v1| ≤ 8k1+2α .Indeed, note that |v2|, |v3| ≤ |v4|. Choosing |x2|, |x3| ≤

|x4|

4 ,k1+α2 ≤ |x4| ≤ k1+α (there are at least k3+3α

32 choices of suchtriples (x2, x3, x4)) implies that

|2x2v2 + 2x3v3| ≤ 2|x2||v2| + 2|x3||v3| ≤ 2|x4|4

(|v2| + |v3|) ≤ |x4||v4|.

Here |v1| ≥ |x4||v4| ≥k1+2α

4 . The inequality |v1| ≤ 8k1+2α is immediate.Moreover, each line ℓ satisfying the above conditions is incident to O(k) different points of P (and can thus be expressed

in O(k) different ways as x + tv | t ∈ R ⊆ S, for |x1| ≤ k2+2α, |x2|, |x3|, |x4| ≤ k1+α). Indeed, parameterize ℓ asx + tv | t ∈ R ⊂ S, where x, v satisfy

|v1| ≥k1+2α

4, |x1| ≤ k2+2α,

and v = (v1, v2, v3, v4) is primitive (i.e., its coordinates do not have a common factor). Notice that if |t| > 8k, then the firstcoordinate of x + tv has absolute value greater than k2+2α , and that if t ∈ Z, then x + tv ∈ Z4 (since v is primitive andx ∈ Z4). In either case, x + tv ∈ P . This implies that

ℓ ∩ P ⊆ x + tv | t ∈ Z, |t| ≤ 8k,

and thus |ℓ ∩ P| ≤ 16k = O(k) as claimed. Therefore, the total number of lines isΩ( k3+3αkα

k ) = Ω(k2+4α).It is easy to see that each line in L is incident toΩ(k) points in P . It follows that |L| = O(k2+4α). Hence |L| = Θ(k2+4α).Since each line hasΘ(k) integer points in P on it, we have

I(P, L) = Θ(k3+4α).

We now bound the number of lines fully contained in any 2-flat, and then bound the number of lines on any hyperplane.The bounds will be uniform (i.e., independent of the specific 2-flat or hyperplane).

Let π denote any 2-flat, and we analyze the number of lines that are fully contained in π ∩ S. We claim that S containsno planes, so π ⊂ S. Assume the contrary, then we parameterize

π = (u1s + r1t + w1, u2s + r2t + w2, u3s + r3t + w3, u4s + r4t + w4) | s, t ∈ R,

for constants ui, ri, wi ∈ R, i = 1, 2, 3, 4 where (u1, u2, u3, u4) and (r1, r2, r3, r4) are both nonzero and not proportional toeach other. Comparing the coefficients of quadratic terms in the identity

u1s + r1t + w1 ≡ (u2s + r2t + w2)2 + (u3s + r3t + w3)2 − (u4s + r4t + w4)2,

we deduce (u2, u3, u4) and (r2, r3, r4) are proportional to each other. Hence we may assume u2 = u3 = u4 = 0. But thisforces u1 = 0, a contradiction. Therefore π is not contained in S. Thus the intersection π ∩ S is a curve of degree at most two,so there are at most two lines fully contained in π ∩ S.

Next, we take any hyperplane H , and analyze the number of lines fully contained in S∩H . The surface S∩H is a quadratic2-surface contained in H . We will use the classification of (real) quadratic surfaces in R3 (see, e.g., Sylvester’s original paper[10]), and distinguish between two cases.

N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) 585–590 589

If the equation of H can be expressed as x1 = ϕ(x2, x3, x4), where ϕ is a linear form, then each point x ∈ H ∩ S satisfiesthe equations

x22 + x23 − x24 = ϕ(x2, x3, x4),x ∈ H. (7)

This is either a cone, i.e., is linearly equivalent to x22 + x23 − x24 = 0, or a hyperboloid of one or two sheets, i.e., is linearlyequivalent to x22 + x23 − x24 = 1 or x22 + x23 − x24 = −1, respectively. It is easy to verify (and well known) that there are nolines on the hyperboloid of two sheets. We therefore assume that S ∩ H is either a cone or a hyperboloid of one sheet. Inthese cases, there are at most two lines of L with any given direction that are fully contained in S ∩ H . Note that if a linex+ tv | t ∈ R ∈ L is fully contained in S ∩H , then v1 = ϕ(v2, v3, v4) (where we let ϕ denote the linear homogeneous partof ϕ), and v24 = v22 + v23 (being the homogeneous part of degree two in t), |v2|, |v3| ≤ kα and |v4| ≥ kα . As observed above,there are O(kα) such triples (v2, v3, v4). Therefore, the number of lines in L that lie in S ∩ H is O(kα).

In the remaining case, the equation ofH is of the form ϕ(x2, x3, x4) = 0, where ϕ is a linear form.We can assume, withoutloss of generality, that the equation of H is x2 = ψ(x3, x4), where ψ is a linear form (the remaining case x4 = 0 is simpler tohandle). In this case, for every point x ∈ S ∩ H , we have

x1 = ψ(x3, x4)2 + x23 − x24,x ∈ H.

The classification of (real) quadratic surfaces implies that this can be an elliptic paraboloid, a parabolic cylinder or ahyperbolic paraboloid. An elliptic paraboloid contains no lines and the corresponding case is trivial. If S ∩ H is a paraboliccylinder, then all lines on it are parallel. It is straightforward that there are O(k2+2α) points in P that lie on it (by countingpossible pairs (x3, x4)). Hence there are O(k1+2α) lines in L that are fully contained in S

⋂H . In the rest of the discussion we

assume S ∩H is a hyperbolic paraboloid. In this case, similarly to the case of the one-sheeted hyperboloid, there are at mosttwo lines with the same direction. Moreover, the direction (v1, v2, v3, v4) of any line on S ∩ H satisfies v2 = ψ(v3, v4) andv24 = v22 + v23 (where we let ψ denote the linear homogeneous part of ψ). Thus once we fix v1 and ‘‘v3 or v4’’ (depending onψ), we have limited the possible direction (v1, v2, v3, v4) in a set with ≤2 elements. Hence there are O(k1+3α) lines that arefully contained in S ∩ H .

Finally, we show that for α < 12 , the number of incidences is (asymptotically) larger than Θ(m2/5n4/5

+ m1/2n1/2q1/4 +

m2/3n1/3+ m + n), which is the bound of Eq. (3), withm = Θ(k3+3α), n = Θ(k2+4α), q = O(k1+3α), and s = O(1). We have

m2/5n4/5= O(k

6+6α+8+16α5 ) = O(k

14+22α5 ),

and the exponent is smaller than 3 + 4α, as α < 12 . Similarly,

m1/2n1/2q1/4 = O(k6+6α+4+8α+1+3α

4 ) = O(k11+17α

4 ),

and the exponent is smaller than 3 + 4α, as α < 12 < 1. Similarly,

m2/3n1/3= O(k

6+6α+2+4α3 ) = O(k

8+10α3 ),

and the exponent is smaller than 3 + 4α for every α. Since bothm and n are O(k3+4α), the claim is proved.

3. Proof of Theorem 1.5

The proof of Theorem 1.5 follows easily by Corollary 1.4, together with the following lemma.

Lemma 3.1. Let L be a set of n lines in R4 such that at most s lines lie on a common 2-flat. There exists a projection from R4 ontoa hyperplane H ⊂ R4, such that at most s lines lie on any common plane in H.

Proof of Lemma 3.1. Let π1, . . . , πk denote the set of 2-flats containing at least two lines in L, then k ≤(n2

). For a generic

hyperplane H ⊂ R4, the projection p : R4→ H maps πi onto a plane π ′

i contained in H . We pick, as we may, a hyperplaneH , so that p is bijective on π1, . . . , πk. Denote by L′ the set of projected lines in R3. It is easy to verify that the set of planes inH containing at least two lines in L′ consists precisely of π ′

1, . . . , π′

k. Moreover, the number of lines in L′ that are containedin π ′

i is equal to the number of lines in L that are contained in πi, thus completing the proof.

4. Discussion and open questions

In Corollary 1.4, we show a concrete irreducible quadratic hypersurface S inR4, together with a set ofm points and n linesthat lie on S, for n9/8 < m < n3/2, such that (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines inany common hyperplane is O(n/m1/3), and (iii) the number of incidences between the points and lines isΩ(m2/3n1/2), whichis asymptotically larger thanΘ(m2/5n4/5

+m1/2n1/2q1/4 +m2/3n1/3+m+ n) in this regime ofm and n. A natural question is

590 N. Solomon, R. Zhang / Discrete Mathematics 340 (2017) 585–590

to extend this result to other regimes by a similar construction. The condition (i) is natural and should not be hard to achieve,since if a plane is not contained in a quadratic hypersurface, then by the generalized version of Bézout’s theorem [3] it cancontain at most two lines. Here are a few natural questions that arise

1. Canwe generalize our construction, such that in (ii) we are allowed to have amore general q, not necessarily∼n/m1/3,s.t. the number of lines in any common hyperplane is O(q), andwe still get a lower bound of incidences asymptoticallylarger thanΘ(m2/5n4/5

+ m1/2n1/2q1/4 + m2/3n1/3+ m + n)?

2. Can we find a similar construction whenm < n9/8?3. How powerful is the natural generalization of this construction for Rd, when d > 4? Notice that for d > 4, finding

the precise bound for the number of incidences between a set P of m points and a set L of n lines in Rd is already aninteresting open question. It is probably too early for us to answer this question before we find the correct bound.

4. In three dimensions, it remains a question to determine if Theorem 1.5 is tight.

Question 4.1. Let P be a set of m distinct points and L a set of n distinct lines in R3, and assume that no plane contains morethan s = O(1) lines of L. Then what is a good or tight upper bound of I(P, L)? Would O(m

23 n

12 + m + n) suffice?

We do not know the answer to this question yet. It seems to require new techniques.

Acknowledgments

We thank Micha Sharir for his invaluable advice, and the anonymous referees for their helpful comments.

References

[1] M. Benito, J.L. Varona, Pythagorian triples with legs less than n, J. Comput. Appl. Math. 143 (1) (2002) 117–126.[2] K. Clarkson, H. Edelsbrunner, L. Guibas,M. Sharir, E.Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput.

Geom. 5 (1990) 99–160.[3] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, in: Expository Lectures from the CBMS Regional Conference Held at George

Mason University, June 27–July 1, 1983, vol. 54, AMS Bookstore, 1984.[4] L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane, Ann. of Math. (2) 181 (2015) 155–190. Also in arXiv:1011.4105.[5] J. Pach, M. Sharir, Geometric incidences, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, in: ContemporaryMathematics, vol. 342, Amer. Math.

Soc., Providence, RI, 2004, pp. 185–223.[6] M. Sharir, N. Solomon, Incidences between points and lines on a two- and three-dimensional varieties, in arXiv:1609.09026.[7] M. Sharir, N. Solomon, Incidences between points and lines in four dimensions, Discrete Comput. Geom. (2016) in press. Also in Proc. 56th IEEE Symp.

on Foundations of Computer Science (2015), 1378–1394, and in arXiv:1411.0777.[8] M. Sharir, N. Solomon, Incidences between points and lines in R4 , in: Proc. 30th Annu. ACM Sympos. Comput. Geom., 2014, pp. 189–197.[9] A. Sheffer, Lower bounds for incidences with hypersurfaces, Discrete Anal. (2016) in press. Also in arXiv:1511.03298.

[10] J.J. Sylvester XIX, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to theform of a sum of positive and negative squares, Lond. Edinb. Dublin Phil. Mag. J. Sci. 4 (23) (1852) 138–142.

[11] L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6 (1997) 353–358.[12] E. Szemerédi, W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983) 381–392.[13] T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices Amer. Math. Soc. 48 (3)

(2001) 294–303.[14] T. Tao, A new bound for finite field Besicovitch sets in four dimensions, Pacific J. Math. 222 (2) (2005) 337–363.[15] R. Zhang, Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem, Selecta Math. (N.S.) (2014)

1–18.

104

Part II

Incidences between points and linesin R3, with applications

105

4 Incidences between points and linesin R3

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Incidences between points and lines in three dimensions∗

Micha Sharir† Noam Solomon‡

January 13, 2015

Abstract

We give a fairly elementary and simple proof that shows that the number of inci-dences between m points and n lines in R3, so that no plane contains more than s lines,is

O(m1/2n3/4 +m2/3n1/3s1/3 +m+ n

)

(in the precise statement, the constant of proportionality of the first and third termsdepends, in a rather weak manner, on the relation between m and n).

This bound, originally obtained by Guth and Katz [9] as a major step in theirsolution of Erdos’s distinct distances problem, is also a major new result in incidencegeometry, an area that has picked up considerable momentum in the past six years. Itsoriginal proof uses fairly involved machinery from algebraic and differential geometry,so it is highly desirable to simplify the proof, in the interest of better understandingthe geometric structure of the problem, and providing new tools for tackling similarproblems. This has recently been undertaken by Guth [7]. The present paper presentsa different and simpler derivation, with better bounds than those in [7], and withoutthe restrictive assumptions made there. Our result has a potential for applications toother incidence problems in higher dimensions.

1 Introduction

Let P be a set of m distinct points in R3 and let L be a set of n distinct lines in R3. LetI(P,L) denote the number of incidences between the points of P and the lines of L; thatis, the number of pairs (p, ℓ) with p ∈ P , ℓ ∈ L, and p ∈ ℓ. If all the points of P and all thelines of L lie in a common plane, then the classical Szemeredi–Trotter theorem [26] yieldsthe worst-case tight bound

I(P,L) = O(m2/3n2/3 +m+ n

). (1)

This bound clearly also holds in three dimensions, by projecting the given lines and pointsonto some generic plane. Moreover, the bound will continue to be worst-case tight by

∗Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the IsraelScience Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–IsraelBinational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (CenterNo. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.

†School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@post.tau.ac.il‡School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com

1

placing all the points and lines in a common plane, in a configuration that yields the planarlower bound.

In the 2010 groundbreaking paper of Guth and Katz [9], an improved bound has beenderived for I(P,L), for a set P of m points and a set L of n lines in R3, provided that nottoo many lines of L lie in a common plane. Specifically, they showed:1

Theorem 1 (Guth and Katz [9]). Let P be a set of m distinct points and L a set of ndistinct lines in R3, and let s ≤ n be a parameter, such that no plane contains more than slines of L. Then

I(P,L) = O(m1/2n3/4 +m2/3n1/3s1/3 +m+ n

).

This bound was a major step in the derivation of the main result of [9], which wasto prove an almost-linear lower bound on the number of distinct distances determined byany finite set of points in the plane, a classical problem posed by Erdos in 1946 [6]. Theirproof uses several nontrivial tools from algebraic and differential geometry, most notablythe Cayley–Salmon theorem on osculating lines to algebraic surfaces in R3, and additionalproperties of ruled surfaces. All this machinery comes on top of the main innovation ofGuth and Katz, the introduction of the polynomial partitioning technique; see below.

In this paper, we provide a simple derivation of this bound, which bypasses most of thetechniques from algebraic geometry that are used in the original proof. A recent relatedstudy by Guth [7] provides another simpler derivation of a similar bound, but (a) the boundobtained in [7] is slightly worse, involving extra factors of the form mε, for any ε > 0, and(b) the assumptions there are stronger, namely that no algebraic surface of degree at mostcε, a (potentially large) constant that depends on ε, contains more than s lines of L (infact, Guth considers in [7] only the case s =

√n). It should be noted, though, that Guth

also manages to derive a (slightly weaker but still) near-linear lower bound on the numberof distinct distances.

As in the classical work of Guth and Katz [9], and in the follow-up study of Guth [7], heretoo we use the polynomial partitioning method, as pioneered in [9]. The main differencebetween our approach and those of [7, 9] is the choice of the degree of the partitioningpolynomial. Whereas Guth and Katz [9] choose a large degree, and Guth [7] chooses aconstant degree, we choose an intermediate degree. This reaps many benefits from both thehigh-degree and the constant-degree approaches, and pays a small price in the bound (albeitmuch better than in [7]). Specifically, our main result is a simple and fairly elementaryderivation of the following result.

Theorem 2. Let P be a set of m distinct points and L a set of n distinct lines in R3, andlet s ≤ n be a parameter, such that no plane contains more than s lines of L. Then

I(P,L) ≤ Am,n

(m1/2n3/4 +m

)+B

(m2/3n1/3s1/3 + n

), (2)

where B is an absolute constant, and, for another suitable absolute constant b > 1,

Am,n = O

(b

log(m2n)

log(n3/m2)

), for m ≤ n3/2, and O

(blog(m3/n4)

log(m2/n3)

), for m ≥ n3/2. (3)

1We skip over certain subtleties in their bound: They also assume that no regulus contains more than sinput lines, but then they are able also to bound the number of intersection points of the lines. Moreover,if one also assumes that each point is incident to at least three lines then the term m in the bound can bedropped.

2

Remarks. (1) Only the range√n ≤ m ≤ n2 is of interest; outside this range, regardless

of the dimension of the ambient space, we have the well known and trivial upper boundO(m+ n).

(2) The term m2/3n1/3s1/3 comes from the planar Szemeredi–Trotter bound (1), and isunavoidable, as it can be attained if we densely “pack” points and lines into planes, inpatterns that realize the bound in (1).

(3) Ignoring this term, the two terms m1/2n3/4 and m “compete” for dominance; the formerdominates when m ≤ n3/2 and the latter when m ≥ n3/2. Thus the bound in (2) isqualitatively different within these two ranges.

(4) The threshold m = n3/2 also arises in the related problem of joints (points incident toat least three non-coplanar lines) in a set of n lines in 3-space; see [8].

A concise rephrasing of the bound in (2) and (3) is as follows. We partition each of theranges m ≤ n3/2, m > n3/2 into a sequence of subranges nαj−1 < m ≤ nαj , j = 0, 1, . . .(for m ≤ n3/2), or nαj−1 > m ≥ nαj , j = 0, 1, . . . (for m ≥ n3/2), so that within eachrange the bound asserted in the theorem holds for some fixed constant of proportionality(denoted as Am,n in the bound), where these constants vary with j, and grow, exponentiallyin j, as prescribed in (3), as m approaches n3/2 (from either side). Informally, if we keep m“sufficiently away” from n3/2, the bound in (2) holds with a fixed constant of proportionality.Handling the “border range” m ≈ n3/2 is also fairly straightforward, although, to bypassthe exponential growth of the constant of proportionality, it results in a slightly differentbound; see below for details.

Our proof is elementary to the extent that, among other things, it avoids any explicithandling of singular and flat points on the zero set of the partitioning polynomial. Whilethese notions are relatively easy to handle in three dimensions (see, e.g., [5, 8]), they becomemore complex notions in higher dimensions (as witnessed, for example, in our companionwork on the four-dimensional setting [22]), making proofs based on them harder to extend.

Additional merits and features of our analysis are discussed in detail in the concludingsection. In a nutshell, the main merits are:

(i) We use two separate partitioning polynomials. The first one is of “high” degree, and isused to prune away some points and lines, and to establish useful properties of the survivingpoints and lines. The second partitioning step, using a polynomial of “low” degree, is thenapplied, from scratch, to the surviving input, exploiting the properties established in thefirst step. This idea seems to have a potential for further applications.

(ii) Because of the way we use the polynomial partitioning technique, we need induction tohandle incidences within the cells of the second partition. One of the nontrivial achievementsof our technique is the ability to retain The “planar” term O(m2/3n1/3s1/3) in the bound in(2) through the inductive process. Without such care, this term does not “pass well” throughthe induction, which has been a sore issue in several recent works on related problems (see[19, 20, 21]). This is one of the main reasons for using two separate partitioning steps.

Background. Incidence problems have been a major topic in combinatorial and compu-tational geometry for the past thirty years, starting with the aforementioned Szemeredi-Trotter bound [26] back in 1983. Several techniques, interesting in their own right, have

3

been developed, or adapted, for the analysis of incidences, including the crossing-lemmatechnique of Szekely [25], and the use of cuttings as a divide-and-conquer mechanism (e.g.,see [3]). Connections with range searching and related algorithmic problems in compu-tational geometry have also been noted, and studies of the Kakeya problem (see, e.g.,[27]) indicate the connection between this problem and incidence problems. See Pach andSharir [16] for a comprehensive (albeit a bit outdated) survey of the topic.

The landscape of incidence geometry has dramatically changed in the past six years,due to the infusion, in two groundbreaking papers by Guth and Katz [8, 9], of new toolsand techniques drawn from algebraic geometry. Although their two direct goals have beento obtain a tight upper bound on the number of joints in a set of lines in three dimensions[8], and a near-linear lower bound for the classical distinct distances problem of Erdos [9],the new tools have quickly been recognized as useful for incidence bounds. See [5, 12, 13,20, 24, 30, 31] for a sample of recent works on incidence problems that use the new algebraicmachinery.

The simplest instances of incidence problems involve points and lines, tackled by Sze-meredi and Trotter in the plane [26], and by Guth and Katz in three dimensions [9]. Otherrecent studies on incidence problems include incidences between points and lines in fourdimensions (Sharir and Solomon [21, 22]), and incidences between points and circles inthree dimensions (Sharir, Sheffer and Zahl [20]), not to mention incidences with higher-dimensional surfaces, such as in [1, 12, 24, 30, 31]. In a companion paper (with Sheffer) [19],we study the general case of incidences between points and curves in any dimension, andderive reasonably sharp bounds (albeit weaker in several respects than the one derivedhere).

That tools from algebraic geometry form the major key for successful solution of difficultproblems in combinatorial geometry, came as a big surprise to the community. It has leadto intensive research of the new tools, aiming to extend them and to find new applications.A major purpose of this study, as well as of Guth [7], is to show that one can still tacklesuccessfully the problems using less heavy algebraic machinery. This offers a new, simplified,and more elementary approach, which we expect to prove potent for other applications too,such as those just mentioned. Looking for simpler, yet effective techniques that would beeasier to extend to more involved contexts (such as incidences in higher dimensions) hasbeen our main motivation for this study.

A more detailed supplementary discussion (which would be premature at this point) ofthe merits and other issues related to our technique is given in a concluding section.

2 Proof of Theorem 2

The proof proceeds by induction on m. As already mentioned, the bound in (2) is qual-itatively different in the two ranges m ≤ n3/2 and m ≥ n3/2. The analysis bifurcatesaccordingly. While the general flow is fairly similar in both cases, there are many differ-ences too.

The case m < n3/2. We partition this range into a sequence of ranges m ≤ nα0 , nα0 <m ≤ nα1 , . . ., where α0 = 1/2 and the sequence αjj≥0 is increasing and converges to

4

3/2. More precisely, as our analysis will show, we can take αj = 32 − 2

j+2 , for j ≥ 0. Theinduction is actually on the index j of the range nαj−1 < m ≤ nαj , and establishes (2) form in this range, with a coefficient Aj (written in (2, 3) as Am,n) that increases with j. Thisparadigm has already been used in Sharir et al. [20] and in Zahl [31], for related incidenceproblems, albeit in a somewhat less effective manner; see the discussion at the end of thepaper.

The base range of the induction is m ≤ √n, where the trivial general upper bound on

point-line incidences, in any dimension, yields I = O(m2 + n) = O(n), so (2) holds for asufficiently large choice of the initial constant A0.

Assume then that (2) holds for all m ≤ nαj−1 for some j ≥ 1, and consider an instanceof the problem with nαj−1 < m ≤ n3/2 (the analysis will force us to constrain this upperbound in order to complete the induction step, thereby obtaining the next exponent αj).

Fix a parameter r, whose precise value will be chosen later (in fact, and this is a majornovelty of our approach, there will be two different choices for r—see below), and applythe polynomial partitioning theorem of Guth and Katz (see [9] and [13, Theorem 2.6]), toobtain an r-partitioning trivariate (real) polynomial f of degree D = O(r1/3). That is, everyconnected component of R3 \ Z(f) contains at most m/r points of P , where Z(f) denotesthe zero set of f . By Warren’s theorem [29] (see also [13]), the number of components ofR3 \ Z(f) is O(D3) = O(r).

Set P1 := P ∩ Z(f) and P ′1 := P \ P1. A major recurring theme in this approach is

that, although the points of P ′1 are more or less evenly partitioned among the cells of the

partition, no nontrivial bound can be provided for the size of P1; in the worst case, all thepoints of P could lie in Z(f). Each line ℓ ∈ L is either fully contained in Z(f) or intersectsit in at most D points (since the restriction of f to ℓ is a univariate polynomial of degreeat most D). Let L1 denote the subset of lines of L that are fully contained in Z(f) and putL′1 = L \ L1. We then have

I(P,L) = I(P1, L1) + I(P1, L′1) + I(P ′

1, L′1).

We first bound I(P1, L′1) and I(P ′

1, L′1). As already observed, we have

I(P1, L′1) ≤ |L′

1| ·D ≤ nD.

We estimate I(P ′1, L

′1) as follows. For each (open) cell τ of R3 \Z(f), put Pτ = P ∩ τ (that

is, P ′1 ∩ τ), and let Lτ denote the set of the lines of L′

1 that cross τ ; put mτ = |Pτ | ≤ m/r,and nτ = |Lτ |. Since every line ℓ ∈ L′

1 crosses at most 1 +D components of R3 \ Z(f), wehave ∑

τ

nτ ≤ n(1 +D), and I(P ′1, L

′1) =

∑

τ

I(Pτ , Lτ ).

For each τ we use the trivial bound I(Pτ , Lτ ) = O(m2τ + nτ ). Summing over the cells, we

get

I(P ′1, L

′1) =

∑

τ

I(Pτ , Lτ ) = O

(r · (m/r)2 +

∑

τ

nτ

)= O

(m2/r + nD

)= O(m2/D3+nD).

For the initial value of D, we take D = m1/2/n1/4 (which we get from a suitable value ofr = Θ(D3)), and get the bound

I(P ′1, L

′1) + I(P1, L

′1) = O(m1/2n3/4).

5

This choice of D is the one made in [9]. It is sufficiently large to control the situation inthe cells, by the bound just obtained, but requires heavy-duty machinery from algebraicgeometry to handle the situation on Z(f).

We now turn to Z(f), where we need to estimate I(P1, L1). Since all the incidencesinvolving any point in P ′

1 and/or any line in L′1 have already been accounted for, we discard

these sets, and remain with P1 and L1 only. We “forget” the preceding polynomial partition-ing step, and start afresh, applying a new polynomial partitioning to P1 with a polynomialg of degree E, which will typically be much smaller than D, but still non-constant.

Before doing this, we note that the set of lines L1 has a special structure, because allits lines lie on the algebraic surface Z(f), which has degree D. We exploit this to derivethe following lemmas. We emphasize, since this will be important later on in the analysis,that Lemmas 3–7 hold for any choice of (r and) D.

We note that in general the partitioning polynomial f may be reducible, and apply someof the following arguments to each irreducible factor separately. Clearly, there are at mostD such factors.

Lemma 3. Let π be a plane which is not a component of Z(f). Then π contains at mostD lines of L1.

Proof. Suppose to the contrary that π contains at least D + 1 lines of L. Every genericline λ in π intersects these lines in at least D + 1 distinct points, all belonging to Z(f).Hence f must vanish identically on λ, and it follows that f ≡ 0 on π, so π is a componentof Z(f), contrary to assumption.

Lemma 4. The number of incidences between the points of P1 that lie in the planar com-ponents of Z(f) and the lines of L1, is O(m2/3n1/3s1/3 + nD).

Proof. Clearly, f can have at most D linear factors, and thus Z(f) can contain at most Dplanar components. Enumerate them as π1, . . . , πk, where k ≤ D. Let P1 denote the subsetof the points of P1 that lie in these planar components. Assign each point of P1 to the firstplane πi, in this order, that contains it, and assign each line of L1 to the first plane thatfully contains it; some lines might not be assigned at all in this manner. For i = 1, . . . , k,let Pi denote the set of points assigned to πi, and let Li denote the set of lines assigned toπi. Put mi = |Pi| and ni = |Li|. Then

∑i mi ≤ m and

∑i ni ≤ n; by assumption, we also

have ni ≤ s for each i. Then

I(Pi, Li) = O(m2/3i n

2/3i +mi + ni) = O(m

2/3i n

1/3i s1/3 +mi + ni).

Summing over the k planes, we get, using Holder’s inequality,

∑

i

I(Pi, Li) =∑

i

O(m2/3i n

1/3i s1/3 +mi + ni)

= O

(∑

i

mi

)2/3(∑

i

ni

)1/3

s1/3 +m+ n

= O

(m2/3n1/3s1/3 +m+ n

).

We also need to include incidences between points p ∈ P1 and lines ℓ ∈ L1 not assignedto the same plane as p (or not assigned to any plane at all). Any such incidence (p, ℓ) can

6

be charged (uniquely) to the intersection point of ℓ with the plane πi to which p has beenassigned. The number of such intersections is O(nD), and the lemma follows.

Lemma 5. Each point p ∈ Z(f) is incident to at most D2 lines of L1, unless Z(f) has anirreducible component that is either a plane containing p or a cone with apex p.

Proof. Fix any line ℓ that passes through p, and write its parametric equation as p+ tv |t ∈ R, where v is the direction of ℓ. Consider the Taylor expansion of f at p along ℓ

f(p+ tv) =

D∑

i=1

1

i!Fi(p; v)t

i,

where Fi(p; v) is the i-th order derivative of f at p in direction v; it is a homogeneouspolynomial in v (p is considered fixed) of degree i, for i = 1, . . . ,D. For each line ℓ ∈ L1 thatpasses through p, f vanishes identically on ℓ, so we have Fi(p; v) = 0 for each i. Assumingthat p is incident to more than D2 lines of L1, we conclude that the homogeneous system

F1(p; v) = F2(p; v) = · · · = FD(p; v) = 0 (4)

has more than D2 (projectively distinct) roots. The classical Bezout’s theorem, applied inthe projective plane where the directions v are represented (e.g., see [4]), asserts that, sinceall these polynomials are of degree at most D, each pair of polynomials Fi(p; v), Fj(p; v)must have a common factor. The following slightly more involved inductive argument showsthat in fact all these polynomials must have a common factor.2

Lemma 6. Let f1, . . . , fn ∈ C[x, y, z] be n homogeneous polynomials of degree at most D.If |Z(f1, . . . , fn)| > D2, then all the fi’s have a nontrivial common factor.

Proof. The proof is via induction on n. The case n = 2 is precisely the classical Bezout’stheorem in the projective plane. Assume that the inductive claim holds for n − 1 poly-nomials. By assumption, |Z(f1, . . . , fn−1)| ≥ |Z(f1, . . . , fn)| > D2, so the induction hy-pothesis implies that there is a polynomial g that divides fi, for i = 1, . . . , n − 1; as-sume, as we may, that g = GCD(f1, . . . , fn−1). If there are more than deg(g)deg(fn)points in Z(g, fn), then again, by the classical Bezout’s theorem in the projective plane,g and fn have a nontrivial common factor, which is then also a common factor of fi, fori = 1, . . . , n, completing the proof. Otherwise, put fi = fi/g, for i = 1, . . . , n − 1. Noticethat Z(f1, . . . , fn−1) = Z(f1, . . . , fn−1) ∪ Z(g), implying that each point of Z(f1, . . . , fn)belongs either to Z(g) ∩ Z(fn) or to Z(f1, . . . , fn−1) ∩ Z(fn). As |Z(f1, . . . , fn)| > D2 and|Z(g, fn)| ≤ deg(g)deg(fn) ≤ deg(g)D, it follows that

|Z(f1, . . . , fn−1)| ≥ |Z(f1, . . . , fn−1, fn)| ≥ (D − deg(g))D > (D − deg(g))2.

Hence, applying the induction hypothesis to the polynomials f1, . . . , fn−1 (all of degree atmost D − deg(g)), we conclude that they have a nontrivial common factor, contradictingthe fact that g is the greatest common divisor of f1, . . . , fn−1.

Continuing with the proof of Lemma 5, there is an infinity of directions v that satisfy(4), so there is an infinity of lines passing through v and contained in Z(f). The union of

2See also [17] for a similar observation.

7

these lines can be shown to be a two-dimensional algebraic variety,3 contained in Z(f), soZ(f) has an irreducible component that is either a plane through p or a cone with apex p,as claimed.

Lemma 7. The number of incidences between the points of P1 that lie in the (non-planar)conic components of Z(f), and the lines of L1, is O(m+ nD).

Proof. Let σ be such an (irreducible) conic component of Z(f) and let p be its apex.We observe that σ cannot contain any line that is not incident to p, because such a linewould span with p a plane contained in σ, contradicting the assumption that σ is irreducibleand non-planar. It follows that the number of incidences between Pσ := P1 ∩ σ and Lσ,consisting of the lines of L1 contained in σ, is thus O(|Pσ| + |Lσ|) (p contributes |Lσ|incidences, and every other point at most one incidence). Applying a similar “first-come-first-serve” assignment of points and lines to the conic components of Z(f), as we did for theplanar components in the proof of lemma 4, and adding the bound O(nD) on the numberof incidences between points and lines not assigned to the same component, we obtain thebound asserted in the lemma.

Remark. Note that in both Lemma 4 and Lemma 7, we bound the number of incidencesbetween points on planar or conic components of Z(f) and all the lines of L1.

Pruning. To continue, we remove all the points of P1 that lie in some planar or coniccomponent of Z(f), and all the lines of L1 that are fully contained in such components.With the choice of D = m1/2/n1/4, we lose in the process

O(m2/3n1/3s1/3 +m+ nD) = O(m1/2n3/4 +m2/3n1/3s1/3)

incidences (recall that the term m is subsumed by the term m1/2n3/4 for m < n3/2). Con-tinue, for simplicity of notation, to denote the sets of remaining points and lines as P1 andL1, respectively, and their sizes as m and n. Now each point is incident to at most D2 lines(a fact that we will not use for this value of D), and no plane contains more than D lines ofL1, a crucial property for the next steps of the analysis. That is, this allows us to replacethe input parameter s, bounding the maximum number of coplanar lines, by D; this is akey step that makes the induction work.

A new polynomial partitioning. We now return to the promised step of constructinga new polynomial partitioning. We adapt the preceding notation, with a few modifications.We choose a degree E, typically much smaller than D, and construct a partitioning poly-nomial g of degree E for P1. With an appropriate value of r = Θ(E3), we obtain O(r) opencells, each containing at most m/r points of P1, and each line of L1 either crosses at mostE + 1 cells, or is fully contained in Z(g).

Set P2 := P1∩Z(g) and P ′2 := P1 \P2. Similarly, denote by L2 the set of lines of L1 that

are fully contained in Z(g), and put L′2 := L1 \ L2. We first dispose of incidences involving

3It is simply the variety given by the equations (4), rewritten as F1(p;x − p) = F2(p;x − p) = · · · =FD(p;x− p) = 0. It is two-dimensional because it is contained in Z(f), hence at most two-dimensional, andit cannot be one-dimensional since it would then consist of only finitely many lines (see, e.g., [22, Lemma2.3]).

8

the lines of L2. (That is, now we first focus on incidences within Z(g), and only then turnto look at the cells.) By Lemma 4 and Lemma 7, the number of incidences involving pointsP2 that lie in some planar or conic component of Z(g), and all the lines of L2, is

O(m2/3n1/3s1/3 +m+ nE) = O(m1/2n3/4 +m2/3n1/3s1/3 + n).

(For E ≪ D, this might be a gross overestimation, but we do not care.) We remove thesepoints from P2, and remove all the lines of L2 that are contained in such components;continue to denote the sets of remaining points and lines as P2 and L2. Now each pointis incident to at most E2 lines of L2 (Lemma 5), so the number of remaining incidencesinvolving points of P2 is O(mE2); for E suitably small, this bound will be subsumed byO(m1/2n3/4).

Unlike the case of a “large” D, namely, D = m1/2/n1/4, here the difficult part is totreat incidences within the cells of the partition. Since E ≪ D, we cannot use the naivebound O(n2 +m) within each cell, because that would make the overall bound too large.Therefore, to control the incidence bound within the cells, we proceed in the followinginductive manner.

For each cell τ of R3 \ Z(g), put Pτ := P ′2 ∩ τ , and let Lτ denote the set of the lines of

L′2 that cross τ ; put mτ = |Pτ | ≤ m/r, and nτ = |Lτ |. Since every line ℓ ∈ L1 (that is, of

L′2) crosses at most 1 + E components of R3 \ Z(g), we have

∑τ nτ ≤ n(1 + E).

It is important to note that at this point of the analysis the sizes of P1 and of L1 mightbe smaller than the original respective values m and n. In particular, we may no longerassume that |P1| > |L1|αj−1 , as we did assume for m and n. Nevertheless, in what follows mand n will denote the original values, which serve as upper bounds for the respective actualsizes of P1 and L1, and the induction will work correctly with these values; see below fordetails.

In order to apply the induction hypothesis within the cells of the partition, we wantto assume that mτ ≤ nτ

αj−1 for each τ . To ensure that, we require that the number oflines of L′

2 that cross a cell be at most n/E2. Cells τ that are crossed by κn/E2 lines, forκ > 1, are treated as if they occur ⌈κ⌉ times, where each incarnation involves all the pointsof Pτ , and at most n/E2 lines of Lτ . The number of subproblems remains O(E3). Arguingsimilarly, we may also assume that mτ ≤ m/E3 for each cell τ (by “duplicating” each cellinto a constant number of subproblems, if needed).

We therefore require thatm

E3≤( n

E2

)αj−1

. (Note that, as already commented above,

these are only upper bounds on the actual sizes of these subsets, but this will have no realeffect on the induction process.) That is, we require

E ≥( m

nαj−1

)1/(3−2αj−1). (5)

With these preparations, we apply the induction hypothesis within each cell τ , recallingthat no plane contains more than D lines4 of L′

2 ⊆ L1, and get

I(Pτ , Lτ ) ≤ Aj−1

(m1/2

τ n3/4τ +mτ

)+B

(m2/3

τ n1/3τ D1/3 + nτ

)

≤ Aj−1

((m/E3)1/2(n/E2)3/4 +m/E3

)+B

(+(m/E3)2/3(n/E2)1/3D1/3 + n/E2

).

4This was the main reason for carrying out the first partitioning step, as already noted.

9

Summing these bounds over the cells τ , that is, multiplying them by O(E3), we get, for asuitable absolute constant b,

I(P ′2, L

′2) =

∑

τ

I(Pτ , Lτ ) ≤ bAj−1

(m1/2n3/4 +m

)+B

(m2/3n1/3E1/3D1/3 + nE

).

We now require that E = O(D). Then the last term satisfies nE = O(nD) = O(m1/2n3/4),and, as already remarked, the preceding term m is also subsumed by the first term. Thesecond term, after substituting D = O(m1/2/n1/4), becomes O(m5/6n1/4E1/3). Hence, witha slightly larger b, we have

I(P ′2, L

′2) ≤ bAj−1m

1/2n3/4 + bBm5/6n1/4E1/3.

Adding up all the bounds, including those for the portions of P and L that were discardedduring the first partitioning step, we obtain, for a suitable constant c,

I(P,L) ≤ c(m1/2n3/4 +m2/3n1/3s1/3 + n+mE2

)+ bAj−1m

1/2n3/4 + bBm5/6n1/4E1/3.

We choose E to ensure that the two E-dependent terms are dominated by the termm1/2n3/4. That is,

m5/6n1/4E1/3 ≤ m1/2n3/4, or E ≤ n3/2/m,

and mE2 ≤ m1/2n3/4, or E ≤ n3/8/m1/4.

Since n3/2/m =(n3/8/m1/4

)4, and both sides are ≥ 1, the latter condition is stricter, and

we ignore the former. As already noted, we also require that E = O(D); specifically, werequire that E ≤ m1/2/n1/4.

In conclusion, recalling (5), the two constraints on the choice of E are

( m

nαj−1

)1/(3−2αj−1) ≤ E ≤ min

n3/8

m1/4,m1/2

n1/4

, (6)

and, for these constraints to be compatible, we require that

( m

nαj−1

)1/(3−2αj−1) ≤ n3/8

m1/4, or m ≤ n

9+2αj−12(7−2αj−1) ,

and that ( m

nαj−1

)1/(3−2αj−1) ≤ m1/2

n1/4,

which fortunately always holds, as is easil;y checked, since m ≤ n3/2 and αj−1 ≥ 1/2. Notethat we have not explicitly stated any concrete choice of E; any value satisfying (6) will do.We put

αj :=9 + 2αj−1

2(7 − 2αj−1),

and conclude that if m ≤ nαj then the bound asserted in the theorem holds, with Aj =bAj−1 + c and B = c. This completes the induction step. Note that the recurrence Aj =bAj−1 + c solves to Aj = O(bj).

10

It remains to argue that the induction covers the entire range m = O(n3/2). Using theabove recurrence for the αj ’s, with α0 = 1/2, it easily follows that

αj =3

2− 2

j + 2,

for each j ≥ 0, showing that αj converges to 3/2, implying that the entire rangem = O(n3/2)is covered by the induction.

To calibrate the dependence of the constant of proportionality on m and n, we notethat, for nαj−1 ≤ m < nαj , the constant is O(bj). We have

3

2− 2

j + 1= αj−1 ≤

logm

log n, or j ≤

12 +

logmlogn

32 −

logmlogn

=log(m2n)

log(n3/m2).

This establishes the expression for Am,n given in the statement of the theorem.

Handling the middle ground m ≈ n3/2. Some care is needed when m approaches n3/2,because of the potentially unbounded growth of the constant Aj. To handle this situation,we simply fix a value j, in the manner detailed below, write m = knαj , solve k separateproblems, each involving m/k = nαj points of P and all the n lines of L, and sum up theresulting incidence bounds. We then get

I(P,L) ≤ akbj((m/k)1/2n3/4 + (m/k)

)+ kB

((m/k)2/3n1/3s1/3 + n

)

= ak1/2bjm1/2n3/4 + abjm+ k1/3Bm2/3n1/3s1/3 + kBn,

for a suitable absolute constant a. Recalling that αj =32 − 2

j+2 , we have

k ≤ m/nαj ≤ n3/2/nαj = n2/(j+2).

Hence the coefficient of the leading term in the above bound is bounded by an1/(j+2)bj, andwe (asymptotically) minimize this expression by choosing

j = j0 :=√

log n/√

log b.

With this choice all the other coefficients are also dominated by the leading coefficient, andwe obtain

I(P,L) = O(22

√log b

√logn

(m1/2n3/4 +m2/3n1/3s1/3 +m+ n

)). (7)

In other words, the bound in (2) and (3) holds for any m ≤ n3/2, but, for m ≥ nαj0 oneshould use instead the bound in (7), which controls the exponential growth of the constantsof proportionality within this range.

The case m > n3/2. The analysis of this case is, in a sense, a mirror image of the precedinganalysis, except for a new key lemma (Lemma 8). For the sake of completeness, we repeata sizeable portion of the analysis, providing many of the relevant (often differing) details.

We partition this range into a sequence of ranges m ≥ nα0 , nα1 ≤ m < nα0 , . . ., whereα0 = 2 and the sequence αjj≥0 is decreasing and converges to 3/2. The induction is on

11

the index j of the range nαj ≤ m < nαj−1 , and establishes (2) for m in this range, with acoefficient Aj (written in (2,3) as Am,n) that increases with j.

The base range of the induction is m ≥ n2, where the trivial general upper bound onpoint-line incidences in any dimension, dual to the one used in the previous case, yieldsI = O(n2 + m) = O(m), so (2) holds for a sufficiently large choice of the initial constantA0.

Assume then that (2) holds for all m ≥ nαj−1 for some j ≥ 1, and consider an instanceof the problem with n3/2 ≤ m < nαj−1 (again, the lower bound will increase, to nαj , tofacilitate the induction step).

For a parameter r, to be specified later, apply the polynomial partition theorem toobtain an r-partitioning trivariate (real) polynomial f of degree D = O(r1/3). That is,every connected component of R3 \Z(f) contains at most m/r points of P , and the numberof components of R3 \ Z(f) is O(D3) = O(r).

Set P1 := P ∩ Z(f) and P ′1 := P \ P1. Each line ℓ ∈ L is either fully contained in Z(f)

or intersects it in at most D points. Let L1 denote the subset of lines of L that are fullycontained in Z(f) and put L′

1 = L \ L1. As before, we have

I(P,L) = I(P1, L1) + I(P1, L′1) + I(P ′

1, L′1).

We haveI(P1, L

′1) ≤ |L′

1| ·D ≤ nD,

and we estimate I(P ′1, L

′1) as follows. For each cell τ of R3 \Z(f), put Pτ = P ∩ τ (that is,

P ′1 ∩ τ), and let Lτ denote the set of the lines of L′

1 that cross τ ; put mτ = |Pτ | ≤ m/r, andnτ = |Lτ |. As before, we have

∑τ nτ ≤ n(1+D), so the average number of lines that cross

a cell is O(n/D2). Arguing as above, we may assume, by possibly increasing the number ofcells by a constant factor, that each nτ is at most n/D2. Clearly, we have

I(P ′1, L

′1) =

∑

τ

I(Pτ , Lτ ).

For each τ we use the trivial dual bound, mentioned above, I(Pτ , Lτ ) = O(n2τ + mτ ).

Summing over the cells, we get

I(P ′1, L

′1) =

∑

τ

I(Pτ , Lτ ) = O(D3 · (n/D2)2 +m

)= O

(n2/D +m

).

For the initial value of D, we take D = n2/m, noting that 1 ≤ D3 ≤ m because n3/2 ≤m ≤ n2, and get the bound

I(P ′1, L

′1) + I(P1, L

′1) = O(n2/D +m+ nD) = O(m+ n3/m) = O(m),

where the latter bound follows since m ≥ n3/2.

It remains to estimate I(P1, L1). Since all the incidences involving any point in P ′1

and/or any line in L′1 have been accounted for, we discard these sets, and remain with P1

and L1 only. As before, we forget the preceding polynomial partitioning step, and startafresh, applying a new polynomial partitioning to P1 with a polynomial g of degree E,which will typically be much smaller than D, but still non-constant.

12

For this case we need the following lemma, which can be regarded, in some sense, as adual (albeit somewhat more involved) version of Lemma 5. Unlike the rest of the analysis,the best way to prove this lemma is by switching to the complex projective setting. Thisis needed for one key step in the proof, where we need the property that the projection ofa complex projective variety is a variety. Once this is done, we can switch back to the realaffine case, and complete the proof.

Here is a very quick review of the transition to the complex projective setup. A realaffine algebraic variety X, defined by a collection of real polynomials, can also be regardedas a complex projective variety. (Technically, one needs to take the projective closure ofthe complexification of X; details about these standard operations can be found, e.g., inBochnak et al. [2, Proposition 8.7.17] and in Cox et al. [4, Definition 8.4.6].) If f is anirreducible polynomial over R, it might still be reducible over C, but then it must have theform f = gg, where g is an irreducible complex polynomial and g is its complex conjugate.(Indeed, if h is any irreducible factor of f , then h is also an irreducible factor of f , andtherefore hh is a real polynomial dividing f . As f is irreducible over R, the claim follows.)

In the following lemma, adapting a notation used in earlier works, we say that a pointp ∈ P1 is 1-poor (resp., 2-rich) if it is incident to at most one line (resp., to at least twolines) of L1.

Recall also that a regulus is a doubly-ruled surface in R3 or in C3. It is the union ofall lines that pass through three fixed pairwise skew lines; it is a quadric, which is either ahyperbolic paraboloid or a one-sheeted hyperboloid.

Lemma 8. Let f be an irreducible polynomial in C[x, y, z], such that Z(f) is not a complexplane nor a complex regulus, and let L1 be a finite set of lines fully contained in Z(f).Then, with the possible exception of at most two lines, each line ℓ ∈ L1 is incident to atmost O(D3) 2-rich points.

Proof. The strategy of the proof is to charge each incidence of ℓ with some 2-rich point pto an intersection of ℓ with another line of L1 that passes through p, and to argue that, ingeneral, there can be only O(D3) such other lines. This in turn will be shown by arguingthat the union of all the lines that are fully contained in Z(f) and pass through ℓ is a one-dimensional variety, of degree O(D3), from which the claim will follow. As we will show,this will indeed be the case except when ℓ is one of at most two “exceptional” lines on Z(f).

Fix a line ℓ as in the lemma, assume for simplicity that it passes through the origin,and write it as tv0 | t ∈ C; since ℓ is a real line, v0 can be assumed to be real. Considerthe union V (ℓ) of all the lines that are fully contained in Z(f) and are incident to ℓ; thatis, V (ℓ) is the union of ℓ with the set of all points p ∈ Z(f) \ ℓ for which there exists t ∈ Csuch that the line connecting p to tv0 ∈ ℓ is fully contained in Z(f). In other words, forsuch a t and for each s ∈ C, we have f((1− s)p+ stv0) = 0. Regarding the left-hand side as

a polynomial in s, we can write it asD∑

i=0

Gi(p; t)si ≡ 0, for suitable (complex) polynomials

Gi(p; t) in p and t, each of total degree at most D. In other words, p and t have to satisfythe system

G0(p; t) = G1(p; t) = · · · = GD(p; t) = 0, (8)

which defines an algebraic variety σ(ℓ) in P4(C). Note that, substituting s = 0, we haveG0(p; t) ≡ f(p), and that the limit points (tv0, t) (corresponding to points on ℓ) also satisfy

13

this system, since in this case f((1− s)tv0 + stv0) = f(tv0) = 0 for all s.

In other words, V (ℓ) is the projection of σ(ℓ) into P3(C), given by (p, t) 7→ p. Foreach p ∈ Z(f) \ ℓ this system has only finitely many solutions in t, for otherwise the planespanned by p and ℓ0 would be fully contained in Z(f), contrary to our assumption.

By the projective extension theorem (see, e.g., [4, Theorem 8.6]), the projection of σ(ℓ)into P3(C), in which t is discarded, is an algebraic variety τ(ℓ). We observe that τ(ℓ) iscontained in Z(f), and is therefore of dimension at most two.

Assume first that τ(ℓ) is two-dimensional. As f is irreducible over C, we must haveτ(ℓ) = Z(f). This implies that each point p ∈ Z(f) \ ℓ is incident to a (complex) line thatis fully contained in Z(f) and is incident to ℓ. In particular, Z(f) is ruled by complex lines.

By assumption, Z(f) is neither a complex plane nor a complex regulus. We may alsoassume that Z(f) is not a complex cone, for then each line in L1 is incident to at most one2-rich point (namely, the apex of Z(f)), making the assertion of the lemma trivial. It thenfollows that Z(f) is an irreducible singly ruled (complex) surface. As argued in Guth andKatz [9] (see also our companion paper [23] for an independent analysis of this situation,which caters more explicitly to the complex setting too), Z(f) can contain at most two linesℓ with this property.

Excluding these (at most) two exceptional lines ℓ, we may thus assume that τ(ℓ) is (atmost) a one-dimensional curve.

Clearly, by definition, each point (p, t) ∈ σ(ℓ), except for p ∈ ℓ, defines a line λ, in theoriginal 3-space, that connects p to tv0, and each point q ∈ λ satisfies (q, t) ∈ σ(ℓ). Hence,the line (q, t) | q ∈ λ is fully contained in σ(ℓ), and therefore the line λ is fully containedin τ(ℓ). Since τ(ℓ) is one-dimensional, this in turn implies (see, e.g., [22, Lemma 2.3]) thatτ(ℓ) is a finite union of (complex) lines, whose number is at most deg(τ(ℓ)). This alsoimplies that σ(ℓ) is the union of the same number of lines, and in particular σ(ℓ) is alsoone-dimensional, and the number of lines that it contains is at most deg(σ(ℓ)).

We claim that this latter degree is at most O(D3). This follows from a well-knownresult in algebra (see, e.g., Schmid [18, Lemma 2.2]), that asserts that, since σ(ℓ) is a one-dimensional curve in P4(C), and is the common zero set of polynomials, each of degreeO(D), its degree is O(D3).

This completes the proof of the lemma. (The passage from the complex projectivesetting back to the real affine one is trivial for this property.)

Corollary 9. Let f be a real or complex trivariate polynomial of degree D, such that (thecomplexification of) Z(f) does not contain any complex plane nor any complex regulus. LetL1 be a set of n lines fully contained in Z(f), and let P1 be a set of m points contained inZ(f). Then I(P1, L1) = O(m+ nD3).

Proof. Write f =∏s

i=1 fi for its decomposition into irreducible factors, for s ≤ D. We ap-ply Lemma 8 to each complex factor fi of the f . By the observation preceding Lemma 8,someof these factors might be complex (non-real) polynomials, even when f is real. That is, re-gardless of whether the original f is real or not, we carry out the analysis in the complexprojective space P3(C), and regard Z(fi) as a variety in that space.

Note also that, by focussing on the single irreducible component Z(fi) of Z(f), weconsider only points and lines that are fully contained in Z(fi). We thus shrink P1 and

14

L1 accordingly, and note that the notions of being 2-rich or 1-poor are now redefined withrespect to the reduced sets. All of this will be rectified at the end of the proof.

Assign each line ℓ ∈ L1 to the first component Z(fi), in the above order, that fullycontains ℓ, and assign each point p ∈ P1 to the first component that contains it. If a point pand a line ℓ are incident, then either they are both assigned to the same component Z(fi),or p is assigned to some component Z(fi) and ℓ, which is assigned to a later component,is not contained in Z(fi). Each incidence of the latter kind can be charged to a crossingbetween ℓ and Z(fi), and the total number of these crossings is O(nD). It therefore sufficesto consider incidences between points and lines assigned to the same component. Moreover,if a point p is 2-rich with respect to the entire collection L1 but is 1-poor with respect tothe lines assigned to its component, then all of its incidences except one are accounted bythe preceding term O(nD), which thus takes care also of the single incidence within Z(fi).

By Lemma 8, for each fi, excluding at most two exceptional lines, the number of inci-dences between a line assigned to (and contained in) Z(fi) and the points assigned to Z(fi)that are still 2-rich within Z(fi), is O(deg(fi)

3) = O(D3). Summing over all relevant lines,we get the bound O(nD3).

Finally, each irreducible component Z(fi) can contain at most two exceptional lines, fora total of at most 2D such lines. The number of 2-rich points on each such line ℓ is at mostn, since each such point is incident to another line, so the total number of correspondingincidences is at most O(nD), which is subsumed by the preceding bound O(nD3). Thenumber of incidences with 1-poor points is, trivially, at most m. This completes the proofof the corollary.

Pruning. In the preceding lemma and corollary, we have excluded planar and regulicomponents of Z(f). Arguing as in the case of small m, the number of incidences involvingpoints that lie on planar components of Z(f) is O(m2/3n1/3s1/3 + m) (see Lemma 4),and the number of incidences involving points that lie on conic components of Z(f) isO(m + nD) = O(m) (see Lemma 7). A similar bound holds for points on the regulicomponents. Specifically, we assign each point and line to a regulus that contain them,if one exists, in the same first-come first-serve manner used above. Any point p can beincident to at most two lines that are fully contained in the regulus to which it is assigned,and any other incidence of p with a line ℓ can be uniquely charged to the intersection of ℓwith that regulus, for a total (over all lines and reguli) of O(nD) incidences.

We remove all points that lie in any such component and all lines that are fully containedin any such component. With the choice of D = n2/m, we lose in the process

O(m2/3n1/3s1/3 +m+ nD) = O(m+m2/3n1/3s1/3)

incidences (recall that nD ≤ m for m ≥ n3/2). For the remainder sets, which we continue todenote as P1 and L1, respectively, no plane contains more than O(D) lines of L1, as arguedin Lemma 3.

A new polynomial partitioning. We adapt the notation used in the preceding case,with a few modifications. We choose a degree E, typically much smaller than D, andconstruct a partitioning polynomial g of degree E for P1. With an appropriate value of

15

r = Θ(E3), we obtain O(r) cells, each containing at most m/r points of P1, and each lineof L1 either crosses at most E + 1 cells, or is fully contained in Z(g).

Set P2 := P1∩Z(g) and P ′2 := P1 \P2. Similarly, denote by L2 the set of lines of L1 that

are fully contained in Z(g), and put L′2 := L1 \ L2. We first dispose of incidences involving

the lines of L2. By Lemma 4 and the preceding arguments, the number of incidencesinvolving points of P2 that lie in some planar, conic, or regulus component of Z(g), and allthe lines of L2, is

O(m2/3n1/3s1/3 +m+ nE).

We remove these points from P2, and remove all the lines of L2 that are contained in suchcomponents. Continue to denote the sets of remaining points and lines as P2 and L2. ByCorollary 9, the number of incidences between P2 and L2 is O(m+ nE3).

To complete the estimation, we need to bound the number of incidences in the cells ofthe partition, which we do inductively, as before. Specifically, for each cell τ of R3\Z(g), putPτ := P ′

2∩τ , and let Lτ denote the set of the lines of L′2 that cross τ ; put mτ = |Pτ | ≤ m/r,

and nτ = |Lτ |. Since every line ℓ ∈ L0 crosses at most 1 + E components of R3 \ Z(g), wehave

∑τ nτ ≤ n(1 + E), and, arguing as above, we may assume that each nτ is at most

n/E2, and each mτ is at most m/E3. To apply the induction hypothesis in each cell, we

therefore require thatm

E3≥( n

E2

)αj−1

. (As before, the actual sizes of P1 and L1 might be

smaller than the respective original values m and n. We use here the original values, andnote, similar to the preceding case, that the fact that these are only upper bounds on theactual sizes is harmless for the induction process.) That is, we require

E ≥(nαj−1

m

)1/(2αj−1−3)

. (9)

With these preparations, we apply the induction hypothesis within each cell τ , recallingthat no plane contains more than D lines of L′

2 ⊆ L1, and get

I(Pτ , Lτ ) ≤ Aj−1

(m1/2

τ n3/4τ +mτ

)+B

(m2/3

τ n1/3τ D1/3 + nτ

)

≤ Aj−1

((m/E3)1/2(n/E2)3/4 +m/E3

)+B

((m/E3)2/3(n/E2)1/3D1/3 + n/E2

).

Summing these bounds over the cells τ , that is, multiplying them by O(E3), we get, for asuitable absolute constant b,

I(P ′2, L

′2) =

∑

τ

I(Pτ , Lτ ) ≤ bAj−1

(m1/2n3/4 +m

)+ bB

(m2/3n1/3E1/3D1/3 + nE

).

Requiring that E ≤ m/n, the last term satisfies nE ≤ m, and the first term is also at mostO(m) (because m ≥ n3/2). The second term, after substituting D = O(n2/m), becomesO(m1/3nE1/3). Hence, with a slightly larger b, we have

I(P ′2, L

′2) ≤ bAj−1m+ bBm1/3nE1/3.

Collecting all partial bounds obtained so far, we obtain

I(P,L) ≤ c(m2/3n1/3s1/3 +m+ nE3

)+ bAj−1m+ bBm1/3nE1/3,

16

for a suitable constant c. We choose E to ensure that the two E-dependent terms aredominated by m. That is,

m1/3nE1/3 ≤ m, or E ≤ m2/n3, and nE3 ≤ m, or E ≤ m1/3/n1/3.

In addition, we also require that E ≤ m/n, but, as is easily seen, both of the aboveconstraints imply that E ≤ m/n, so we get this latter constraint for free, and ignore it inwhat follows.

As is easily checked, the second constraint E ≤ m1/3/n1/3 is stricter than the firstconstraint E ≤ m2/n3 for m ≥ n8/5, and the situation is reversed when m ≤ n8/5. So inour inductive descent of m, we first consider the second constraint, and then switch to thefirst constraint.

Hence, in the first part of this analysis, the two constraints on the choice of E are

(nαj−1

m

)1/(2αj−1−3)

≤ E ≤ m1/3

n1/3,

and, for these constraints to be compatible, we require that

(nαj−1

m

)1/(2αj−1−3)

≤ m1/3

n1/3, or m ≥ n

5αj−1−3

2αj−1 .

We start the process with α0 = 2, and take α1 :=5α0 − 3

2α0= 7/4. As this is still larger than

8/5, we perform two additional rounds of the induction, using the same constraints, leadingto the exponents

α2 =5α1 − 3

2α1=

23

14, and α3 =

5α2 − 3

2α2=

73

46<

8

5.

To play it safe, we reset α3 := 8/5, and establish the induction step for m ≥ n8/5. We canthen proceed to the second part, where the two constraints on the choice of E are

(nαj−1

m

)1/(2αj−1−3)

≤ E ≤ m2

n3,

and, for these constraints to be compatible, we require that

(nαj−1

m

)1/(2αj−1−3)

≤ m2

n3, or m ≥ n

7αj−1−9

4αj−1−5 .

We define, for j ≥ 4, αj =7αj−1 − 9

4αj−1 − 5. Substituting α3 = 8/5 we get α4 = 11/7, and in

general a simple calculation shows that

αj =3

2+

1

4j − 2,

for j ≥ 3. This sequence does indeed converge to 3/2 as j → ∞, implying that the entirerange m = Ω(n3/2) is covered by the induction.

17

In both parts, we conclude that if m ≥ nαj then the bound asserted in the theoremholds with Aj = bAj−1 + c,, and B = c. This completes the induction step.

Finally, we calibrate the dependence of the constant of proportionality on m and n, bynoting that, for nαj ≤ m < nαj−1 , the constant is O(bj). We have

3

2+

1

4j − 6= αj−1 ≥

logm

log n, or j ≤

3 logmlogn − 4

2 logmlogn − 3

=log(m3/n4

)

log (m2/n3).

(Technically, this only handles the range j ≥ 3, but, for an asymptotic bound, we canextend it to j = 1, 2 too.) This establishes the explicit expression for Am,n for this range,as stated in the theorem, and completes its proof.

Again, as in the case of a small m, we need to be careful when m approaches n3/2. Herewe can fix a j, assume that n3/2 ≤ m < nαj , and set k := m/nα′

j , where α′j = 3/2−2/(j+2)

is the j-th index in the hierarchy for m ≤ n3/2. That is,

k ≤ nαj−α′j =

1

4j − 2+

2

j + 2.

As before, we now solve k separate subproblems, each with m/k points of P and all thelines of L, and sum up the resulting incidence bounds. The analysis is similar to the oneused above, and we omit its details. It yields almost the same bound as in (7), where theslightly larger upper bound on k leads to the slightly larger bound

I(P,L) = O(2√4.5

√log b

√logn

(m1/2n3/4 +m2/3n1/3s1/3 +m+ n

)),

with a slightly different absolute constant b.

3 Discussion

In this paper we derived an asymptotically tight bound for the number of incidences betweena set P of points and a set L of lines in R3. This bound has already been established byGuth and Katz [9], where the main tool was the use of partitioning polynomials. As alreadymentioned, the main novelty here is to use two separate partitioning polynomials of differentdegrees; the one with the higher degree is used as a pruning mechanism, after which themaximum number of coplanar lines of L can be better controlled (by the degree D of thepolynomial), which is a key ingredient in making the inductive argument work.

The second main tool of Guth and Katz was the Cayley–Salmon theorem. This theoremsays that a surface in R3 of degree D cannot contain more than 11D2 − 24D lines, unless itis ruled by lines. This is an “ancient” theorem, from the 19th century, combining algebraicand differential geometry, and its re-emergenece in recent years has kindled the interest ofthe combinatorial geometry community in classical (and modern) algebraic geometry. Newproofs of the theorem were obtained (see, e.g., Terry Tao’s blog [28]), and generalizationsto higher dimensions have also been developed (see Landsberg [15]). However, the theoremonly holds over the complex field, and using it over the reals requires some care.

There is also an alternative way to bound the number of point-line incidences usingflat and singular points. However, as already remarked, these two, as well as the Cayley–Salmon machinery, are non-trivial constructs, especially in higher dimensions, and their

18

generalization to other problems in combinatorial geometry (even incidence problems withcurves other than lines or incidences with lines in higher dimensions) seem quite difficult(and are mostly open). It is therefore of considerable interest to develop alternative, moreelementary interfaces between algebraic and combinatorial geometry, which is a primarygoal of the present paper (as well as of Guth’s recent work [7]).

In this regard, one could perhaps view Lemma 5 and Corollary 9 as certain weakeranalogs of the Cayley–Salmon theorem, which are nevertheless easier to derive, withouthaving to use differential geometry. Some of the tools in Guth’s paper [7] might also beinterpreted as such weaker variants of the Cayley–Salmon theory. It would be interestingto see suitable extensions of these tools to higher dimensions.

Besides the intrinsic interest in simplifying the Guth–Katz analysis, the present workhas been motivated by our study of incidences between points and lines in four dimensions.This has begun in a year-old companion paper [21], where we have used the the polynomialpartitioning method, with a polynomial of constant degree. This, similarly to Guth’s workin three dimensions [7], has resulted in a slightly weaker bound and considerably stricterassumptions concerning the input set of lines. In a more involved follow-up study [22],we have managed to improve the bound, and to get rid of the restrictive assumptions,using two partitioning steps, with polynomials of non-constant degrees, as in the presentpaper. However, the analysis in [22] is not as simple as in the present paper, because, eventhough there are generalizations of the Cayley–Salmon theorem to higher dimensions (dueto Landsberg, as mentioned above), it turns out that a thorough investigation of the varietyof lines fully contained in a given hypersurface of non-constant degree, is a fairly intricateand challenging problem, raising many deep questions in algebraic geometry, some of whichare still unresolved.

One potential application of the techniques used in this paper, mainly the interplaybetween partitioning polynomials of different degrees, is to the problem, recently studiedby Sharir, Sheffer and Zahl [20], of bounding the number of incidences between points andcircles in R3. That paper uses a partitioning polynomial of constant degree, and, as aresult, the term that caters to incidences within lower-dimensional spaces (such as our termm2/3n1/3s1/3) does not go well through the induction mechanism, and consequently thebound derived in [20] was weaker. We believe that our technique can improve the bound of[20] in terms of this “lower-dimensional” term.

A substantial part of the present paper (half of the proof of the theorem) was devotedto the treatment of the case m > n3/2. However, under the appropriate assumptions, thenumber of points incident to at least two lines was shown by Guth and Katz [9] to bebounded by O(n3/2). A recent note by Kollar [14] gives a simplified proof, including anexplicit multiplicative constant. In his work, Kollar does not use partitioning polynomials,but employs more advanced algebraic geometric tools, like the arithmetic genus of a curve,which serves as an upper bound for the number of singular points. If we accept (pedagogi-cally) the upper bound O(n3/2) for the number of 2-rich points as a “black box”, the regimein which m > n3/2 becomes irrelevant, and can be discarded from the analysis, thus greatlysimplifying the paper.

A challenging problem is thus to find an elementary proof that the number of pointsincident to at least two lines is O(n3/2) (e.g., without the use of the Cayley–Salmon theoremor the tools used by Kollar). Another challenging (and probably harder) problem is to

19

improve the bound of Guth and Katz when the bound s on the maximum number ofmutually coplanar lines is ≪ n1/2: In their original derivation, Guth and Katz [9] considermainly the case s = n1/2, and the lower bound constrcution in [9] also has s = n1/2. Anothernatural further research direction is to find further applications of partitioning polynomialsof intermediate degrees.

References

[1] S. Basu and M. Sombra, Polynomial partitioning on varieties and point-hypersurfaceincidences in four dimensions, in arXiv:1406.2144.

[2] J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Springer Verlag, Hei-delberg, 1998.

[3] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorialcomplexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5(1990), 99–160.

[4] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction toComputational Algebraic Geometry and Commutative Algebra, Springer Verlag, Hei-delberg, 2007.

[5] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimen-sions, J. Combinat. Theory, Ser. A 118 (2011), 962–977. Also in arXiv:0905.1583.

[6] P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

[7] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, inarXiv:1404.2321.

[8] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem,Advances Math. 225 (2010), 2828–2839. Also in arXiv:0812.1043v1.

[9] L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, AnnalsMath. 181 (2015), 155–190. Also in arXiv:1011.4105.

[10] J. Harris, Algebraic Geometry: A First Course, Vol. 133. Springer-Verlag, New York,1992.

[11] R. Harshorne, Algebraic Geometry, Springer-Verlag, New York. 1983.

[12] H. Kaplan, J. Matousek, Z. Safernova and M. Sharir, Unit distances in three dimen-sions, Combinat. Probab. Comput. 21 (2012), 597–610. Also in arXiv:1107.1077.

[13] H. Kaplan, J. Matousek and M. Sharir, Simple proofs of classical theorems in dis-crete geometry via the Guth–Katz polynomial partitioning technique, Discrete Com-put. Geom. 48 (2012), 499–517. Also in arXiv:1102.5391.

[14] J. Kollar, Szemeredi–Trotter-type theorems in dimension 3, in arXiv:1405.2243.

[15] J. M. Landsberg, is a linear space contained in a submanifold? On the number ofderivatives needed to tell, J. Reine Angew. Math. 508 (1999), 53–60.

20

[16] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of Geometric Graphs(J. Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Providence,RI, 2004, pp. 185–223.

[17] O. Raz, M. Sharir, and F. De Zeeuw, Polynomials vanishing on Cartesian products:The Elekes–Szabo Theorem revisited, manuscript, 2014.

[18] J. Schmid, On the affine Bezout inequality, Manuscripta Mathematica 88(1) (1995),225–232.

[19] M. Sharir, A. Sheffer, and N. Solomon, Incidences with curves in Rd, manuscript, 2014.

[20] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between pointsand circles, Combinat. Probab. Comput., in press. Also in Proc. 29th ACM Symp. onComputational Geometry (2013), 97–106, and in arXiv:1208.0053.

[21] M. Sharir and N. Solomon, Incidences between points and lines in R4, Proc. 30th Annu.ACM Sympos. Comput. Geom., 2014, 189–197.

[22] M. Sharir and N. Solomon, Incidences between points and lines in four dimensions, inarXiv:1411.0777.

[23] M. Sharir and N. Solomon, Incidences between points and lines on a two-dimensionalvariety, manuscript, 2014.

[24] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput.Geom. 48 (2012), 255–280.

[25] L. Szekely, Crossing numbers and hard Erdos problems in discrete geometry, Combinat.Probab. Comput. 6 (1997), 353–358.

[26] E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combina-torica 3 (1983), 381–392.

[27] T. Tao, From rotating needles to stability of waves: Emerging connections betweencombinatorics, analysis, and PDE, Notices AMS 48(3) (2001), 294–303.

[28] T. Tao, The Cayley–Salmon theorem via classical differential geometry,http://terrytao.wordpress.com, March 2014.

[29] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer.Math. Soc. 133 (1968), 167–178.

[30] J. Zahl, An improved bound on the number of point-surface incidences in three dimen-sions, Contrib. Discrete Math. 8(1) (2013). Also in arXiv:1104.4987.

[31] J. Zahl, A Szemeredi-Trotter type theorem in R4, in arXiv:1203.4600.

21

5 Ramsey-type theorems for lines in3-space

129

arX

iv:1

512.

0323

6v2

[mat

h.C

O]

15 S

ep 2

016

Discrete Mathematics and Theoretical Computer Science DMTCS vol. 18:3, 2016, #14

Ramsey-Type Theorems for Lines in 3-space

Jean Cardinal1 Michael S. Payne2 Noam Solomon3

1 Universite libre de Bruxelles (ULB), Belgium2 Monash University, Melbourne, Australia3 Tel-Aviv University, Israel

received 26thJan. 2016, revised 20thAug. 2016, accepted 21stAug. 2016.

We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees onthe size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, andreguli in 3-space. Among other things, we prove the following:

• The intersection graph ofn lines inR3 has a clique or independent set of sizeΩ(n1/3).

• Every set ofn lines in R3 has a subset of√n lines that are all stabbed by one line, or a subset of

Ω((n/ log n)1/5

)such that no6-subset is stabbed by one line.

• Every set ofn lines in general position inR3 has a subset ofΩ(n2/3) lines that all lie on a regulus, or a subsetof Ω(n1/3) lines such that no4-subset is contained in a regulus.

The proofs of these statements all follow from geometric incidence bounds – such as the Guth-Katz bound on point-line incidences inR3 – combined with Turan-type results on independent sets in sparse graphs and hypergraphs. Asan intermediate step towards the third result, we also show that for a fixed family of plane algebraic curves withsdegrees of freedom, every set ofn points in the plane has a subset ofΩ(n1−1/s) points incident to a single curve, ora subset ofΩ(n1/s) points such that at mosts of them lie on a curve. Although similar Ramsey-type statements canbe proved using existing generic algebraic frameworks, thelower bounds we get are much larger than what can beobtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimedsize.

Keywords: Geometric Ramsey theory, Erdos-Hajnal property, incidence bounds

1 IntroductionRamsey theory studies the conditions under which particular discrete structures must contain certain sub-structures. Ramsey’s theorem says that for everyn, every sufficiently large graph has either a clique oran independent set of sizen. Early geometric Ramsey-type statements include the HappyEnding Prob-lem on convex quadrilaterals in planar point sets, and the Erdos-Szekeres Theorem on subsets in convexposition [9].

We prove a number of Ramsey-type statements involving linesin R3. The combinatorics of linesin space is a widely studied topic which arises in many applications such as computer graphics, motion

ISSN 1365–8050 c© 2016 by the author(s) Distributed under a Creative Commons Attribution 4.0 International License

2 Jean Cardinal, Michael S. Payne, Noam Solomon

planning, and solid modeling [4]. Our proofs combine two main ingredients: geometric information in theform of bounds on the number of incidences among the objects,and a Turan-type theorem that convertsthis information into a Ramsey-type statement. We establish a general lemma that allows us to streamlinethe proofs.

Ramsey’s Theorem for graphs and hypergraphs only guarantees the existence of rather small cliquesor independent sets. However, as discussed below, for the geometric relations we study the bounds areknown to be much larger. Therefore we are interested in finding the correct asymptotics. In particular,we are interested in theErdos-Hajnal property. A class of graphs has this property if each member withn vertices has either a clique or an independent set of sizenδ for some constantδ > 0. This comes fromtheErdos-Hajnal conjecturewhich states that, for each graphH , the family of graphs excludingH as aninduced subgraph has this property. Our results yield new Erdos-Hajnal exponents for each of the classesof (hyper)graphs studied.

The results presented here make use of important recent advances in combinatorial geometry. The keyexample is the bound on the number of incidences between points and lines inR3 given by Guth andKatz [12] in their recent solution of the Erdos distinct distances problem. Such results have sparked a lotof interest in the field, and it can be expected that further progress will yield further Ramsey-type results.

1.1 A general framework

In general we consider two classes of geometric objectsP andQ in Rd and a binary incidence relationcontained inP × Q. For a finite setP ⊆ P and a fixed integert ≥ 2, we say that at-subsetS ∈

(Pt

)

is degeneratewhenever there existsq ∈ Q such that everyp ∈ S is incident toq. Hence the incidencerelation together with the integert induces at-uniform hypergraphH = (P,E), whereE ⊆

(Pt

)is the

set of all degeneratet-subsets ofP . A clique in this hypergraph is a subsetS ⊆ P such that(St

)⊆ E.

Similarly, an independent set is a subsetS ⊆ P such that(St

)∩E = ∅.

In what follows, the familiesP andQ will mostly consist of lines or points in 3-space. We are interestedin Ramsey-type statements stating that thet-uniform hypergraphH induced by a setP ⊂ P of sizen haseither a clique of sizeω(n) or an independent set of sizeα(n).

1.2 Previous results

We first briefly survey some known results that fit into this framework. In many cases, eitherP orQ is aset of points. WhenP is a set of points, finding a large independent set amounts to finding a large subsetof points in some kind of general position defined with respect to Q. WhenQ is the set of points, we aredealing with intersections between the objects inP . In particular, the caset = 2 corresponds to the studyof geometric intersection graphs.

General position subset problems

A set inRd is usually said to be in general position whenever nod + 1 points lie on a hyperplane. Forpoints and lines in the plane, Payne and Wood proved that the Erdos-Hajnal property essentially holdswith exponent1/2 [19]. Cardinal et al. proved an analogous result inRd [3].

Theorem 1.1([19, 3]). Fix d ≥ 2. Every set ofn points inRd contains√n cohyperplanar points or

Ω((n/ logn)1/d) points in general position.

Ramsey-Type Theorems for Lines in 3-space 3

In both cases, the proofs rely on incidence bounds, in particular the Szemeredi-Trotter Theorem [24] inthe plane, and the point-hyperplane incidence bounds due toElekes and Toth [8] inRd. In this paper weformalise the technique used in those proofs in order to easily apply it to other incidence relations.

Erdos-Hajnal properties for geometric intersection graphsA survey of Erdos-Hajnal properties for geometric intersection graphs was produced by Fox and Pach [10].A general Ramsey-type statement for the case whereP is the set of plane convex sets has been known fora long time. In what follows, avertically convexset is a set whose intersection with any vertical line is aline segment.

Theorem 1.2(Larman et al. [16]). Any family ofn compact, connected and vertically convex sets in theplane contains at leastn1/5 members that are either pairwise disjoint or pairwise intersecting.

Larman et al. also showed that there exist arrangements ofk2.3219 line segments with at mostk pairwisecrossing and at mostk pairwise disjoint segments. This lower bound was improved successively byKarolyi et al. [14], and Kyncl [15].

More recently Fox and Pach studied intersection graphs of a large variety of other geometric ob-jects [11]. For example they proved the following about families of s-intersecting curves in the plane– families such that no two curves cross more thans times.

Theorem 1.3(Fox-Pach [11]). For eachǫ > 0 and positive integers, there isδ = δ(ǫ, s) > 0 such that ifG is an intersection graph of as-intersecting family ofn curves in the plane, thenG has a clique of sizeat leastnδ or an independent set of size at leastn1−ǫ.

Erdos-Hajnal properties for hypergraphs have been provedby Conlon, Fox, and Sudakov [6].

Semi-algebraic sets and relationsA very general version of the problem for the caset = 2 has been studied by Alon et al. [1]. Here Ramsey-type results are provided for intersection relations between semialgebraic sets of constant descriptioncomplexity inRd. It was shown that intersection graphs of such objects always have the Erdos-Hajnalproperty. The proof combines a linearisation technique with a space decomposition theorem due to Yaoand Yao [27]. The following general statement can be extracted from their proof.

Theorem 1.4. Consider a relationR on elements of a familyF of semi-algebraic sets of constant de-scription complexity. Suppose that each elementf ∈ F can be parameterized by a pointf∗ ∈ Rd,and that the relationR can be mapped into a semi-algebraic setR∗ in R2d. For eachg ∈ F , letΣg = f∗ ∈ Rd : (f∗, g∗) ∈ R∗. Let Q be the smallest dimension of a spaceRQ in which the de-scription ofΣg becomes linear, and letk be the number of bilinear inequalities in the definition ofR∗ inRQ. Then the graph of the relationR satisfies the Erdos-Hajnal property with exponent1/(2k(Q+ 1)).

A similar result is given for the so-calledstrongversion of the Erdos-Hajnal property: for every suchintersection relation, there exists a constantǫ and a pair of subfamiliesF1,F2 ⊆ F , each of size at leastǫ|F|, such that either every element ofF1 intersects every element ofF2, or no element ofF1 intersectsany element ofF2. The exponent for the usual Erdos-Hajnal statement is a function of thisǫ.

As an example, Alon et al. applied their machinery to prove the following result on arrangement oflines inR3.

Theorem 1.5(Alon et al. [1]). Every family ofn pairwise skew lines inR3 contains at leastk ≥ n1/6

elementsℓ1, ℓ2, . . . , ℓk such thatℓi passes aboveℓj for all i < j.

4 Jean Cardinal, Michael S. Payne, Noam Solomon

For the problems we consider, however, the exponents we obtain are significantly larger than what canbe obtained from Theorem 1.4.

A general version of this problem in which degeneratet-tuples are defined by a finite number of poly-nomial equations and inequalities of bounded description complexity has recently been studied by Conlonet al. [5]. They show that the Ramsey numbers in this general setting grow like towers of heightt − 1,and that this is asymptotically tight. Such a setting is relevant here, since we also consider Erdos-Hajnalstatements for some geometric hypergraphs.

1.3 Summary of our results

In Section 2 we give a simple lemma that allows to convert geometric incidence bounds into bounds onthe number of degenerate subsets, hence on the number of hyperedges of the hypergraphs of interest. Wealso recall the statements of the Turan bound for hypergraphs due to Spencer.

Section 3 deals with the case whereP andQ are lines and points inR3. A natural object to consider isthe intersection graph of lines inR3, for which we prove the Erdos-Hajnal property with exponent 1/3.

Theorem 3.7. The intersection graph ofn lines inR3 has a clique or independent set of sizeΩ(n1/3).

This makes use of the Guth-Katz incidence bound between points and lines inR3 [13]. We further showthat this exponent can be raised to1/2 if we consider lines in the projective 3-space. We also show howto obtain bounds on the size of independent sets fort = 3, in which a subset of lines in general position isdefined as a set of lines with no three intersecting in the samepoint.

Section 4 deals with the setting where bothP andQ are lines inR3. We prove the following theorem.

Theorem 4.1. LetL be a set ofn lines inR3. Then either there is a subset of√n lines ofL that are all

stabbed by one line, or there is a subset ofΩ((n/ logn)

1/5)

lines ofL such that no6-subset is stabbed

by one line.

The proof involves lifting the set of lines to a set of points and hyperplanes inR5, and applying theRamsey-type result on points and hyperplanes due to Cardinal et al. [3]. The latter in turn relies on apoint-hyperplane incidence bound due to Elekes and Toth [8].

Finally, in Section 5 we introduce the notion of a subset of lines in general position inR3 with respectto reguli, defined as loci of lines intersecting three pairwise skew lines. We use the Pach-Sharir bound onincidences between points and curves in the plane [18] to obtain the following result.

Theorem 5.5. LetL be a set ofn pairwise skew lines inR3. Then there areΩ(n2/3) lines on a regulus,or Ω(n1/3) lines, no 4-subset of which lie on a regulus.

We also explain how to use a line-regulus incidence bound dueto Aronov et al. [2] for an alternativeproof of this result.

The large subsets whose existence our results guarantee canbe found in polynomial time. In each case,a degeneratet-subset is incident to only one element ofQ (for example, three collinear points lie on onlyone line). Furthermore, the cliques given by our results areof a particular type: all the elements intersect asingle element ofQ (for example, a collinear set of points). Thus the largest such clique in the hypergraphH can be found in polynomial time by checking all the elements of Q that determine a degeneratet-subset (for example, all lines determined by the point set).If the clique size is small, Turan-type theoremsyield an independent set of a guaranteed minimum size. Thesetheorems are constructive, hence the largeindependent set can be found efficiently.

Ramsey-Type Theorems for Lines in 3-space 5

2 PreliminariesIn order to prove the existence of large independent sets in hypergraphs with no large clique, we proceedin two steps. First, we use incidence bounds to get upper bounds on the density of the (hyper)graph. Thenwe apply Turan’s Theorem or its hypergraph analogue to find alower bound on the size of the independentset. This is an extension of the method used to prove Theorem 1.1 in [19, 3]. The use of incidence boundsis also reminiscent from the technique used by Pach and Sharir for the repeated angle problem [17].

The following lemma will allow us to quickly convert incidence bounds into density conditions. Recallthat we consider two familiesP andQ with an incidence relation inP ×Q, and that at-subsetS of P issaid to be degenerate whenever there existsq ∈ Q such that everyp ∈ S is incident toq.

Lemma 2.1. LetP be a subset ofP with |P | = n, such that no element ofQ is incident to more thanℓelements ofP . Let us denote byP≥k the number of elements ofQ incident to at leastk elements ofP , andsupposeP≥k . g(n)/ka for some functiong and some real numbera. Then the number of degeneratet-subsets induced byP is at most

m .

g(n) if t < a,

g(n) log ℓ if t = a,

g(n)ℓt−a if t > a.

Proof: Let Pj be the number of elements ofQ incident toexactlyj elements ofP . Then

m =

ℓ∑

j=t

Pj

(j

t

)<

ℓ∑

j=1

Pjjt <

ℓ∑

j=1

Pj

(t

j∑

k=1

kt−1

)≃

ℓ∑

k=1

kt−1

ℓ∑

j=k

Pj

=ℓ∑

k=1

kt−1P≥k . g(n)ℓ∑

k=1

kt−1−a,

where we use that∑j

k=1 kt−1 = jt/t + O(jt−1), andt = O(1). The final sum simplifies differently

depending on the relative values oft anda.

We recall the statement of Turan’s Theorem.

Theorem 2.2(Turan [25]). LetG be a graph withn vertices andm edges. Thenα(G) ≥ n2mn +1

. Thus if

m < n/2 thenα(G) > n/2. Otherwiseα(G) ≥ n2/4m.

The hypergraph version of this result was proved by Spencer.

Theorem 2.3(Spencer [23]). LetH be at-uniform hypergraph withn vertices andm edges. Ifm < n/tthenα(H) > n/2. Otherwise

α(H) ≥ t− 1

tt/(t−1)

n

(m/n)1/(t−1).

3 Points and lines in R3

The recent resolution of Erdos’ distinct distance problemby Guth and Katz involves new bounds on thenumber of incidences between points and lines inR3 [12]. This breakthrough has fostered research onpoint-line incidence bounds in space. In this section and the next, we exploit those recent results to obtainvarious new Ramsey-type statements on point-line incidence relations in space.

6 Jean Cardinal, Michael S. Payne, Noam Solomon

3.1 General position with respect to linesTheorem 1.1 ford = 2 states that in a setP of n points in the plane there exist either

√n collinear

points, orΩ(√n/ logn) points with no three collinear. Payne and Wood [19] conjectured that the true

size should beΩ(√n), but this small improvement has proven elusive.

Here we consider the same question but withP = R3, Q defined as the set of lines inR3, andt = 3.Hence we consider that a setP ⊂ R3 is in general position when no three points are collinear. Sofarthis is the same question as in the planar case, since a point set in higher dimensional space can alwaysbe projected to the plane in a way that maintains the collinearity relation. However, under a small extraassumption, namely that among then points inR3, at mostn/ logn are coplanar, we are able to removethe logn factor in the independent set. This sheds some light on the nature of potential counterexamplesto the conjecture of Payne and Wood.

We will use the following result of Dvir and Gopi [7], which isdeduced from Guth and Katz [13].

Theorem 3.1. Given a setP of n points inR3, such that at mosts points are contained in a plane, thenumberP≥k of lines containing at leastk points is

P≥k . n2

k4+

ns

k3+

n

k.

Theorem 3.2. Any set ofn points inR3 such that at mostn/ logn of the points lie in a plane containseither

√n collinear points orΩ(

√n) with no three collinear.

Proof: We apply Lemma 2.1 on each term of the bound in Theorem 3.1. We obtain that the number ofdegenerate 3-subsets of points is

m . n2 + ns log ℓ+ nℓ2,

whereℓ =√n ands = n/ logn. Hence the dominating term isn2. Applying Theorem 2.3 yields an

independent set of sizeΩ(√n).

In fact, this theorem holds inRd for d > 3. To see this, we take a generic projection ofRd ontoR3.The condition that at mostn/ logn lines are coplanar remains true under a generic projection.

3.2 Line intersection graphs in R3

We now consider the setting in which the familyP is the set of lines inR3 andQ = R3. The first subcasewe consider ist = 2, or in other words, intersection graphs of lines. Note that in an intersection graphof lines inR3, every clique of sizek ≥ 2 corresponds either to a subset ofk lines having a commonintersection point, or to a subset ofk lines lying in a plane. However,k lines lying in a plane do not forma clique if some of them are parallel.

We consider a setL of n lines inR3, such that no more thanℓ lines intersect in a point, and at mosts lines lie in a common plane or aregulus. We recall that a regulus is a degree two algebraic surface,which is the union of all the lines inR3 that intersect three pairwise-skew lines inR3. It is adoubly-ruledsurface; each point on a regulus is incident to precisely twolines fully contained in the regulus. Moreover,there are tworulings for the regulus; every line from one ruling intersects everyline from the other ruling,and does not intersect any line from the same ruling.

We first recall two important theorems of Guth and Katz [13]. In what follows,P≥k denotes the numberof points incident to at leastk lines inL.

Ramsey-Type Theorems for Lines in 3-space 7

Theorem 3.3([13, Theorem 4.5]). If L is a set ofn lines, so that no plane contains more thans lines,then fork ≥ 3 we have

P≥k . n3/2

k2+

ns

k3+

n

k.

Theorem 3.4([13, Theorem 2.11],[21]). If L is a set ofn lines, so that no plane or regulus contains morethans lines, thenP≥2 . n3/2 + ns.

Note the difference between the two statements: the assumption that no regulus contains more thanslines is required for the casek = 2 only.

Applying Lemma 2.1 to the bounds in Theorems 3.3 and 3.4 yields the following.

Proposition 3.5. Given a setL of n lines, so that no plane or regulus contains more thans lines, and nopoint is incident to more thanℓ lines ofL, the number of line-line incidences isO(n3/2 log ℓ+ ns+ nℓ).

Lemma 3.6. Consider a setL of n lines inR3, such that no plane contains more thans lines, and nopoint is incident to more thanℓ lines ofL. Let G be the intersection graphL. If s, ℓ . n1/2, thenα(G) & √

n/ log ℓ. Moreover, ifr := maxs, ℓ & n12+ǫ for someǫ > 0, thenα(G) & n/r.

Proof: If there is some regulus containing at leastn1/2 lines, we divide the lines into the two rulings ofthe regulus. One ruling contains at least half the lines, andas the lines in one ruling do not intersect oneanother, it follows thatα(G) & n1/2. We may therefore assume that the number of lines contained in acommon regulus is at mostn1/2.

If s, ℓ ≤ n1/2, the first term in the bound in Proposition 3.5 dominates, andapplying Theorem 2.2 givesα(G) & √

n/ log ℓ. If r ≥ n12+ǫ, one of the latter terms dominates, and we apply Theorem 2.2 to get

α(G) & n/r.

Theorem 3.7. The intersection graph ofn lines inR3 has a clique or independent set of sizeΩ(n1/3).

Proof: Suppose that such a graphG hasα(G) ≪ n1/3. Then by Lemma 3.6,maxs, ℓ & n2/3. Ifℓ & n2/3 we are done, sos & n2/3. Therefore, we may assume that there is a plane containingn2/3 lines.Divide these lines into classes of pairwise parallel lines.If some class contains at leastn1/3 lines, we haveα(G) & n1/3. Otherwise, there are at leastn1/3 different parallel classes. Choosing one line from eachclass yields a clique of sizen1/3.

Note that the Erdos-Hajnal property for intersection graphs of lines inR3 can be directly establishedfrom Theorem 1.4 by Alon et al. [1], but with a much smaller exponent. In their setting, we can representthe intersection relation between lines using Plucker coordinates inR5, and using two inequalities. Thisyields k = 2 andQ = 5, and an Erdos-Hajnal exponent of1/24. Although it is likely that it can beimproved by shortcutting steps in the general proof, any exponent we would get would still be far from1/3.

We now make a connection with intersection graphs of lines inspace and line graphs. Recall that theline graph of a graphG has the set of edgesE(G) as vertex set, and an edge between two edges ofGwhenever they are incident to the same vertex ofG. Observe that for every graphG, the line graph ofG can be represented as the intersection graph of lines inR3 by drawingG on a vertex set in generalenough position inR3, and extending the edges of the drawing to lines. By applyingVizing’s Theorem,which says that the edge chromatic number of every graph is atmost∆ + 1, we may see that the class

8 Jean Cardinal, Michael S. Payne, Noam Solomon

of line graphs has the Erdos–Hajnal property with exponent1/2. The question of the exact Erdos–Hajnalexponent for intersection graphs of lines inR3 remains open – it lies somewhere between1/3 and1/2.

Finally we note that for sets of lines in projective space, coplanar sets of lines always form a clique.The following stronger result can be directly obtained.

Theorem 3.8. For every intersection graphG of n lines in P3, either ω(G) ≥ √n or α(G) =

Ω(√n/ logn).

Hence intersection graphs of lines in the projective plane satisfy the Erdos-Hajnal property with expo-nent roughly1/2.

3.3 Independent Sets of Lines for t = 3

We now consider the case in whichP is the set of lines inR3, Q = R3 andt = 3. This can be seen as akind of three-dimensional version of the dual of the result of Payne and Wood (Theorem 1.1 withd = 2).

Theorem 3.9. Consider a collectionL of n lines inR3, such that at mosts lie in a plane, withs ≤n/ logn. Then there exists a point incident to

√n lines, or a subset ofΩ(

√n) lines such that at most two

intersect in one point.

Proof: We letℓ be the largest number of lines intersecting in one point, andsupposeℓ <√n. Applying

Lemma 2.1 and Theorem 3.3, we get that the number of triples sharing a point is at most

m . ℓn3/2 + ns log ℓ+ nℓ2 . n2.

Then by Theorem 2.3 we have an independent set of sizeΩ(√n).

If the above theorem is stated with dependence onℓ, we getΩ(n3/4/√ℓ). If s is allowed to be as large

asn, we are back in the dual of general position subset selection, and we getΩ(√n/ logn), the same as

Theorem 1.1.

4 Stabbing lines in R3

Three lines inR3 are typically intersected by a fourth line, except in certain degenerate cases. Thus itmakes sense to study configurations of lines inR3, and to consider a set of4 or more lines degenerate ifall its elements are intersected by another line. Here we provide a result for6-tuples of lines.

We define a 6-tuple of lines to be degenerate if all six lines are intersected (or “stabbed”) by a singleline inR3. We call this line astabbing linefor the6-tuple of lines. So in our framework this is the settingin which bothP andQ are the set of lines inR3, andt = 6.

We make use of the Plucker coordinates and coefficients for lines inR3, which are a common tool fordealing with incidences between lines, see e.g. Sharir [20]. Leta = (a0 : a1 : a2 : a3), b = (b0 : b1 : b2 :b3) be two points on a lineℓ, given in projective coordinates. By definition, the Plucker coordinates ofℓare given by

(π01 : π02 : π12 : π03 : π13 : π23) ∈ P5,

whereπij = aibj − ajbi for 0 ≤ i < j ≤ 3. Similarly, the Plucker coefficients ofℓ are given by

(π23 : −π13 : π03 : π12 : −π02 : π01) ∈ P5,

Ramsey-Type Theorems for Lines in 3-space 9

i.e., these are the Plucker coordinates written in reverseorder with two signs flipped. The importantproperty is that two linesℓ1 andℓ2 are incident if and only if the Plucker coordinates ofℓ1 lie on thehyperplane defined by the Plucker coefficients ofℓ2 and vice versa. Therefore, we defineP , andQ tobe the points inP5 defined by the Plucker coordinates of the lines inL, and the hyperplanes defined bythe Plucker coefficients of the lines inR3, respectively. The incidence relation between points inP andhyperplanes inQ is the standard incidence relation between points and hyperplanes. The integert is setto 6, and a 6-tuple of points inP is degenerate whenever there is a hyperplane inQ which is incident toall six points in the 6-tuple.

We prove the following Ramsey-type result for stabbing lines inR3.

Theorem 4.1. LetL be a set ofn lines inR3. Then either there is a subset of√n lines ofL that are all

stabbed by one line, or there is a subset ofΩ((n/ logn)1/5

)lines ofL such that no6-subset is stabbed

by one line.

Theorem 4.1 is an immediate consequence of the following generalisation of Theorem 1.1. The differ-ence is that the set of hyperplanesH is arbitrary instead of being the set of all hyperplanes inRd.

Theorem 4.2. LetH be a set of hyperplanes inRd. Then, every set ofn points inRd with at mostℓ points

on any hyperplane inH, whereℓ = O(n1/2), contains a subset ofΩ((n/ log ℓ)

1/d)

points so that every

hyperplane inH contains at mostd of these points.

Theorem 4.2, withd = 5, applied to the points and hyperplanes given by the Pluckercoordinates andcoefficients, implies Theorem 4.1. Theorem 4.2 follows fromthe following generalized version of Lemma4.5 of Cardinal et al. [3].

Lemma 4.3. Fix d ≥ 2 and a setH of hyperplanes inRd. LetP be a set ofn points inRd with no morethan l points in a hyperplane inH, for somel = O(n1/2). Then, the number of(d + 1)-tuples inP thatlie in a hyperplane inH isO(nd log l).

The difference between this lemma and the original version in [3] is that the set of hyperplanesH isarbitrary, rather than being the set of all hyperplanes. Theproof is similar to that of Cardinal et al., and isgiven in Appendix A.

The following result provides a simple upper bound.

Theorem 4.4. For every constant integert ≥ 4, there exists an arrangementL of n lines inR3 such thatthere is no subset of more thanO(

√n) lines that are all stabbed by one line, nor any subset of more than

O(√n) lines with not stabbed by one line.

Proof: ConstructL as follows: pick√n parallel planes, each containing

√n lines, with no three inter-

secting and no two parallel. Consider a subset stabbed by oneline. Either it has three coplanar lines; thenit must be fully contained in one of the planes and contains atmost

√n lines; or it has no three coplanar

lines, hence contains at most two lines from each plane, and has at most2√n lines. Now consider a subset

such that not lines are stabbed by one. Then it contains at mostt − 1 lines from each plane, and has atmost(t− 1)

√n lines.

10 Jean Cardinal, Michael S. Payne, Noam Solomon

5 Lines and reguli in R3

Consider the case in whichP is the class of lines inR3, Q is the class of reguli, andt = 4. Let P bea set ofn lines, and assume that the lines inP are pairwise skew. Every triple of lines inP thereforedetermines a single regulus, and we may consider the set of all reguli determined byP . We consider thecontainment relation rather than intersection – we are interested in4-tuples that all lie in the same regulus.In order to prove our result, we first reformulate previouslyknown incidence bounds between points andcurves in the plane.

5.1 General position with respect to algebraic curvesWe first consider the case whereP = R2 andQ is a family of algebraic curves of bounded degree. Wedefine the number of degrees of freedom of a family of algebraic curvesC to be the minimum valuessuch that for anys points inR2 there are at mostc curves passing through all of them, for some constantc. Moreover,C has multiplicity typer if any two curves inC intersect in at mostr points. We consider aset of points to be in general position with respect toC when nos+ 1 points lie on a curve inC.

It is possible to extract Ramsey-type statements for this situation directly from Theorem 1.1 via lineari-sation. For example, let us consider the special case of circles, wheres = 3. Given a set of points in theplane, we can lift it onto a paraboloid inR3 in such a way that a subset of the original set lies on a circle(possibly degenerated into a line) if and only if the corresponding lifted points lie on a hyperplane inR3.By applying Theorem 1.1 on the lifted set, we can show that there exists a subset of

√n points incident

to a circle, or a subset ofΩ((n/ logn)1/3) points such that at most three of them lie on a circle. We showhow we can improve on this.

In order to apply our technique, we need Szemeredi-Trotter-type bounds on the number of incidencesbetween points and curves. This has been studied by Pach and Sharir [18].

Theorem 5.1([18]). Let P be a set ofn points in the plane and letC be a set ofm bounded degreeplane algebraic curves withs degrees of freedom and multiplicity typer. Then the number of point-curveincidences is at most

I(P, C) ≤ C(r, s)(ns/(2s−1)m(2s−2)/(2s−1) + n+m

)

whereC(r, s) is a constant depending only onr ands.

Pach and Sharir proved Theorem 5.1 for simple curves withs degrees of freedom and multiplicitytyper. It is well known that one may replace simple curves with bounded degree algebraic curves, sincesuch curves may be cut into a constant number of simple pieces. Note that a set of bounded degreealgebraic curves has constant multiplicity type if no two curves share a common component. Wang etal. [26] recently proved another result for incidences between points and algebraic curves, though for ourpurposes Theorem 5.1 is stronger.

Theorem 5.2. Consider a familyC of bounded degree algebraic curves inR2 with constant multiplicitytype ands degrees of freedom, for somes > 2. Then in any set ofn points inR2, there exists a subsetof Ω(n1−1/s) points incident to a single curve ofC, or a subset ofΩ(n1/s) points such that at mosts ofthem lie on a curve ofC.

Proof: Sett = s + 1 and count the number of degeneratet-subsets. We denote byP≥k the number ofcurves ofC containing at leastk points ofP . A direct corollary of Theorem 5.1 is that, for values ofk

Ramsey-Type Theorems for Lines in 3-space 11

larger than some constant,

P≥k . ns

k2s−1+

n

k.

On the other hand, for smaller values ofk, the trivial boundP≥k . ns holds since for anys points, thereare at most a constant number of curves passing through all ofthem. Suppose now that no curve containsmore thanℓ . n1−1/s points ofP . Sinces > 2, it follows thatt < 2s− 1. Using Lemma 2.1, we deducethat the number of degeneratet-subsets is

m . ns + nℓs . ns.

Thus by Theorem 2.3 there exists an independent set of size atleast

t− 1

tt/(t−1)

n

(m/n)1/(t−1)= Ω(n1/s).

As an example, we can instantiate the result as follows for circles in the plane.

Corollary 5.3. In any set ofn points inR2, there exists a subset ofΩ(n2/3) points incident to a circle, ora subset ofΩ(n1/3) points such that no four of them lie on a circle.

Using the standard point-line duality, Theorem 1.1 states that for every arrangement ofn lines inR2,either there exists a point contained in

√n lines, or there exists a set ofΩ((n/ logn)1/2) lines inducing

a simple arrangement, that is, such that no point belongs to more than two lines. We provide a similardual version of Theorem 5.2. This corresponds to the case whereP is a family of algebraic curves withsdegrees of freedom,Q = R2, andt = 3. As mentioned before, the caset = 2, or intersection graphs, hasbeen studied previously [10, 11]. The proof is very similar to that of Theorem 5.2 and omitted.

Theorem 5.4. Consider a familyC of bounded degree algebraic curves inR2 with constant multiplicitytype ands degrees of freedom, for somes > 2. Then in any arrangementC ofm such curves, there existsa subset ofΩ(m1−1/s) curves intersecting in one point, or a subset ofΩ(m1/s) curves inducing a simplesubarrangement, that is, such that at most two intersect in one point.

5.2 Ramsey-type results for lines and reguli in R3

We now come back to our original problem in whichP is the class of lines inR3, Q is the class ofreguli, andt = 4. Here we restrict the finite arrangementP ⊂ P to be pairwise skew, that is, pairwisenonintersecting and nonparallel. We also consider the containment relation, that is,ℓ ∈ P is incident toR ∈ Q if it is fully contained in it.

Recall that a regulus can be defined as a quadratic ruled surface which is the locus of all lines that areincident to three lines in general position. This surface isa doubly ruledsurface, that is, every point ona regulus is incident to precisely two lines fully containedin it. There are only two kinds of reguli, bothof which are quadrics – hyperbolic paraboloids and hyperboloids of one sheet; see for instance Sharir andSolomon [22] for more details.

Theorem 5.5. LetL be a set ofn pairwise skew lines inR3. Then there areΩ(n2/3) lines on a regulus,or Ω(n1/3) lines, no 4-subset of which lie on a regulus.

12 Jean Cardinal, Michael S. Payne, Noam Solomon

Proof: We map the lines inL to a setP of points inR4. This can be done for instance by associating witheach line thex- andy-coordinates of the two points of intersection with the planesz = 0 andz = 1. (Wemay assume no line is parallel to these planes). Under this mapping, a ruling of a regulus correspondsto an algebraic curve inR4. Let C be the finite set of all curves corresponding to a ruling of a regulusdetermined by three lines inL. Note thatanytriple of points inR4 is contained in at most one such curve,because three lines inR3 lie in at most one ruling of one regulus. (A pair of parallel orintersecting linesare not contained in a ruling of any regulus, even though theyare contained in many reguli).

Apply a generic projectionπ from R4 to R2, and consider the arrangement of pointsP ′ = π(P ) to-gether with the set of projected curvesC′ = π(C). Such a projection preserves the incidences betweenpoints and curves inR4, and only creates new intersections between pairs of curves(i.e. ‘simple’ cross-ings). Three or more curves inC′ intersect in a point if and only if their preimages inC intersect in apoint.

The set of curvesC′ has three degrees of freedom, since for any three points inR2 there are at most twocurves passing through all of them. Otherwise, if three curves pass through three points, the correspondingcurves inC also intersect in three points inR4, a contradiction.

Moreover, the curves inC′ are algebraic of bounded degree, do not share common components, andthus have constant multiplicity type. Applying Theorem 5.2with s = 3, we obtain that there areΩ(n2/3)points ofπ(P ) on one curve, orΩ(n1/3) points ofπ(P ), no four of which lie on a curve. The resultfollows.

The bounds can be shown to be tight in the following sense.

Theorem 5.6. There exists a setP of n pairwise skew lines inR3 such that there is no subset of morethanO(n2/3) lines on a regulus, and no more thanO(n1/3) lines such that no 4-subset lie on a regulus.

Proof: The setP is constructed as follows: take a set ofn1/3 distinct reguli, and for each regulus taken2/3 lines in one of its rulings, givingn pairwise skew lines. Consider a subset ofP contained in aregulus. Either it is one of the chosen reguli, and it contains at mostn2/3 lines, or it contains at most twolines from each regulus, and has size at most2n1/3. On the other hand, consider a subset of lines with nofour on a regulus. It can contain at most three lines from eachchosen regulus, and therefore has size atmost3n1/3.

Alternative proof. Aronov et al. [2] proved the following bound on the number of incidences betweenlines and reguli in 3-space.

Theorem 5.7(Aronov et al.[2]). LetL be a set ofn lines inR3, and letR be a set ofm reguli inR3. Thenthe number of incidences between the lines ofL and the reguli ofR isO(n4/7m17/21+n2/3m2/3+m+n).

From this bound, one may derive an alternative proof of Theorem 5.5, of which we now give a briefsketch. First boundP≥k, defined as the number of reguli containing at leastk lines. From the aboveTheorem, we getP≥k . n3/k21/4 + n2/k3 + n/k. Then from Lemma 2.1 we know that if no reguluscontains more thanℓ lines, then the number of degenerate 4-tuples of lines ism . n3+n2ℓ+nℓ3. Henceeitherℓ is larger thann2/3, orm . n3 and from Theorem 2.3 there exists an independent set of linesofsizeΩ(n1/3).

Ramsey-Type Theorems for Lines in 3-space 13

AcknowledgmentsThe authors wish to thank the reviewers for their comments, including those on earlier, preliminary ver-sions of this paper.

References[1] Noga Alon, Janos Pach, Rom Pinchasi, Rados Radoicic, and Micha Sharir. Crossing patterns of

semi-algebraic sets.J. Comb. Theory, Ser. A, 111(2):310–326, 2005.

[2] Boris Aronov, Vladlen Koltun, and Micha Sharir. Incidences between points and circles in three andhigher dimensions.Discrete & Computational Geometry, 33(2):185–206, 2005.

[3] Jean Cardinal, Csaba D. Toth, and David R. Wood. GeneralPosition Subsets and IndependentHyperplanes in d-Space.ArXiv e-prints, 2014, 1410.3637. To appear inJournal of Geometry.

[4] Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Jorge Stolfi. Linesin space: Combinatorics and algorithms.Algorithmica, 15(5):428–447, 1996.

[5] David Conlon, Jacob Fox, Janos Pach, Benny Sudakov, andAndrew Suk. Ramsey-type results forsemi-algebraic relations. InProc. Symposium on Computational Geometry (SoCG), pages 309–318,2013.

[6] David Conlon, Jacob Fox, and Benny Sudakov. Erdos-Hajnal-type theorems in hypergraphs.J.Comb. Theory, Ser. B, 102(5):1142–1154, 2012.

[7] Zeev Dvir and Sivakanth Gopi. On the number of rich lines in truly high dimensional sets. InProc.31st International Symposium on Computational Geometry (SoCG), 2015.

[8] Gyorgy Elekes and Csaba D. Toth. Incidences of not-too-degenerate hyperplanes. InProceedingsof the 21st ACM Symposium on Computational Geometry (SoCG), pages 16–21, 2005.

[9] Paul Erdos and George Szekeres. A combinatorial problem in geometry.Compositio Mathematica,2:463–470, 1935.

[10] Jacob Fox and Janos Pach. Erdos-Hajnal-type resultson intersection patterns of geometric objects.Bolyai Society Mathematical Studies – Horizon of Combinatorics, 17:79–103, 2008.

[11] Jacob Fox and Janos Pach. Coloring Kk-free intersection graphs of geometric objects in the plane.Eur. J. Comb., 33(5):853–866, 2012.

[12] Larry Guth and Nets H. Katz. Algebraic methods in discrete analogs of the Kakeya problem.Ad-vances Math., 225:2828–2839, 2010.

[13] Larry Guth and Nets H. Katz. On the Erdos distinct distances problem in the plane.Annals Math.,181:155–190, 2015.

[14] Gyula Karolyi, Janos Pach, and Geza Toth. Ramsey-type results for geometric graphs, I.Discrete& Computational Geometry, 18(3):247–255, 1997.

14 Jean Cardinal, Michael S. Payne, Noam Solomon

[15] Jan Kyncl. Ramsey-type constructions for arrangements of segments.Eur. J. Comb., 33(3):336–339,2012.

[16] David Larman, Jiri Matousek, Janos Pach, and Jeno Torocsik. A Ramsey-type result for convex sets.Bull. London Math. Soc., 26(2):132–136, 1994.

[17] Janos Pach and Micha Sharir. Repeated angles in the plane and related problems.J. Comb. Theory,Ser. A, 59(1):12–22, 1992.

[18] Janos Pach and Micha Sharir. On the number of incidences between points and curves.Combina-torics, Probability & Computing, 7(1):121–127, 1998.

[19] Michael S. Payne and David R. Wood. On the general position subset selection problem.SIAM J.Discrete Math., 27(4):1727–1733, 2013.

[20] Micha Sharir. On joints in arrangements of lines in space and related problems.J. Comb. Theory,Ser. A, 67.1:89–99, 1994.

[21] Micha Sharir and Noam Solomon. Incidences between points and lines in three dimensions.ArXive-prints, 2015, 1501.02544.

[22] Micha Sharir and Noam Solomon. Incidences between points and lines on a two-dimensional variety.ArXiv e-prints, 2015, 1502.01670.

[23] Joel Spencer. Turan’s theorem for k-graphs.Discrete Mathematics, 2(2):183 – 186, 1972.

[24] Endre Szemeredi and William T. Trotter. Extremal problems in discrete geometry.Combinatorica,3(3):381–392, 1983.

[25] Paul Turan. On an extremal problem in graph theory.Mat. Fiz. Lapok, 48:436–452, 1941.

[26] Hong Wang, Ben Yang, and Ruixiang Zhang. Bounds of incidences between points and algebraiccurves.ArXiv e-prints, 2013, 1308.0861.

[27] Andrew Chi-Chih Yao and F. Frances Yao. A general approach to d-dimensional geometric queries(extended abstract). InProceedings of the 17th Annual ACM Symposium on Theory of Computing(STOC), pages 163–168, 1985.

A Proof of Lemma 4.3For the proof we need the following observation regarding generic projection maps.

Lemma A.1. LetP be a finite set of points inRd, and letA be a finite set of(d − 2)-flats inRd. Letπbe a generic projection fromRd to a hyperplane. Then a pointp ∈ P lies on a(d− 2)-flatA ∈ A if andonly if π(p) ∈ π(A).

Proof: The forward implication is clear. For the other direction, supposep /∈ A. Then the affine span ofp∪A is a hyperplane, that is, it is(d− 1)-dimensional. By the genericity ofπ, the imageπ(span(p∪A)) must also be(d− 1)-dimensional, soπ(p) /∈ π(A).

We also need the following result of Elekes and Toth [8]. Given a point setP , a hyperplaneh is said tobeγ-degenerateif at mostγ|P ∩ h| points ofP ∩ h lie on a(d− 2)-flat.

Ramsey-Type Theorems for Lines in 3-space 15

Theorem A.2. For everyd ≥ 3 there exist constantsCd > 0 andγd > 0 such that for every set ofnpoints inRd, the numberh≥k of γd-degenerate hyperplanes containing at leastk points ofP is at most

Cd

(nd

kd+1+

nd−1

kd−1

).

For convenience we restate Lemma 4.3.

Lemma 4.3. Fix d ≥ 2 and a setH of hyperplanes inRd. LetP be a set ofn points inRd with no morethanℓ points in a hyperplane inH, for someℓ = O(n1/2). Then, the number of(d+ 1)-tuples inP thatlie in a hyperplane inH isO(nd log ℓ).

Proof: The proof is an adaptation of the proof of Lemma 4.5 in Cardinal et al. [3]. It proceeds by inductionond ≥ 2. The base case isd = 2. We wish to bound the number of triples of points ofP , lying on a linein H. Let hk (resp.,h≥k) denote the number of lines ofH containing exactly (resp., at least)k points ofP . The number of triples of points lying on a line ofH is

∑ℓk=3 hk

(k3

)≤∑ℓ

k=3 k2h≥k

.∑ℓ

k=3 k2(

n2

k3 + nk

). n2 log ℓ+ ℓ2n . n2 log ℓ,

(1)

whereh≥k . n2

k3 + nk follows by the Szemeredi-Trotter Theorem [24].

We now consider the general cased ≥ 3. LetP be a set ofn points inRd, with no more thanℓ points ina hyperplane inH, whereH is a given set of hyperplanes inRd, andℓ = O(n1/2). Letγ := γd > 0 be theconstant specified in Theorem A.2. We distinguish between the following three types of(d+ 1)-tuples:Type 1: (d + 1)-tuples of P contained in a (d − 2)-flat in a hyperplane in H. Let F be the set of(d − 2)-flats that are contained in some hyperplane inH and spanned by the pointsP . Let sk denote thenumber of flats inF that contain exactlyk points ofP . We projectP onto a(d− 1)-flatK via a genericprojectionπ to obtain a set of pointsP ′ := π(P ) in Rd−1. Let H′ be the set of hyperplanesπ(Γ) foreachΓ ∈ F . By Lemma A.1,|P ∩ Γ| = |P ′ ∩ π(Γ)| for eachΓ ∈ F . Thussk is also the number ofhyperplanes inH′ containingk points ofP ′. Moreover, the hyperplanes inH′ contain at mostℓ points ofP ′.

Applying the induction hypothesis onP ′ with respect toH′ we deduce that the number ofd-tuples inP ′ that lie in a hyperplane inH′ is

ℓ∑

k=d

sk

(k

d

). nd−1 log ℓ.

Therefore, the number of(d+ 1)-tuples ofP lying on a(d− 2)-flat in F is

ℓ∑

k=d+1

sk

(k

d+ 1

)≤

ℓ∑

k=d+1

ksk

(k

d

). ℓnd−1 log ℓ . nd log ℓ.

16 Jean Cardinal, Michael S. Payne, Noam Solomon

Type 2: (d + 1)-tuples ofP that span aγ-degenerate hyperplane inH. Let hk denote the number ofγ-degenerate hyperplanes inH containing exactlyk points ofP . Using Theorem A.2, we get

∑ℓk=d+1 hk

(k

d+1

)≤∑ℓ

k=d+1 kdh≥k

.∑ℓ

k=d+1 kd(

nd

kd+1 + nd−1

kd−1

). nd log ℓ+ ℓ2nd−1 . nd log ℓ.

(2)

Type 3: (d + 1)-tuples of P that span a hyperplane inH that is not γ-degenerate.Recall that if ahyperplaneH spanned byP is not γ-degenerate, then more than aγ fraction of its points lie in some(d−2)-flat. Consider a(d−2)-flatL containing exactlyk points ofP . A point inP \L can be on at mostone hyperplane containingL. Letnr denote the number of hyperplanes inH containingL and exactlyrpoints ofP \ L. Then

∑r nrr ≤ n, and by assumption on the hyperplanes inH, we haver ≤ ℓ.

We will assign each tuple of Type 3 to a(d − 2)-flat that causes it to be Type 3. Fix a(d − 2)-flat Lwith k points and consider a hyperplaneH ∈ H that is notγ-degenerate because it containsL. That is,supposeH containsr + k points, andk > γ(r + k), sor < O(k). All tuples that span H contain at leastone point not inL. Hence the number of tuples that spanH is O(rkd). Assign these tuples toL. Thetotal number of tuples of Type 3 that will be assigned toL in this way is therefore at most

O

(∑

r

nrrkd

). nkd.

Let F be the set of(d − 2)-flats that have at least one Type 3 tuple assigned to them. Thus F is afinite set. Letsk denote the number of flats inF that contain exactlyk points ofP . We projectP onto a(d− 1)-flat K via a generic projectionπ to obtain a set of pointsP ′ := π(P ) in Rd−1. LetH′ be the setof hyperplanesπ(Γ) for eachΓ ∈ F . By Lemma A.1,|P ∩ Γ| = |P ′ ∩ π(Γ)| for eachΓ ∈ F . Thussk isalso the number of hyperplanes inH′ containingk points ofP ′. Moreover, the hyperplanes inH′ containat mostℓ points ofP ′. Applying the induction hypothesis onP ′ with respect toH′ we deduce that thenumber ofd-tuples inP ′ that lie in a hyperplane inH′ is

ℓ∑

k=d

sk

(k

d

). nd−1 log ℓ.

Moreover,∑d−1

k=1 skkd . nd−1. Therefore, the number of(d+ 1)-tuples of Type 3 is at most

ℓ∑

k=1

sknkd ≤ n

ℓ∑

k=1

skkd . nd log ℓ.

Summing over all three cases, the proof is complete.

146

Part III

Incidences between points and lineson varieties

147

6 Incidences between points and lineson two- and three-dimensional va-rieties

149

Incidences between points and lines on two- and

three-dimensional varieties∗

Micha Sharir† Noam Solomon‡

August 31, 2017

Abstract

Let P be a set of m points and L a set of n lines in R4, such that the points of P lieon an algebraic three-dimensional variety of degree D that does not contain hyperplaneor quadric1 components, and no 2-flat contains more than s lines of L. We show thatthe number of incidences between P and L is

I(P,L) = O(m1/2n1/2D +m2/3n1/3s1/3 + nD +m

),

for some absolute constant of proportionality. This significantly improves the bound ofthe authors [37], for arbitrary sets of points and lines in R4, when D is not too large.Moreover, when D and s are constant, we get a linear bound. The same bound holdswhen the three-dimensional surface is embedded in any higher-dimensional space.

The bound extends (with a slight deterioration, when D is large) to the complexfield too. For a complex three-dimensional variety, of degree D, embedded in C4 (or inany higher-dimensional Cd), under the same assumptions as above, we have

I(P,L) = O(m1/2n1/2D +m2/3n1/3s1/3 +D6 + nD +m

).

For the proof of these bounds, we revisit certain parts of [37], combined with thefollowing new incidence bound, for which we present a direct and fairly simple proof.Going back to the real case, let P be a set of m points and L a set of n lines in Rd, ford ≥ 3, which lie in a common two-dimensional algebraic surface of degree D that doesnot contain any 2-flat, so that no 2-flat contains more than s lines of L (here we requirethat the lines of L also be contained in the surface). Then the number of incidencesbetween P and L is

I(P,L) = O(m1/2n1/2D1/2 +m2/3D2/3s1/3 +m+ n

).

∗Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the IsraelScience Foundation. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–IsraelBinational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (CenterNo. 4/11), by the Blavatnik Computer Science Research Fund at Tel Aviv University, and by the HermannMinkowski-MINERVA Center for Geometry at Tel Aviv University. An earlier version of the paper, whichonly contains some of the results and in a weaker form, is: M. Sharir and N. Solomon, Incidences betweenpoints and lines on a two-dimensional variety, in arXiv:1501.01670.

†School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@post.tau.ac.il‡School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com1A quadric is an algebraic variety of degree two.

1

When d = 3, this improves the bound of Guth and Katz [15] for this special case, whenD ≪ n1/2. Moreover, the bound does not involve the term O(nD). This term arises inmost standard approaches, and its removal is a significant aspect of our result. Again,the bound is linear when D = O(1).

This bound too extends (with a slight deterioration, when D is large) to the complexfield. For a complex two-dimensional variety, of degree D, when the ambient space isC3 (or any higher-dimensional Cd), under the same assumptions as above, we have

I(P,L) = O(m1/2n1/2D1/2 +m2/3D2/3s1/3 +D3 +m+ n

).

These new incidence bounds are among the very few bounds, known so far, that holdover the complex field. The bound for two-dimensional (resp., three-dimensional) vari-eties coincides with the bound in the real case whenD = O(m1/3) (resp., D = O(m1/6)).

1 Introduction

Let P be a set of m points and L a set of n lines in Rd or in Cd. Let I(P,L) denote thenumber of incidences between the points of P and the lines of L; that is, the number ofpairs (p, ℓ) with p ∈ P , ℓ ∈ L, and p ∈ ℓ. If all the points of P and all the lines of L lie ina common 2-flat, then, in the real case, the classical Szemeredi–Trotter theorem [44] yieldsthe worst-case tight bound

I(P,L) = O(m2/3n2/3 +m+ n

). (1)

The same bound also holds in the complex plane, as shown later by Toth [45] and Zahl [47].

This bound clearly also holds in Rd and in Cd, for any d, by projecting the given linesand points onto some generic 2-flat. Moreover, the bound will continue to be worst-casetight by placing all the points and lines in a common 2-flat, in a configuration that yieldsthe planar lower bound.

In the 2010 groundbreaking paper of Guth and Katz [15], an improved bound for thereal case has been derived for I(P,L), for a set P of m points and a set L of n lines in R3,provided that not too many lines of L lie in a common plane. Specifically, they showed:2

Theorem 1.1 (Guth and Katz [15]). Let P be a set of m points and L a set of n lines inR3, and let s ≤ n be a parameter, such that no plane contains more than s lines of L. Then

I(P,L) = O(m1/2n3/4 +m2/3n1/3s1/3 +m+ n

).

This bound (or, rather, an alternative formulation thereof) was a major step in thederivation of the main result of [15], which was an almost-linear lower bound on the numberof distinct distances determined by any finite set of points in the plane, a classical problemposed by Erdos in 1946 [8]. Guth and Katz’s proof uses several nontrivial tools fromalgebraic and differential geometry, most notably the Cayley–Salmon–Monge theorem onosculating lines to algebraic surfaces in R3, and various properties of ruled surfaces. Allthis machinery comes on top of the major innovation of Guth and Katz, the introductionof the polynomial partitioning technique.

2Actually, Theorem 1.1 is not stated explicitly in [15], but it follows immediately from the bounds thatthey derive.

2

For the purpose of the analysis in this paper, it is important to recall, right away, thatthe polynomial partitioning technique holds only over the reals. This will be the majorstumbling block that we will face as we handle the complex case. We overcome (or ratherbypass) it by exploiting the assumption that in this case all the lines are also contained inthe given variety; see Section 3 for details.

In four dimensions, and for the real case, the authors established in [37] a sharper and(almost) optimal bound. More precisely, they have shown:

Theorem 1.2. Let P be a set of m points and L a set of n lines in R4, and let s ≤ q ≤ nbe parameters, such that (i) no hyperplane or quadric contains more than q lines of L, and(ii) no 2-flat contains more than s lines of L. Then,

I(P,L) ≤ 2c√logm

(m2/5n4/5 +m

)+A

(m1/2n1/2q1/4 +m2/3n1/3s1/3 + n

), (2)

for suitable (absolute) constant parameters A and c. Moreover, except for the factor 2c√logm,

the bound is tight in the worst case, for all m and n and suitable ranges of q and s. Forcertain ranges of m and n the bound holds without that factor.

Our results. In this work we “contest” the leading terms O(m1/2n3/4

)(for d = 3) and

2c√logmm2/5n4/5 (for d = 4), and present situations in which they can be significantly

improved. A major feature of this work is that, in the setups considered here, the analysiscan also be carried over to the complex domain, except for a small penalty that we pay forbypassing the polynomial partitioning technique, which, as noted, only holds over the reals.

Concretely, we assume that the points of P lie on some algebraic variety, and derivesignificantly improved bounds when the degree of the variety is not too large. In the formercase we assume that the points and the lines lie on a two-dimensional variety, which isallowed to be embedded in any Rd, for d ≥ 3. In the latter case we assume that the points(but not necessarily the lines) lie on a three-dimensional variety, embedded in any Rd, ford ≥ 4. In the former (resp., latter) case we also assume that the variety contains no plane(resp., no hyerplane or quadric). Thus, in addition to improving the respective bounds inTheorems 1.1 and 1.2, for the special cases under consideration, and extending them to thecomplex domain, we obtain an extra bonus by extending the results to two-dimensional andthree-dimensional varieties embedded in any higher dimension.

Points on a two-dimensional variety. We derive two closely related results, one thatholds over the real field and one that holds also over the complex field. It is simplest tothink of the variety as embedded in R3 or in C3. The real case is a special case of the setupof Guth and Katz [15], where there is no need to use the polynomial partitioning method,because we assume that the points and lines all lie in a common surface (the zero set ofa polynomial) of degree3 D. This very assumption is also the one that lets us derive the(slightly weaker) version that holds over C, thereby constituting a significant progress overthe existing theory of incidences in three (and higher) dimensions. To be more precise, overthe reals we do apply the polynomial partitioning technique (as a step in the application ofthe Guth-Katz bound), but only to a small subset of the lines.

Concretely, our first main result, for this setup, is the following theorem.

3See later, in Section 2, for a discussion of the notions of degree and dimension over the reals.

3

Theorem 1.3. (a) The real case: Let P be a set of m points and L a set of n lines inRd, for any d ≥ 3, and let 2 ≤ s ≤ D be two integer parameters, so that all the pointsand lines lie in a common two-dimensional algebraic variety V of degree D that does notcontain any 2-flat, and so that no 2-flat contains more than s lines of L. Then

I(P,L) = O(m1/2n1/2D1/2 +m2/3D2/3s1/3 +m+ n

). (3)

(b) The complex case: Under exactly the same assumptions, when the ambient space isCd, for any d ≥ 3, we have

I(P,L) = O(m1/2n1/2D1/2 +m2/3D2/3s1/3 +D3 +m+ n

). (4)

The assumption that s is at most D can be dropped, because, for any 2-flat π, theintersection π ∩ V is a plane algebraic curve of degree at most D in π (this holds since Vdoes not contain any 2-flat), and can therefore contain at most D lines.

We also have the following easy and interesting corollary.

Corollary 1.4. Let P be a set of m points and L a set of n lines in Rd or in Cd, for anyd ≥ 3, such that all the points lie in a common two-dimensional algebraic variety of constantdegree that does not contain any 2-flat. Then I(P,L) = O (m+ n), where the constant ofproportionality depends on the degree of the surface.

For d = 3, the corollary can also be derived, for the real case, from the analysis in Guthand Katz [15], using a somewhat different approach. Moreover, although not explicitlystated there, it seems that the argument in [15] also works over C. As a matter of fact,the corollary can also be extended (with a different bound though) to the case where thecontaining surface may have planar components. See a remark to that effect in Corollary 5.1in the concluding section.

We remark that the fact that V does not have planar components (and all the lines ofL are contained in V ) is what enables us to get rid of the term O(nD) in the bound. Thisterm is unavoidable, so to speak, in a general setting. For example, if we only assume thatthe points lie on V but the lines are arbitrary, we will incur the term O(nD) that we aretrying to avoid.

In the case of three-dimensional varieties, though, discussed later in the introduction(see Remark (2) following Theorem 1.5), we cannot avoid this term even when the lines lieon the variety, which is why we do not impose this property in that context.

We also exploit the proof technique of Theorem 1.3 to derive an upper bound of O(nD)on the number of 2-rich points determined by a set of n lines contained in a variety, asabove, in both the real and complex cases. See Section 5 for details.

The significance of Theorem 1.3 is fourfold:

(a) First and foremost, the theorem yields a new incidence result for points and lines on atwo-dimensional variety over the complex field, in three and higher dimensions. Incidenceresults over the complex domain are rather rare. They include (as already mentioned)Toth’s extension of the Szemeredi-Trotter bound to the complex plane [45], which was theonly result of that kind that predated the introduction of the algebraic machinery by Guth

4

and Katz, and several more recent works [39, 41, 42, 47] (where the latter work [47] providesan alternative algebraic derivation of Toth’s bound).

(b) In the real three-dimensional case, the bound improves the Guth–Katz bound whenD ≪ n1/2, for points and lines in two-dimensional varieties V that do not contain planes.Note that the threshold n1/2 is a natural one because, as is well known and easy to show,any set of n lines in R3 admits a polynomial of degree O(n1/2) whose zero set containsall the lines; a simple modification of the construction applies in higher dimensions too.Of course the comparison is far from perfect, because this polynomial may have manylinear components, in which case our bound does not apply. Still, it offers some basis forevaluating the quality of our bound. In three dimensions, this threshold is in fact largerthan the standard degree O(m1/2/n1/4) used in the analysis of Guth and Katz [15], whenm < n3/2.

(c) Another significant feature of our bound is that it does not contain the term nD, whicharises naturally in [15] and other works, and seems to be unavoidable when P is an arbitraryset of points. When D is not a constant, this becomes a crucial feature of the new bound,which has already been exploited in the analysis in [37], and is also used in the second mainresult of this paper, Theorem 1.5 below.4 See an additional discussion of this feature at theend of the paper.

(d) Our result offers a sharper point-line incidence bound in arbitrary dimensions, for thespecial case assumed in the theorem (which again holds over the complex field too).

Theorem 1.3(a) has been used, as one of the key tools, in the analysis in our paper [37] onincidences between points and lines in four dimensions. In this application, the absence ofthe term nD is a crucial feature of our result, which was required in the scenario consideredin [37].

The proof of Theorem 1.3 makes extensive use of several properties of ruled surfaces in R3

or in C3. While these results exist as folklore in the literature, and short proofs are providedfor some of them, e..g., in [15], we include here detailed and rigorous proofs thereof, makingthem more accessible to the combinatorial geometry community. Other recent expositionsinclude Guth’s recent survey [12] and book [13], and a survey by Kollar [20].5

Points on a three-dimensional variety. Our second main result deals with the casewhere the points lie on a three-dimensional variety, embedded in R4 or in C4, or in anyhigher dimension. Similar to the case of two-dimensional varieties discussed above, we haveto be careful here too, because hyperplanes and 3-quadrics (in R4, and, a posteriori, inC4 too) admit “too many” incidences in the worst case. That is, by a generalization ofElekes’s construction [6], there exists a configuration of m points and n lines in a 3-flat withΘ(m1/2n3/4) incidences. More recently, Solomon and Zhang [40] established an analogousstatement for three-dimensional quadrics, when n9/8 < m < n3/2. Concretely, for suchvalues of m and n, they have constructed a quadric S ⊂ R4, a set P of m points on S,and a set L of n lines contained in S, so that (i) the number of lines in any common 2-flatis O(1), (ii) the number of lines in any hyperplane is O(n/m1/3), and (iii) the number of

4Although the bound in Theorem 1.5 does contain the term nD, it still crucially relies on the absence ofthis term in the bound for two-dimensional varieties.

5While there is (naturally) some overlap between these surveys and our exposition, the main technicalproperties that we present do not seem to be rigorously covered in the other works.

5

incidences between the points and lines is Ω(m2/3n1/2), which is asymptotically larger thanthe corresponding bound in (2) (for s = O(1)), when n9/8 ≪ m ≤ n3/2.

In other words, when studying incidences with points on a variety in R4 or in C4, thecases where the variety is (or contains) a hyperplane or a quadric are special and do notyield the sharper bounds that we derive below. In the real case, the case of a hyperplaneputs us back in R3, where the best bound is Guth and Katz’s in Theorem 1.1, and the case ofa quadric reduces to the same setup via a suitable generic projection onto R3. In the secondmain result of this paper, we show that if all the points lie on a three-dimensional algebraicvariety of degree D without 3-flat or 3-quadric components, and if no 2-flat contains morethan s lines, then, if D and s are not too large, the bound becomes significantly smaller.Moreover, here too we get a real version and a complex version of the theorem which nearlycoincide. Specifically, we show:

Theorem 1.5. (a) The real case: Let P be a set of m points and L a set of n linesin Rd, for any d ≥ 4, and let s and D be parameters, such that (i) all the points of Plie on a three-dimensional algebraic variety of degree D, without any linear or quadraticthree-dimensional components, and (ii) no 2-flat contains more than s lines of L. Then

I(P,L) = O(m1/2n1/2D +m2/3n1/3s1/3 + nD +m

). (5)

When D and s are constants, we get the linear bound O(m+ n).

(b) The complex case: Under exactly the same assumptions, when the ambient space isCd, for any d ≥ 4, we have

I(P,L) = O(m1/2n1/2D +m2/3n1/3s1/3 +D6 + nD +m

). (6)

Remarks. (1) Note that for D < minm1/2/n1/4, n1/4, our bound for the real case issharper than the bound of Guth and Katz [15] (note that m1/2/n1/4 is the degree of thepartitioning polynomial used in the analysis of [15] for m ≤ n3/2). On the other hand,when D > minm1/2/n1/4, n1/4, our bound is not the best possible. Indeed, in this casewe can project P and L onto some generic 3-flat, and apply instead the bound of [15] to theprojected points and lines (we also show that a generic choice of the image 3-flat ensuresthat no 2-flat contains more than s of the projected lines), which is sharper than ours forthese values of D. In the complex case, we also need to assume that D is small enough sothat the term D6 does not dominate the other terms, so as to make the bound look like thebound in the real case.

(2) As already noted earlier, here we do not have to insist that the lines of L be containedin the variety. A line not contained in a variety of degree D can intersect it in at most Dpoints, so the number of incidences with such lines is at most nD. The actual argumentthat yields this term is more involved, because we apply the argument to a variety of largerdegree; see Section 4 for details. For two-dimensional varieties, since we want to avoid thisterm, we require the lines to be contained in the variety. Here, since we allow this term,the lines can be arbitrary.

(3) The assumption that the points of P lie on a variety is not as restrictive as it mightsound, because, in four dimensions, one can always construct a polynomial f of degree

6

O(m1/4) whose zero set contains all the given points, or alternatively, a polynomial fof degree O(n1/3) whose zero set contains all the given lines. The assumptions becomerestrictive, and the bound becomes more interesting, when D is significantly smaller. Inaddition, the constructed polynomial f could be one for which Z(f) does contain 3-flatsor 3-quadrics, so another restrictive aspect of our assumptions is that they exclude thesesituations.

(4) The assumption that Z(f) does not contain any linear or quadratic component isalso significant, because having such components would make the problem behave like theproblem of point-line incidences in three dimensions, as studied in Guth and Katz [15]. Thebound would then “deteriorate” to their bound, and hold only over the reals. Additionaldiscussion is provided in Section 5.

Similar to the case of a two-dimensional variety, we also have here the following easyand interesting corollary (it does not hold when V contains 3-flats or 3-quadrics, in light ofthe lower bound constructions in [37, 40]).

Corollary 1.6. Let P be a set of m points and L a set of n lines in Rd or in Cd, for anyd ≥ 4, such that all the points lie in a common three-dimensional algebraic variety V ofconstant degree that does not contain any 3-flats or 3-quadrics, and no 2-flat contains morethan O(1) lines of L. Then I(P,L) = O (m+ n), where the constant of proportionalitydepends on the degree of V .

Theorem 1.3 is a key technical ingredient in the proof of Theorem 1.5. The proofs of boththeorems are somewhat technical, and use a battery of sophisticated tools from algebraicgeometry. Some of these tools are borrowed and adapted from our previous work [37].Other tools involve properties of ruled surfaces, which, as already said, are established hererigorously, for the sake of completeness. Since most of the presentation and derivation ofthese results is within the scope of algebraic and differential geometry, we give the necessarybackground in the following section, to make the presentation in this paper easier to followfor non-experts. Reader might consider skipping Section 2 on first reading, using it as areference for the various algebraic tools that are used later in the paper.

The proof of Theorem 1.3 is then presented in Section 3, and the proof of Theorem 1.5is presented in Section 4. The concluding Section 5 discusses our results, establishes a fewconsequences thereof, and raises several related open problems.

2 Algebraic tools and ruled surfaces

In this section we review the preliminary algebraic (and differential) geometry infrastructureneeded for our analysis, and then go on to establish the properties of ruled surfaces thatwe will use. These properties are considered folklore in the literature; having failed to findrigorous proofs of them (except for several short proofs or proof sketches for some of them),we provide here such proofs for the sake of completeness. Some of the notions covered inthis section are also discussed in our study [37] on point-line incidences in four dimensions.

Degree and dimension of real variaties. In principle, all the varieties considered inthe paper are regarded as complex algebraic varieties, for which the notions of degree and

7

dimension are classical well known concepts (discussed, e.g., in Harris [16]). When we talkabout a real variety V , we mean a variety defined as the zero set of real polynomials, makingthe degree and dimension of V still well defined. When we consider such a variety in anincidence problem over the reals, though, we replace V by the subset VR of its real points.

Singularity. The notion of singularities is a major concept, treated in full generality inalgebraic geometry (see, e.g., Kunz [21, Theorem VI.1.15] and Cox et al. [3]). Here we onlyrecall some of their properties, and only for a few special cases that are relevant to ouranalysis.

Let V be a two-dimensional variety in R3 or C3 of degree D, given as the zero set Z(f)of some trivariate polynomial f . Assuming f to be square-free, a point p ∈ Z(f) is singularif ∇f(p) = 0. For any point p ∈ Z(f), let

f(p+ x) = fµ(x) + fµ+1(x) + . . .

be the Taylor expansion of f near p, where fj is the j-th order term in the expansion (whichis a homogeneous polynomial of x of degree j), and where we assume that there are no termsof order (i.e., degree) smaller than µ. The terms fj also depend on p, which we regard asfixed in the present discussion. In general, we have f1(x) = ∇f(p) · x, f2(x) = 1

2xTHf (p)x,

where Hf is the Hessian matrix of f , and the higher-order terms are similarly defined, albeitwith more involved expressions.

If p is singular, we have µ ≥ 2. In this case, we say that p is a singular point ofV = Z(f) of multiplicity µ = µV (p). For any point p ∈ Z(f), we call the hypersurfaceZ(fµ) the tangent cone of Z(f) at p, and denote it by CpZ(f). If µ = 1, then p is non-singular and the tangent cone coincides with the (well-defined) tangent plane TpZ(f) toZ(f) at p. We denote by Vsing the locus of singular points of V . This is a subvariety ofdimension at most 1; see, e.g., Solymosi and Tao [41, Proposition 4.4]. We say that a lineℓ is a singular line for V if all of its points are singular points of V .

Similarly, let γ be a one-dimensional algebraic curve in R2 or in C2, specified as Z(f),for some bivariate square-free polynomial f . Then p ∈ Z(f) is singular if ∇f(p) = 0. Themultiplicity µ of a point p ∈ γ is defined as in the three-dimensional case, and we denote itas µγ(p); the multiplicity is at least 2 when p is singular. The singular locus γsing of γ isnow a discrete set. Indeed, the fact that f is square-free guarantees that f has no commonfactor with any of its first-order derivatives, and Bezout’s Theorem (see, e.g., [3, Theorem8.7.7]) then implies that the common zero set of f , fx, fy, and fz is a (finite) discrete set.

Still in two dimensions, a line ℓ, not contained in the curve γ, can intersect it in at mostDpoints, counted with multiplicity. To define this concept formally, as in, e.g., Beltrametti [2,Section 3.4], let ℓ be a line and let p ∈ ℓ∩γ, such that ℓ is not contained in the tangent coneof γ at p. The intersection multiplicity of γ and ℓ at p is the smallest order of a nonzeroterm of the Taylor expansion of f at p in the direction of ℓ. As it happens, the intersectionmultiplicity is also equal to µγ(p) (informally, this is the number of branches of γ that ℓcrosses at p, counted with multiplicity; see [3, Section 8.7] for a treatment on the intersectionmultiplicity in the plane). The intersection between a line ℓ and a curve γ (not containingℓ) consists of at most deg(γ) points, counted with their intersection multiplicities.

This standard property is a crucial ingredient of one of the key lemmas (Lemma 3.1) inthe proof of Theorem 1.3 in Section 3.

8

Assume that V is irreducible. By Guth and Katz [14] (see also Elekes et al. [7, Corollary2]), the number of singular lines contained in V is at most D(D − 1).

Flatness. We say that a non-singular point x ∈ V is flat if the second-order Taylorexpansion of f at x vanishes on the tangent plane TxV , or alternatively, if the secondfundamental form of V vanishes at x (see, e.g., Pressley [26]). As argued, e.g., in Elekeset al. [7], if x is a non-singular point of V and there exist three lines incident to p thatare contained in V (this property is captured by calling p a linearly flat point) then x isa flat point. Following Guth and Katz [14], Elekes et al. [7, Proposition 6] proved that anon-singular point x ∈ V is flat if and only if certain three polynomials, each of degree atmost 3D − 4, vanish at p. A non-singular line ℓ is said to be flat if all of its non-singularpoints are flat. By Guth and Katz [14] (see also Elekes et al. [7, Proposition 7]), the numberof flat lines contained in V is at most D(3D − 4), unless V is a plane.

As in the proof of Theorem 1.5, the notions of linear flatness and flatness can be extendedto any higher dimension. For example, for a three-dimensional surface V in R4 or in C4,which is the zero set of some (square-free) polynomial f of degree D, a non-singular pointx ∈ V is said to be linearly flat, if it is incident to at least three 2-flats that are containedin V = Z(f) (and thus also in the tangent hyperplane TpZ(f)). Linearly flat points canthen be shown to be flat, meaning that the second fundamental form of f vanishes at them.This property, at a point p, can be expressed by several polynomials of degree at most3D − 4 vanishing at p (see [37, Section 2.5]). As in the three-dimensional case, the secondfundamental form vanishes identically on Z(f) if and only if Z(f) is a hyperplane. Thisproperty holds in any dimension; see, e.g. [18, Exercise 3.2.12.2]). As in three dimensions,we call a line contained in V flat if all its non-singular points are flat.

Ruled surfaces. For a modern approach to ruled surfaces, there are many references;see, e.g., Hartshorne [17, Section V.2], or Beauville [1, Chapter III]; see also Salmon [30]and Edge [5] for earlier treatments of ruled surfaces. Three relevant very recent additionsare the survey [12] and book [13] of Guth, as well as a survey in Kollar [20], where thistopic is addressed in detail.

We say that a real (resp., complex) surface V is ruled by real (resp., complex ) lines ifevery point p in a Zariski-open6 dense subset of V is incident to a real (complex) line thatis contained in V . This definition has been used in several recent works, see, e.g., [15, 20];it is a slightly weaker condition than the classical condition where it is required that everypoint of V be incident to a line contained in V (e.g., as in [30]). Nevertheless, similarlyto the proof of Lemma 3.4 in Guth and Katz [15], a limit argument implies that the twodefinitions are in fact equivalent. We give, in Lemma 2.2 below, a short algebraic proof ofthis fact, for the sake of completeness.

Flecnodes in three dimensions and the Cayley-Salmon-Monge Theorem. Wefirst recall the classical theorem of Cayley and Salmon, also due to Monge. Consider apolynomial f ∈ C[x, y, z] of degree D ≥ 3. A flecnode of f is a point p ∈ Z(f) for whichthere exists a line that is incident to p and osculates to Z(f) at p to order three. Thatis, if the direction of the line is v then f(p) = 0, and ∇vf(p) = ∇2

vf(p) = ∇3vf(p) = 0,

6See Cox et al. [3, Section 4.2] for details concerning the Zariski topology.

9

where ∇vf,∇2vf,∇3

vf are, respectively, the first, second, and third-order derivatives of fin the direction v (compare with the definition of singular points, as reviewed earlier, forthe explicit forms of ∇vf and ∇2

vf). The flecnode polynomial of f , denoted FLf , is thepolynomial obtained by eliminating v from these three homogeneous equations (where p isregarded as a fixed parameter). We thus have a system of three equations in six variables,which is homogeneous in the three variables defining v. Eliminating those variables resultsin a single polynomial equation in p = (x, y, z). Using standard techniques, as in Cox etal. [4], the resulting polynomial FLf is the multipolynomial resultant7 Res3(F1, F2, F3) ofF1, F2, F3, regarding these as polynomials in v (where the coefficients are polynomials in p).

As shown in Salmon [30, Chapter XVII, Section III], the degree of FLf is at most11D− 24. By construction, the flecnode polynomial of f vanishes on all the flecnodes of f ,and in particular on all the lines contained in Z(f).

Theorem 2.1 (Cayley and Salmon [30], Monge [24]). Let f ∈ C[x, y, z] be a polynomial ofdegree D ≥ 3. Then Z(f) is ruled by (complex) lines if and only if Z(f) ⊆ Z(FLf ).

Note that the correct formulation of Theorem 2.1 is over C; earlier applications, overR, as the one in Guth and Katz [15], require some additional arguments to establish theirvalidity; see Katz [19] for a discussion of this issue.

Lemma 2.2. Let f ∈ C[x, y, z] be an irreducible polynomial such that there exists anonempty Zariski open dense set in Z(f) so that each point in the set is incident to aline that is contained in Z(f). Then FLf vanishes identically on Z(f), and Z(f) is ruledby lines.

Proof. Let U ⊂ Z(f) be the set assumed in the lemma. By assumption and definition, FLfvanishes on U , so U , and its Zariski closure, are contained in Z(f,FLf ). Since U is open,it must be two-dimensional. Indeed, otherwise its complement would be a (nonempty)two-dimensional subvariety of Z(f) (a Zariski closed set is a variety). In this case, thecomplement must be equal to Z(f), since f is irreducible, which is impossible since U isnonempty. Hence Z(f,FLf ) is also two-dimensional, and thus, by the same argument justused, must be equal to Z(f). Theorem 2.1 then implies that Z(f) is ruled by (complex)lines, as claimed.8

The notions of flecnodes and of the flecnode polynomial can be extended to four di-mensions, as done in [37]. Informally, the four-dimensional flecnode polynomial FL4f of f isdefined analogously to the three-dimensional variant FLf , and captures the property thata point on Z(f) is incident to a line that osculates to Z(f) up to the fourth order. It isobtained by eliminating the direction v of the osculating line from the four homogeneousequations given by the vanishing of the first four terms of the Taylor expansion of f(p+ tv)near p. Clearly, FL4f vanishes identically on every line that is contained in Z(f). As in thethree-dimensional case, its degree can be shown to be O(D).

Landsberg [22] derives an analog of Theorem 2.1 that holds for three-dimensional sur-faces (see [37, Theorem 2.11]). Specifically, Landsberg’s theorem asserts that if FL4f vanishes

7The resultant method, or Macaulay’s resultant method dates back to the 19th century. It is thoroughlycovered in Van der Waerden [46, Chapter XI] and in Cox et al. [4].

8An alternative proof, based on a compactness argument, has been suggested to us by Martin Sombra(personal communication).

10

identically on Z(f), then Z(f) is ruled by (possibly complex) lines. We will discuss thisin more detail in Section 4. See also another result of Landsberg, given as Theorem 4.2 inSection 4, that we will also require. These theorems, in three and four dimensions, play animportant role in the proofs of the main theorems.

Theorem of the fibers and related tools. The main technical tool for the analysisis the following so-called Theorem of the Fibers. Both Theorem 2.3 and Theorem 2.4 hold(only) for the complex field C.

Theorem 2.3 (Harris [16, Corollary 11.13]). Let X be a projective variety and π : X → Pd

be a homogeneous polynomial map (i.e., the coordinate functions x0 π, . . . , xd π arehomogeneous polynomials); let Y = π(X) denote the image of X. For any p ∈ Y , let9

λ(p) = dim(π−1(p)). Then λ(p) is an upper semi-continuous function of p in the Zariskitopology on Y ; that is, for any m, the locus of points p ∈ Y such that λ(p) ≥ m is Zariskiclosed in Y . Moreover, if X0 ⊂ X is any irreducible component, Y0 = π(X0) its image, andλ0 the minimum value of λ(p) on Y0, then

dim(X0) = dim(Y0) + λ0.

We also need the following theorem and lemma from Harris [16].

Theorem 2.4 (Harris [16, Proposition 7.16]). Let f : X → Y be the map induced by thestandard projection map π : Pd → Pr (which retains the first r coordinates and discards therest), where r < d, where X ⊂ Pd and Y ⊂ Pr are projective varieties, X is irreducible, andY is the image of X (which is also irreducible). Then the general fiber10 of the map f isfinite if and only if dim(X) = dim(Y ). In this case, the number of points in a general fiberof f is constant (which depends on the degree of X).

In particular, when Y is two-dimensional (and d > r ≥ 2 are arbitrary), there exist aninteger cf and an algebraic curve Cf ⊂ Y , such that for any y ∈ Y \Cf , we have |f−1(y)| = cf .With the notations of Theorem 2.4, the set of points y ∈ Y , such that the fiber of f over yis not equal to cf , is a Zariski closed proper subvariety of Y . For more details, we refer thereader to Shafarevich [34, Theorem II.6.4], and to Hartshorne [17, Exercise II.3.7].

Lemma 2.5 (Harris [16, Theorem 11.14]). Let π : X → Y be a polynomial map between twoprojective varieties X, Y , with Y = f(X) irreducible. Suppose that all the fibers π−1(p)of π, for p ∈ Y , are irreducible and of the same dimension. Then X is also irreducible.

Reguli. We rederive here the following (folklore) characterization of doubly ruled surfacesin R3 or C3, namely, irreducible algebraic surfaces, each of whose points is incident to atleast two lines that are contained in the surface. Recall that a regulus is the surface spannedby all lines that meet three pairwise skew lines in 3-space.11 For an elementary proof that

9The dimension of an algebraic variety is defined in any textbook in algebraic geometry (see, e.g., Coxet al. [3, Definitions 7, 10]).

10The meaning of this statement is that the assertion holds for the fiber at any point outside some lower-dimensional exceptional subvariety.

11Technically, in some definitions (cf., e.g., Edge [5, Section I.22]) a regulus is a one-dimensional familyof generator lines of the actual surface, i.e., a curve in the Plucker or Grassmannian space of lines in theKlein quadric, but we use here the alternative notion of the surface spanned by these lines. See below, under“Lines in a variety”, for a comment to that effect.

11

a doubly ruled surface must be a regulus, we refer the reader to Fuchs and Tabachnikov [9,Theorem 16.4]. Their proof however is analytic and works only over the reals.

Lemma 2.6. Let V be an irreducible ruled surface in R3 or in C3 which is not a plane, andlet C ⊂ V be an algebraic curve, such that every non-singular point p ∈ V \ C is incident toat least two lines that are contained in V . Then V is a regulus.12

Proof. As mentioned above (see also [14]), the number of singular lines in V is finite (itis smaller than deg(V )2). For any non-singular line ℓ, contained in V , but not in C, theunion of lines Uℓ intersecting ℓ and contained in V is a subvariety of V (see, e.g., Sharirand Solomon [35, Lemma 8] for the easy proof). By assumption, each non-singular pointin ℓ \ C is incident to another line (other than ℓ) contained in V , and thus Uℓ is the unionof infinitely many lines, and is therefore two-dimensional. Since V is irreducible, it followsthat Uℓ = V . Next, pick any triple of non-singular and non-concurrent lines ℓ1, ℓ2, ℓ3 thatare contained in V and intersect ℓ at distinct non-singular points of ℓ \ C. There has toexist such a triple, for otherwise we would have an infinite family of concurrent (or parallel)lines incident to ℓ and contained in V (where the point of concurrency lies outside ℓ), andthe plane that they span would then have to be contained in (the irreducible) V , contraryto assumption. See Figure 1 for an illustration. The argument given for ℓ applies equallywell to ℓ1, ℓ2, and ℓ3 (by construction, neither of them is contained in C), and implies thatUℓ1 = Uℓ2 = Uℓ3 = V .

Assume that there exists some line ℓ ⊂ V intersecting ℓ1 at some non-singular pointp ∈ ℓ1 \C, and that ℓ∩ℓ2 = ∅. We treat lines here as projective varieties, so this assumptionmeans that ℓ and ℓ2 are skew to one another; parallel lines are considered to be intersecting.Since p ∈ ℓ1 ⊂ V = Uℓ2 , there exists some line ℓ intersecting ℓ2, such that ℓ ∩ ℓ1 = p.Hence there exist three lines, namely ℓ1, ℓ and ℓ, that are incident to p and contained inV . Since p is non-singular, it must be a flat point (as mentioned above; see [7]). Repeatingthis argument for 3deg(V ) non-singular points p ∈ ℓ1, it follows that ℓ1 contains at least3deg(V ) flat points, and is therefore, by the properties of flat points noted earlier, a flatline. As is easily checked, ℓ1 can be taken to be an arbitrary non-singular line among thoseincident to ℓ, so it follows that every non-singular point on V is flat, and therefore, as shownin [7, 14], V is a plane, contrary to assumption.

Therefore, every non-singular line that intersects ℓ1 at a non-singular point also intersectsℓ2, and, similarly, it also intersects ℓ3. This implies that the intersection of V and the surfaceR generated by the lines intersecting ℓ1, ℓ2, and ℓ3 is two-dimensional, and is therefore equalto V , since V is irreducible. Since ℓ1, ℓ2 and ℓ3 are pairwise skew, R = V is a regulus, asasserted.

Most of the basic algebraic geometry tools have been developed over the complex fieldC, and some care has to be exercised when applying them over the reals. A major partof the theory developed in this section is of this nature. For example, both Theorems 2.3and 2.4 hold only over the complex field. As another important example, one of the maintools at our disposal is the Cayley–Salmon–Monge theorem (Theorem 2.1), whose originalformulation also applies only over C. Expanding on a previously made comment, we note

12Over R, a regulus is either a hyperbolic paraboloid or a one-sheeted hyperboloid. Over C, balls (equiva-lent to hyperboloids) and paraboloids (equivalent to hyperbolic paraboloids) are also reguli, and are indeeddoubly ruled by complex lines. Not all complex quadrics are reguli, though: for example, the cylinder y = x2

is not a regulus.

12

ℓ

ℓ1ℓ2

ℓ3

ℓ

p ℓ

Figure 1: The structure of Uℓ in the proof of Lemma 2.6.

that even when V is a variety defined as the zero set of a real polynomial f , the vanishingof the flecnode polynomial FLf only guarantees that the set of complex (and real) points ofV is ruled by complex lines.

A very simple example that illustrates this issue is the unit sphere σ, given by x2 +y2 + z2 = 1, which is certainly not ruled by real lines, but the flecnode polynomial off(x, y, z) = x2 + y2 + z2 − 1 vanishes on σ (since the equation ∇3

vf(p) = 0 that participatesin its construction is identically zero for any quadratic polynomial f). This is the conditionin the Cayley–Salmon–Monge theorem that guarantees that σ is ruled by (complex) lines,and indeed it does, as is easily checked; in fact, for the same reason, every quadric is ruledby complex lines.

This issue has not been directly addressed in Guth and Katz [15], although their theorycan be adjusted to hold for the real case too, as noted later in Katz [19].

This is just one example of many similar issues that one must watch out for. It is afairly standard practice in algebraic geometry that handles a real algebraic variety V , definedby real polynomials, by considering its complex counterpart VC, namely the set of complexpoints at which the polynomials defining V vanish. The rich toolbox that complex algebraicgeometry has developed allows one to derive various properties of VC, but some care might beneeded when transporting these properties back to the real variety V , as the preceding noteconcerning the Cayley–Salmon–Monge theorem illustrates. Fortunately, though, passingto the complex domain (and sometimes also to the projective setting) does not pose anydifficulties for deriving upper bounds in incidence problems—every real incidence will bepreserved, and at worst we will be counting additional incidences, on the non-real portionof the extended varieties. With this understanding, and with the appropriate caution, wewill move freely between the real and complex domains, as convenient.

We note that most of the results developed in Section 3 of this paper also apply overC, except for one crucial step (where we resort to the application of the result of Guth andKatz [15], which holds only over the reals), due to which we do not know how to extendTheorem 1.3(a) to the complex domain. Nevertheless, we can derive the weaker variant ofit, Theorem 1.3(b), for the complex case—see a remark to that effect in Section 5.

Lines on a variety. In preparation for the key technical Theorem 2.7, given below,we make the following comments. Lines in three dimensions are parameterized by theirPlucker coordinates, as follows (see, e.g., Griffiths and Harris [11, Section 1.5]). For two

13

distinct points x, y ∈ P3, given in projective coordinates as x = (x0, x1, x2, x3) and y =(y0, y1, y2, y3), let ℓx,y denote the (unique) line in P3 incident to both x and y. The Pluckercoordinates of ℓx,y are given in projective coordinates in P5 as (π0,1, π0,2, π0,3, π2,3, π3,1, π1,2),where πi,j = xiyj − xjyi. Under this parameterization, the set of lines in P3 correspondsbijectively to the set of points in P5 lying on the Klein quadric (also known as the Grassman-nian manifold of lines in P3) given by the quadratic equation π0,1π2,3+π0,2π3,1+π0,3π1,2 = 0(which is indeed always satisfied by the Plucker coordinates of a line).

Given a surface V in P3, the set of lines contained in V , represented by their Pluckercoordinates in P5, is a subvariety of the Klein quadric (or the Grassmaniann of lines in P4),which is denoted by F (V ), and is called the Fano variety of V ; see Harris [16, Lecture 6,page 63] for details, and [16, Example 6.19] for an illustration, and for a proof that F (V )is indeed a variety. The Plucker coordinates are continuous, in the sense that if one takestwo points ℓ, ℓ′ on the Klein quadric that are near each other, the lines in P3 that theycorrespond to are also near to one another, in an obvious sense whose precise details areomitted here.

Remark. This representation of lines, which can be extended to higher dimensions too,is also useful in this study of ruled surfaces, as it offers an alternative definition of a ruledsurface in terms of its Fano variety.

That is, in the context of the Plucker coordinates, a ruled surface can be defined as aone-dimensional family of lines, that is, a curve on the Klein quadric. With this point ofview, the union of lines in the projective three-space, which are the Klein pre-images ofpoints on this curve is referred to as the point set of the ruled surface. We refer the readerto Pottmann and Wallner’s textbook [25, Chapters 5,6] for a thorough exposition of thisrepresentation, and to Selig [32, Chapter 6], for a more concise exposition. This viewpointwas also used by Rudnev [29]. Here, though, we stick to the other point of view, thinkingof a ruled surface as the surface itself (the point set), and not as its Fano variety.

We note again that our analysis is carried out in the complex projective setting, whichmakes it simpler, and facilitates the application of numerous tools from algebraic geometrythat are developed in this setting. In the particular context discussed here, the passage fromthe complex projective setup back to the real affine one is straightforward—the former is ageneralization of the latter.

Given a plane π by a homogeneous equation A0x0+A1x1+A2x2+A3x3 = 0, and a lineℓ not contained in π, given in Plucker coordinates as (π0,1, π0,2, π0,3, π2,3, π3,1, π1,2), theirpoint of intersection is given in homogeneous coordinates by (A ·m,A ×m − A0d), whered = (π0,1, π0,2, π0,3), m = (π2,3, π3,1, π1,2), and where · stands for the scalar product, and ×for the vector product; see, e.g., [43, p. 29]. This, together with the continuity argumentstated above, implies that, if the Fano variety F (V ) is one-dimensional, and ℓ is a linerepresented by a non-singular point of F (V ), then the cross section of the union of the linesthat lie near ℓ in F (V ) with a generic plane π is a simple arc13. When ℓ is a singular point ofF (V ), then the corresponding cross section is a union of simple arcs meeting at ℓ∩π, wheresome of these arcs might appear with multiplicity; the number of these arcs is determinedby the multiplicity of the singularity of ℓ.

13 In this context, a simple arc is a connected piece of an algebraic curve that is irreducible as a topologicalspace. For more details, we refer the reader to any classical textbook on the geometry of curves, such asBeltarmetti et al. [2, Section 3].

14

Singly ruled surfaces. Ruled surfaces that are neither planes nor reguli are called singlyruled surfaces (a terminology justified by Theorem 2.7, given below). A line ℓ, containedin an irreducible singly ruled surface V , such that every point of ℓ is “doubly ruled”, i.e.,every point on ℓ is incident to another line contained in V , is called an exceptional line14

of V . A point pV ∈ V that is incident to infinitely many lines contained in V is called anexceptional point of V .

The following result is another folklore result in the theory of ruled surfaces, used in manystudies (such as Guth and Katz [15]). It justifies the terminology “singly-ruled surface”, byshowing that the surface is generated by a one-dimensional family of lines, and that eachpoint on the surface, with the possible exception of points lying on some curve, is incidentto exactly one generator line. It also shows that there are only finitely many exceptionallines; the property that their number is at most two (see [15]) is presented later. We give adetailed and rigorous proof, to make our presentation as self-contained as possible; we arenot aware of any similarly detailed argument in the literature.

Theorem 2.7. (a) Let V be an irreducible ruled two-dimensional surface of degree D > 1in R3 (or in C3), which is not a regulus. Then, except for finitely many exceptional lines,the lines that are contained in V are parameterized by an irreducible algebraic curve Σ0 (inthe parametric Plucker space P5, or rather in the Klein quadric contained in that space,that represents lines in 3-space), and thus yielding a 1-parameter family of lines (referred toas generators) ℓ(t), for t ∈ Σ0, that depend continuously on the real or complex parametert. Moreover, if t1 6= t2, and ℓ(t1) 6= ℓ(t2), then there exist sufficiently small and disjointneighborhoods ∆1 of t1 and ∆2 of t2, such that all the lines ℓ(t), for t ∈ ∆1 ∪ ∆2, aredistinct.

(b) There exists a one-dimensional curve C ⊂ V , such that any point p in V \ C is incidentto exactly one line contained in V .

Remark. For a detailed description of the algebraic representation of V by generators, asin part (a) of the theorem, see Edge [5, Section II] (and see also the remark made earlier).

Proof. Assume first that we are working over C. Consider the Fano variety F (V ) of V ,as defined above. We claim that all the irreducible components of F (V ) are at most one-dimensional. Informally, if any component Σ0 of F (V ) were two-dimensional, then the set(p, ℓ) ∈ V ×F (V ) | p ∈ ℓ would be three-dimensional, so, “on average”, the set of lines ofF (V ) incident to a point p ∈ V would be one-dimensional, implying that most points of Vare incident to infinitely many lines that are contained in V , which can happen only whenV is a plane (or a non-planar cone, which cannot arise with a non-singular point p as anapex), contrary to assumption.

To make this argument formal, consider the set (already mentioned above)

W := (p, ℓ) | p ∈ ℓ, ℓ ∈ F (V ) ⊂ V × F (V ),

and the two projectionsΨ1 : W → V, Ψ2 : W → F (V )

14In Guth and Katz [15], a line ℓ contained in an irreducible singly ruled surface V , is called exceptionalif it contains infinitely many “doubly ruled” points, each incident to another line contained in V . Ourdefinition appears to be stricter, but, as the proof below will reveal, the two notions are equivalent.

15

to the first and second factors of the product V × F (V ), respectively.

W can formally be defined as the zero set of suitable homogeneous polynomials; briefly,with the Plucker parameterization of lines in P3, and putting the point p into homogeneouscoordinates, the condition p ∈ ℓ can be expressed as the vanishing of two suitable homo-geneous polynomials, and the other defining polynomials of W are those that define theprojective variety F (V ). Therefore, W is a projective variety.

Consider an irreducible component Σ0 of F (V ) (which is also a projective variety); put

W0 := Ψ−12 (Σ0) = (p, ℓ) ∈ W | ℓ ∈ Σ0.

Since W and Σ0 are projective varieties, so is W0. As is easily verified, Ψ2(W0) = Σ0 (thatis, Ψ2 is surjective). We claim that W0 is irreducible. Indeed, for any ℓ ∈ Σ0, the fiber ofthe map Ψ2|W0 : W0 → Σ0 over ℓ is (p, ℓ) | p ∈ ℓ which is (isomorphic to) a line, and istherefore irreducible of dimension one. As Σ0 is irreducible, Lemma 2.5 implies that W0 isalso irreducible, as claimed.

For a point p ∈ Ψ1(W0), consider the set Σ0,p = Ψ1|−1W0

(p), put λ(p) = dim(Σ0,p), andlet λ0 := minp∈Ψ1(W0) λ(p). By the Theorem of the Fibers (Theorem 2.3), applied to themap Ψ1|W0 : W0 → V , we have

dim(W0) = dim(Ψ1(W0)) + λ0. (7)

We claim that λ0 = 0. In fact, λ(p) = 0 for all points p ∈ V , except for at most one point.Indeed, if λ(p) ≥ 1 for some point p ∈ V , then Σ0,p is (at least) one-dimensional, and V ,being irreducible, is thus a cone with apex at p; since V can have at most one apex, theclaim follows. Hence λ0 = 0, and therefore

dim(W0) = dim(Ψ1(W0)) ≤ dim(V ) = 2. (8)

Next, assume, for a contradiction, that dim(Σ0) = 2. For a point (i.e., a line in P3)ℓ ∈ Ψ2(W0), the set Ψ2|−1

W0(ℓ) = (p, ℓ) | p ∈ ℓ is one-dimensional (the equality follows

from the way W0 is defined). Conforming to the notations in the Theorem of the Fibers,

we have µ(ℓ) := dim(Ψ2|−1

W0(ℓ)

)= 1, and thus µ0 := minℓ∈Ψ2(W0) µ(ℓ) = 1. Also, by

assumption, dim(Ψ2(W0)) = dim(Σ0) = 2. By the Theorem of the Fibers, applied this timeto Ψ2|W0 : W0 → Σ0, we thus have

dim(W0) = dim(Ψ2(W0)) + µ0 = 3, (9)

contradicting Equation (8). Therefore, every irreducible component of F (V ) is at mostone-dimensional, as claimed.

Let Σ0 be such an irreducible component, and let W0 := Ψ−12 (Σ0), as above. As argued,

for every p ∈ V , the fiber of Ψ1|W0 over p is non-empty and finite, except for at most onepoint p (the apex of V if V is a cone). Since W0 is irreducible, Theorem 2.4 implies thatthere exists a Zariski open set O ⊆ V , such that for any point p ∈ O, the fiber of Ψ1|W0 overp has fixed cardinality cf . Put C := V \ O. Being the complement of a Zariski open subsetof the two-dimensional irreducible variety V , C is (at most) a one-dimensional variety. Ifcf ≥ 2, then, by Lemma 2.6, V is a regulus. Otherwise, cf = 1 (cf cannot be zero fora ruled surface), meaning that, for every p ∈ V \ C, there is exactly one line ℓ, such that(p, ℓ) ∈ W0, i.e., Σ0 contains exactly one line incident to p and contained in V .

16

Moreover, we observe that the union of lines of Σ0 is the entire variety V . Indeed,by Equations (7) and (8), we have dim(W0) = dim(Ψ1(W0)) = 2. That is, the varietyΨ1(W0), which is the union of the lines of Σ0, must be the entire variety V , because it istwo-dimensional and is contained in the irreducible variety V .

To recap, we have proved that if Σ0 is a one-dimensional component of F (V ), then theunion of lines that belong to Σ0 covers V , and that there exists a one-dimensional subvariety(a curve) C ⊂ V such that, for every p ∈ V \ C, Σ0 contains exactly one line incident to pand contained in V .

Since V is a ruled surface, some component of F (V ) has to be one-dimensional, forotherwise we would only have a finite number of lines contained in V . We claim that thereis exactly one irreducible component of F (V ) which is one-dimensional. Indeed, assumeto the contrary that Σ0,Σ1 are two (distinct) one-dimensional irreducible components ofF (V ). As we observed, the union of lines parameterized by Σ0 (resp., Σ1) covers V . LetC0, C1 ⊂ V denote the respective excluded curves, so that, for every p ∈ V \ C0 (resp.,p ∈ V \C1) there exists exactly one line in Σ0 (resp., Σ1) that is incident to p and containedin V .

Next, notice that the intersection Σ0 ∩ Σ1 is a subvariety strictly contained in the irre-ducible one-dimensional variety Σ0 (since Σ0 and Σ1 are two distinct irreducible componentsof F (V )), so it must be zero-dimensional, and thus finite. Let C01 denote the union of thefinitely many lines in Σ0 ∩ Σ1, and put C := C0 ∪ C1 ∪ C01. For any point p ∈ V \ C, thereare two (distinct) lines incident to p and contained in V (one belongs to Σ0,p and the otherto Σ1,p). Lemma 2.6 (with C as defined above) then implies that V is a regulus, contraryto assumption.

In other words, the unique one-dimensional irreducible component Σ0 of F (V ) serves asthe desired 1-parameter family of generators for V . The local parameterization of Σ0 canbe obtained, e.g., by using a suitable Plucker coordinate to represent its lines. In additionto Σ0, there is a finite number of zero-dimensional components (i.e., points) of F (V ). Theycorrespond to a finite number of lines, contained in V , and not parameterized by Σ0. Sincethe union of the lines in Σ0 covers V , any of these additional lines ℓ is exceptional, sinceeach point on ℓ is also incident to a generator (different from ℓ), and is thus “doubly ruled”.

This establishes part (a) of the theorem, when V is defined over C. We remark thatGuth and Katz [15, Corollary 3.6] argue that there are at most two such exceptional lines, sothere are at most two zero-dimensional components of F (V ). For the sake of completeness,we sketch a proof of our own of this fact, in Lemma 2.8 below.

If V is defined over R, we proceed as above, i.e., consider instead the complex varietyVC corresponding to V . As we have just proven, the unique one-dimensional irreduciblecomponent Σ0 of F (V ) (regarded as a complex variety) is a (complex) 1-parameter familyof generators for the set of complex points of V . Since V is real, the (real) Fano variety ofV consists of the real points of F (V ), i.e., it is F (V )∩P5(R). As we have mentioned above,the (complex) F (V ) is the union of Σ0 with at most two other points. If Σ0|R := Σ0∩P5(R)were zero-dimensional, the real F (V ) would also be discrete, as there is only one one-dimensional component Σ0, so V would contain only finitely many (real) lines, contradictingthe assumption that V is ruled by real lines. Therefore, Σ0|R is a one-dimensional irreduciblecomponent of the real Fano variety of V . It is irreducible, since otherwise the complex Σ0

would be reducible too, as is easily checked.

17

Summarizing, we have shown that there exists exactly one irreducible one-dimensionalcomponent Σ0 of F (V ), and a corresponding one-dimensional subvariety C ⊂ V , such that,for each point p ∈ V \C, Σ0 contains exactly one line that is incident to p (and contained inV ). In addition to Σ0, F (V ) might also contain up to two zero-dimensional (i.e., singleton)components, whose elements are the exceptional lines mentioned above. Let D denote theunion of C and of the at most two exceptional lines; D is clearly a one-dimensional subvarietyof V . Then, for any point p ∈ V \D, there is exactly one line incident to p and contained inV , as claimed. This establishes part (b), and thus completes the proof of the theorem.

Exceptional lines on a singly ruled surface. In view of the proofs of Theorem 2.7 andLemma 2.2, every point on a singly ruled surface V is incident to at least one generator.Hence an exceptional (non-generator) line is a line ℓ ⊂ V such that every point on ℓ isincident to a generator (which is different from ℓ).

Lemma 2.8. Let V be an irreducible ruled surface in R3 or in C3, which is neither a planenor a regulus. Then (i) V contains at most two exceptional lines, and (ii) V contains atmost one exceptional point.

Proof. (i) We use the property, established in [35] and already used in the proof ofLemma 2.6, that for a line ℓ contained in V , the union τ(ℓ) of the lines that meet ℓ andare contained in V is a variety in the complex projective space P3(C). Moreover, if ℓ is anexceptional line of V , then it follows by [35, Lemma 8] that τ(ℓ) = V . Indeed, τ(ℓ) mustbe two-dimensional, since otherwise it would consist of only finitely many lines. Since V isirreducible, τ(ℓ) must then be equal to V .

If V contained three exceptional lines, ℓ1, ℓ2 and ℓ3, then V would have to be either aplane or a regulus. Indeed, otherwise, by Theorem 2.7 (whose proof does not depend onthe number of exceptional lines), there would exist a one-dimensional curve C ⊂ V (thatincludes ℓ1 ∪ ℓ2 ∪ ℓ3), such that every point p ∈ V \ C is incident to exactly one line ℓpcontained in V . As p ∈ V \ C and σ(ℓi) = V , for i = 1, 2, 3, it follows that ℓp intersectsℓ1, ℓ2, and ℓ3.

If ℓ1, ℓ2, and ℓ3 are pairwise skew, p belongs to the regulus Rℓ1,ℓ2,ℓ3 of all lines intersectingℓ1, ℓ2, and ℓ3. We have thus proved that V \ C is contained in Rℓ1,ℓ2,ℓ3 , and as Rℓ1,ℓ2,ℓ3 isirreducible, it follows that V = Rℓ1,ℓ2,ℓ3 .

If ℓ1, ℓ2, and ℓ3 are concurrent but not coplanar then, arguing similarly, V is a cone withtheir common intersection point as an apex. Since a (non-planar) cone has no exceptionallines, as is easily checked, we may ignore this case.

Finally if any pair among ℓ1, ℓ2, ℓ3, say ℓ1, ℓ2, are parallel then V must be the planethat they span, contrary to assumption. If ℓ1 and ℓ2 intersect at a point ξ, disjoint from ℓ3,then V is the union of the plane spanned by ℓ1 and ℓ2 and the plane spanned by ξ and ℓ3,again a contradiction.

Having exhausted all possible cases, the proof of (i) is complete.

(ii) By Theorem 2.7 and (i), all the lines that are contained in V , except for possibly twosuch lines, are parameterized by an irreducible algebraic curve Σ0 in the Plucker space P5.Let p be an exceptional point of V . The set Σ′ of lines incident to p is an algebraic curvecontained in the irreducible curve Σ0, implying that Σ′ = Σ0. This clearly implies that

18

there is at most one exceptional point (and then it does not contain any exceptional line),and the proof of (ii) is complete too.

Remark. We refer the reader to Guth and Katz [15, Lemma 3.5, Corollary 3.6], for yetanother (somewhat more compact) proof of this lemma.

Generic projections preserve non-planarity. Most of the analysis in Section 3 handlestwo-dimensional varieties embedded in R3, and then handles the general case in Rd, ford > 3, by projecting Rd onto some generic 3-flat, so that non-coplanar triples of lines do notproject to coplanar triples. This is easily achieved by repeated applications of the followingtechnical result, reducing the dimension one step at a time.

Lemma 2.9. Let ℓ1, ℓ2, ℓ3 be three non-coplanar lines in Rd, for d ≥ 4. Then, under ageneric projection of Rd onto some hyperplane H, the respective images ℓ∗1, ℓ

∗2, ℓ

∗3 of these

lines are still non-coplanar.

Proof. Assume without loss of generality that the (generic) hyperplane H onto which weproject passes through the origin of Rd, and let w denote the unit vector normal to H. Theprojection h : Rd 7→ H is then given by h(v) = v − (v · w)w.

Assume first that two of the three given lines, say ℓ1, ℓ2, are skew (i.e., not coplanar).Let ℓ1, ℓ2 denote their projection onto H. If ℓ1, ℓ2 are coplanar they are either intersectingor parallel. If they are intersecting, then there are points p1 ∈ ℓ1, p2 ∈ ℓ2 that projectto the same point, i.e., p1 − p2 has the same direction as w. Then w belongs to the set p1−p2‖p1−p2‖ | p1 ∈ ℓ1, p2 ∈ ℓ2. Since this is a two-dimensional set, it will be avoided for a

generic choice of w, which is a generic point in Sd−1, a set that is at least three-dimensional.

If ℓ1, ℓ2 are parallel, let v1, v2 denote the directions of ℓ1, ℓ2. Since v1 − (v1 · w)w andv2 − (v2 · w)w are vectors in the directions of ℓ1, ℓ2, and are thus parallel, it follows thatw must be a linear combination of v1 and v2. Since ‖w‖ = 1, the resulting set of possibledirections is only one-dimensional, and, again, it will be avoided with a generic choice of w.

We may therefore assume that every pair of lines among ℓ1, ℓ2, ℓ3 are coplanar. Sincethese three lines are not all coplanar, the only two possibilities are that either they are allmutually parallel, or all concurrent.

Assume first that they are concurrent, say they all pass through the origin (even thoughthe origin belongs to H, this still involves no less of generality). Their projections are inthe directions vi− (vi ·w)w, for i = 1, 2, 3. If these projections are coplanar then there existcoefficients α1, α2, α3, not all zero, such that

∑i αi(vi − (vi · w)w) = 0. That is, putting

u :=∑

i αivi, we have u = (u · w)w, so u is parallel to w. In this case w belongs to the set ∑i αivi

‖∑i αivi‖ | α1, α2, α3 ∈ R or C. Again, being a two-dimensional set, it will be avoided by

a generic choice of w.

In the remaining case, the lines ℓ1, ℓ2, ℓ3 are mutually parallel, i.e., they all have the samedirection v. Put, for i = 1, 2, 3, ℓi = pi+ tvt∈R, and choose pi so that pi ·v = 0. The planeπ0 spanned by p1, p2, p3 is projected to the plane π spanned by the points p∗i = pi−(pi ·w)w,for i = 1, 2, 3 (since p1, p2, p3 are not collinear, they will not project into collinear points ina generic projection), and the three lines project into a common plane if and only if theirprojections are contained in π, meaning that the projection v∗ = v − (v · w)w is parallel

19

to π, so it must be a linear combination of p∗1, p∗2, and p∗3. A similar argument to those

used above shows that a generic choice of w will avoid the resulting two-dimensional set offorbidden directions. This completes the proof.

3 Proof of Theorem 1.3

In most of the analysis in this section, we will consider the case d = 3. The reduction froman arbitrary dimension to d = 3 will be presented at the end of the section.

We will prove both parts of the theorem “hand in hand”, bifurcating (in a significantmanner) only towards the end of the analysis.

For a point p on an irreducible singly ruled surface V , which is not the exceptionalpoint of V (see Section 2 for its definition), we let ΛV (p) denote the number of generatorlines incident to p and contained in V (so if p is incident to an exceptional line, we do notcount that line in ΛV (p)). We also put Λ∗

V (p) := max0,ΛV (p)− 1. Finally, if V is a coneand pV is its exceptional point (that is, apex), we put ΛV (pV ) = Λ∗

V (pV ) := 0. We alsoconsider a variant of this notation, where we are also given a finite set L of lines (wherenot all lines of L are necessarily contained in V ), which does not contain any of the (atmost two) exceptional lines of V . For a point p ∈ V , we let λV (p;L) denote the number oflines in L that pass through p and are contained in V , with the same provisions as above,namely that we do not count incidences with exceptional lines, nor do we count incidencesoccurring at an exceptional point, and put λ∗

V (p;L) := max0, λV (p;L)− 1. If V is a conewith apex pV , we put λV (pV ;L) = λ∗

V (pV ;L) = 0. We clearly have λV (p;L) ≤ ΛV (p) andλ∗V (p;L) ≤ Λ∗

V (p), for each point p.

Lemma 3.1. Let V be an irreducible singly ruled two-dimensional surface of degree D > 1in R3 or in C3. Then, for any line ℓ, except for the (at most) two exceptional lines of V ,we have

∑

p∈ℓ∩VΛV (p) ≤ D if ℓ is not contained in V ,

∑

p∈ℓ∩VΛ∗V (p) ≤ D if ℓ is contained in V .

Proof. To streamline the analysis and avoid degenerate situations that might arise overthe reals, we confine ourselves to the complex case; as already mentioned, the incidencebounds that we will obtain will automatically hold over the reals too. That is, every realline that we need to count is also a complex line, and if the real line is contained in the realpart of the variety, its complex counterpart is contained in the complex variety, and thearguent then carries over. We note that the difference between the two cases arises becausewe do not want to count ℓ itself—the former sum would be infinite when ℓ is contained inV . Note also that if V is a cone and pV ∈ ℓ, we ignore in the sum the infinitely many linesincident to pV and contained in V .

The proof is a variant of an observation due to Salmon [30] and repeated in Guth andKatz [15] over the real numbers, and later in Kollar [20] over the complex field and othergeneral fields.

20

By Theorem 2.7(a), excluding the exceptional lines of V , the set of lines contained inV can be parameterized as a (real or complex) 1-parameter family of generator lines ℓ(t),represented by the irreducible curve Σ0 ⊆ F (V ). Let V (2) denote the locus of points of Vthat are incident to at least two generator lines contained in V . By Theorem 2.7(b), V (2)

is contained in some one-dimensional curve C ⊂ V .

Let p ∈ V ∩ ℓ be a point incident to k generator lines of V , other than ℓ, for some k ≥ 1.In case V is a cone, we assume that p 6= pV . Denote the generator lines incident to p (otherthan ℓ, if ℓ ⊂ V , in which case it is assumed to be a generator) as ℓi = ℓ(ti), for ti ∈ Σ0 andfor i = 1, . . . , k. If ℓi is a singular point of F (V ), it may arise as ℓ(ti) for several values of ti,and we pick one arbitrary such value. Let π be a generic plane containing ℓ, and considerthe curve γ0 = V ∩π, which is a plane curve of degree D. Since V (2) ⊆ C is one-dimensional,a generic choice of π will ensure that V (2) ∩ π is a discrete set (since ℓ is non-exceptional,it too meets V (2) in a discrete set).

There are two cases to consider: If ℓ is contained in V (and is thus a generator), then γ0contains ℓ. In this case, let γ denote the closure of γ0 \ℓ; it is also a plane algebraic curve, ofdegree at most D− 1. In case ℓ is not contained in V , we put γ := γ0. By Theorem 2.7(a),we can take, for each i = 1, . . . , k, a sufficiently small open (real or complex) neighborhood∆i along Σ0 containing ti, so that all the lines ℓ(t), for t ∈ ⋃k

i=1∆i are distinct. PutVi :=

⋃t∈∆i

ℓ(t). It follows from the discussion in Section 2 (see “Lines on a variety” there)that Vi ∩ π is either a simple arc or a union of simple arcs meeting at p (depending onwhether or not ℓi is a regular point of Σ0); in the latter case, take γi to be any one of thesearcs. Each of the arcs γi is incident to p and is contained in γ. Moreover, since π is generic,the arcs γi are all distinct. Indeed, for any i 6= j, and any point q ∈ γi ∩ γj , there existti ∈ ∆i, tj ∈ ∆j such that ℓ(ti) ∩ π = ℓ(tj) ∩ π = q, and ℓ(ti) 6= ℓ(tj) (by the properties ofthese neighborhoods). Therefore, any point in γi ∩ γj is incident to (at least) two distinctgenerator lines contained in V . Again, the generic choice of π ensures that γi∩γj ⊆ V (2)∩πis a discrete set, so, in particular, γi and γj are distinct.

We have therefore shown that (i) if ℓ is not contained in V then p is a singular point ofγ of multiplicity at least k (for k ≥ 2; when k = 1 the point does not have to be singular),and (ii) if ℓ is contained in V then p is singular of multiplicity at least k + 1. We havek ≥ ΛV (p) (resp., k ≥ Λ∗

V (p)) if ℓ is not contained (resp., is contained) in V . As arguedat the beginning of Section 2, the line ℓ can intersect γ in at most D points, counted withmultiplicity, and the result follows.

We also need the following result (see, e.g., Guth and Katz [15, Corollary 3.3]), whichis an immediate consequence of the Cayley–Salmon–Monge theorem (Theorem 2.1) and asuitable extension of Bezout’s theorem for intersecting surfaces (see Fulton [10, Proposition2.3]).

Proposition 3.2. Let V be an irreducible two-dimensional variety in C3 of degree D. If Vcontains more than 11D2 − 24D lines then V is ruled by (complex) lines.

Corollary 3.3. Let V be a two-dimensional variety in C3 of degree D. Then the numberof lines that are contained in the union of the non-ruled components of V is O(D2).

Proof. Let V1, . . . , Vk denote those irreducible components of V that are not ruled by lines.By Proposition 3.2, for each i, the number of lines contained in Vi is at most 11deg(Vi)

2 −

21

24deg(Vi). Summing over i = 1, . . . , k, the number of lines contained in the union of thenon-ruled components of V is at most

∑ki=1 11deg(Vi)

2 = O(D2).

The following theorem, which we believe to be of independent interest in itself, is themain technical ingredient of our analysis in this section. Note that it holds over both realand complex fields.

Theorem 3.4. Let V be a possibly reducible algebraic surface of degree D > 1 in R3 or inC3, with no planar components. Let P be a set of m points on V and let L be a set of nlines contained in V . Then there exists a subset L0 ⊆ L of at most O(D2) lines, such thatthe number of incidences between P and L \ L0 satisfies

I(P,L \ L0) = O(m1/2n1/2D1/2 +m+ n

). (10)

Remarks. (a) We can always assume that D = O(n1/2), since there always exists atrivariate polynomial of degree O(n1/2) that identically vanishes on all the lines of L. Notethat Theorem 3.4 becomes vacuous when D = Ω(n1/2), as then L0 could be all of L. Inthe real case we indeed assume that D ≪ n1/2 (see Remark (b) following the statement ofTheorem 1.3), for otherwise the bound in Theorem 1.3 is inferior to that in Theorem 1.1.In the complex case we do allow D to be Θ(n1/2)— and as just observed, there is no needto consider larger values of D.

(b) An important feature of the theorem, already noted for the more general Theorem 1.3,and discussed in more detail later on, is that the bound in (10) avoids the term nD, whicharises naturally in many earlier works, e.g., when bounding the number of incidences be-tween points on V and lines not contained in V . This is significant when D is large.

Proof. We first present a quick informal sketch of the proof, to help the reader navigatingthrough it. Skipping over the definition of the subset L0, we consider incidences with thesurviving set L1 := L \ L0. The key technical step is Lemma 3.5, which roughly assertsthat each line ℓ ∈ L1 is incident to at most O(D) other lines of L1. We want to countthe incidences of ℓ with the points of P , so we introduce a parameter ξ, and distinguishbetween ξ-rich points, incident to more than ξ lines of L1, and ξ-poor points, incident toat most ξ lines. The overall number of incidences with the poor points is O(mξ), and thenumber of incidences of a line with the rich points is O(D/ξ), for a total of O(mξ+nD/ξ).Optimizing ξ gives the main term in the bound we seek.

We now proceed to the full proof. As in the proof of Lemma 3.1, we only work over C,and the results are then easily transported to the real case too. Consider the irreduciblecomponents W1, . . . ,Wk of V . By Corollary 3.3, the number of lines contained in the unionof the non-ruled components of V is O(D2), and we place all these lines in the exceptionalset L0. In what follows we thus consider only ruled components of V . For simplicity,continue to denote them as W1, . . . ,Wk, and note that k ≤ D/2.

We further augment L0 as follows. We first dispose of lines of L that are containedin more than one ruled component Wi. We claim that their number is O(D2). Indeed,for any pair Wi, Wj of distinct components, the intersection Wi ∩Wj is a curve of degree(at most) deg(Wi)deg(Wj), which can therefore contain at most deg(Wi)deg(Wj) lines (bythe generalized version of Bezout’s theorem [10, Proposition 2.3], already mentioned in

22

connection with Proposition 3.2). Since∑k

i=1 deg(Wi) ≤ D, we have

∑

i 6=j

deg(Wi)deg(Wj) ≤(∑

i

deg(Wi)

)2

= O(D2),

as claimed. We add to L0 all the O(D2) lines in L that are contained in more than oneruled component, and all the exceptional lines of all singly ruled components. The numberof lines of the latter kind is at most 2k ≤ 2 · (D/2) = D, so the size of |L0| is still O(D2).Hence, each line of L1 := L\L0 is contained in a unique (singly or doubly) ruled componentof V , and is a generator of that component.

The strategy of the proof is to consider each line ℓ of L1, and to estimate the number ofits incidences with the points of P in an indirect manner, via Lemma 3.1, applied to ℓ and toeach of the ruled components Wj of V . We recall that ℓ is contained in a unique componentWi, and treat that component in a somewhat different manner than the treatment of theother components.

In more detail, we proceed as follows. We first ignore, for each singly ruled coniccomponent Wi, the incidences between its apex (exceptional point) pWi and the lines of L1

that are contained in Wi. We refer to these incidences as conical point-line incidences andto the other incidences as non-conical. When we talk about a line ℓ incident to anotherline ℓ′ at a point p, we will say that ℓ is conically incident to ℓ′ (at p) if p is the apex ofsome conic component Wi and ℓ′ is contained in Wi (and thus incident to p). In all othercases, we will say that ℓ is non-conically incident to ℓ′ (at p). Note that this definition isasymmetric in ℓ and ℓ′; in particular, ℓ does not have to lie in the cone Wi. We also notethat the number of conical point-line incidences is at most n, because each line of L1 iscontained in a unique component Wi, so it can be involved in at most one conical incidence(at the apex of Wi, when Wi is a cone).

We next prune away points p ∈ P that are non-conically incident to at most three linesof L1. Note that p might be an apex of some conic component(s) of V ; in this case p isremoved if it is incident to at most three lines of L1 that are not contained in any of thesecomponents. We lose O(m) (non-conical) incidences in this process. Let P1 denote thesubset of the remaining points.

Lemma 3.5. Each line ℓ ∈ L1 is non-conically incident, at points of P1, to at most 4Dother lines of L1.

Proof. Fix a line ℓ ∈ L1 and let Wi denote the unique ruled component that contains ℓ.Let Wj be any of the other ruled components. We estimate the number of lines of L1 thatare non-conically incident to ℓ and are contained in Wj .

If Wj is a regulus, there are at most four such lines, since ℓ meets the quadratic surfaceWj in at most two points, each incident to exactly two generators (and to no other linescontained in Wj). In this case, we write the bound 4 as deg(Wj)+2. Assume then that Wj

is singly ruled. By Lemma 3.1, we have∑

p∈ℓ∩Wj

λWj (p;L1) ≤∑

p∈ℓ∩Wj

ΛWj (p) ≤ deg(Wj).

Note that, by definition, the above sum counts only non-conical incidences (and only withgenerators of Wj , but the exceptional lines of Wj have been removed from L1 anyway).

23

We sum this bound over all components Wj 6= Wi, including the reguli. Denoting thenumber of reguli by ρ, which is at most D/2, we obtain a total of

∑

j 6=i

deg(Wj) + 2ρ ≤ D + 2ρ ≤ 2D.

Consider next the component Wi containing ℓ. Assume first that Wi is a regulus. Eachpoint p ∈ P1 ∩ ℓ can be incident to at most one other line of L1 contained in Wi (the othergenerator of Wi through p). Since p is in P1, it is non-conically incident to at least 3−2 = 1other line of L1, contained in some other ruled component of V . That is, the number oflines that are (non-conically) incident to ℓ and are contained in Wi, which apriorily can bearbitrarily large, is nevertheless at most the number of other lines (not contained in Wi)that are non-conically incident to ℓ, which, as shown above, is at most 2D.

If Wi is not a regulus, Lemma 3.1 implies that

∑

p∈ℓ∩Wi

Λ∗Wi

(p) ≤ deg(Wi) ≤ D,

where again only non-conical incidences are counted in this sum (and only with generators).That is, the number of lines of L1 that are non-conically incident to ℓ (at points of P1) andare contained in Wi is at most D. Adding the bound for Wi, which has just been shown tobe either D or 2D, to the bound 2D for the other components, the claim follows.

To proceed, choose a threshold parameter ξ ≥ 3, to be determined shortly. Each pointp ∈ P1 that is non-conically incident to at most ξ lines of L contributes at most ξ (non-conical) incidences, for a total of at most mξ incidences. Recall that the overall number ofconical incidences is at most n. For the remaining non-conical incidences, let ℓ be a line inL1 that is incident to t points of P1, so that each such point p is non-conically incident toat least ξ+1 lines of L1 (one of which is ℓ). It then follows from Lemma 3.5 that t ≤ 4D/ξ.Hence, summing this over all ℓ ∈ L1, we obtain a total of at most 4nD/ξ incidences. Wecan now bring back the removed points of P \P1, since the non-conical incidences that theyare involved in are counted in the bound mξ. That is, we have

I(P,L1) ≤ mξ + n+4nD

ξ.

We now choose ξ = (nD/m)1/2. For this choice to make sense, we want to have ξ ≥ 3,which will be the case if 9m ≤ nD. In this case we get the bound O

(m1/2n1/2D1/2 + n

). If

9m > nD we take ξ = 3 and obtain the bound O(m). Combining these bounds, and addingthe at most n conical incidences, the theorem follows.

Having established Theorem 3.4, we now proceed to the proof of Theorem 1.3.

The final stretch for non-singular points: The real case. To complete this proof,we need to bound the number I(P,L0) of incidences involving the lines in the exceptionalset L0 yielded by Theorem 3.4. We remark that, in both the real and the complex cases, nospecial properties need to be assumed in the forthcoming analysis for the lines of L0; theonly property that matters is that the size of L0 is small, i.e., |L0| = minn,O(D2). In

24

the real case, we estimate I(P,L0) using Guth and Katz’s bound ([15]; see Theorem 1.1),recalling that no plane contains more than s lines of L0. We thus obtain

I(P,L0) = O(m1/2|L0|3/4 +m2/3|L0|1/3s1/3 +m+ |L0|

)(11)

= O(m1/2n1/2D1/2 +m2/3D2/3s1/3 +m+ n

).

Combining the bounds in Theorem 3.4 and in (11) yields the asserted bound on I(P,L).

We remark that in the first term in (11) we have estimated L3/40 by O(n1/2D1/2) instead

of the sharper estimate O(D3/2). This is because the term O(m1/2n1/2D1/2) appears inTheorem 3.4 anyway, so the sharper estimate has no effect on the overall asymptotic bound.

The final stretch for non-singular points: The complex case. We next estimateI(P,L0) in the complex case. Again, we have n0 := |L0| = O(D2). We may ignore thepoints of P that are incident to fewer than three lines of L0, as they contribute altogetheronly O(m) incidences. Continue to denote the set of surviving points as P . A point p thatis incident to at least three lines of L0 is either a singular point of V (when not all itsincident lines are coplanar) or a flat point of V ; see Section 2, and also Guth and Katz [15],and Elekes et al. [7].

We decompose V into its irreducible components, sort them in some arbitrary fixedorder, and assign each point p ∈ P (resp., line ℓ ∈ L0) to the first component that containsit. Similar to what has been observed above, the number of “cross-incidences”, betweenpoints and lines assigned to different components of V , is O(n0D) = O(D3). We thereforeassume, as we may, that V is irreducible (over C), and write V = Z(f), for an irreducibletrivariate polynomial f of degree D.

As in Section 2, we call a line flat if all its non-singular points are flat. As argued inSection 2 and in earlier works (see, e.g., [7, 15]), since Z(f) is not a plane, there exists acertain polynomial Π satisfying (i) deg(Π) ≤ 3D − 4, (ii) Z(f,Π) is a curve, and (iii) theflat points of P and the flat lines of L0 are contained in Z(f,Π).

We need the following lemma, adapted (with almost the same proof, which we omit)from a similar lemma that was established in our earlier work [37] for the four-dimensionalcase (see also Section 4).

Lemma 3.6 ([37, Lemma 2.15]). Let f ∈ C[x, y, z] be an irreducible polynomial. If a lineℓ ⊂ Z(f) is flat, then all the tangent planes TpZ(f), for all the non-singular points p ∈ ℓ,coincide.

All this implies, arguing as in previous works [7, 15], that a line ℓ ∈ L0 that is non-singular and non-flat contains at most 4D−4 points of P (each of which is either singular orflat). We prune away these lines from L0, losing at most (4D − 4)|L0| = O(D3) incidenceswith the points of P . Continue to denote the subset of surviving lines as L0.

Therefore, it remains to bound the number of incidences between the surviving pointsand the surviving lines, each of which is either singular or flat. Write P as the union ofthe subset Pf of flat points and the subset Ps of singular points. Similarly, write L0 asthe union of the subset Lf of flat lines and the subset Ls of singular lines. A singular line

25

contains no flat points, and a flat line contains at most D − 1 singular points. Thus,

I(P,L0) ≤ I(Pf , Lf ) + I(Ps, Ls) + n0D.

By Lemma 3.6, all the non-singular points of a flat line have the same tangent plane. Assigneach point p ∈ Pf (resp., line in Lf ) to its tangent plane TpZ(f) (resp., TpZ(f) for somenon-singular point p ∈ Pf ∩ ℓ; we only consider lines in Lf that are incident to at least onepoint in Pf ). We have therefore partitioned the points in Pf and the lines in Lf amongplanes in some finite set H = h1, . . . , hk, and we only need to count incidences betweenpoints and lines assigned to the same plane. Within each h ∈ H, we have a set Ph ⊆ Pf

of mh points in h, and a set Lh ⊆ Lf of nh lines contained in h. Using the planar bound(1), which also holds in the complex plane [45, 47], the number of incidences within h is

O(m2/3h n

2/3h +mh+nh). Summing these bounds over h ∈ H, and using Holder’s inequality,

and the fact that nh ≤ s for each h, we obtain a total of O(m2/3n1/30 s1/3+m+n0) incidences.

Bounding incidences involving singular points and lines. To complete the proof,when the ambient space is three-dimensional, we continue with both the real and the com-plex cases. We actually work over C, and obtain the results over the reals as an immediateconsequence. Bounding the number of incidences between the singular points and linesis done via a procedure that may be referred to as “degree reduction”, to be describednext. Assuming, without loss of generality, that fx (namely, the partial derivative of fwith respect to x) does not vanish identically on Z(f), the points of Ps and the lines ofLs are then contained in Z(fx), and deg(fx) ≤ D − 1. We thus construct a sequence ofpartial derivatives of f that are not identically zero on Z(f). For this we assume, as wemay, that f , and each of its derivatives, are square-free; whenever this fails, we replace thecorresponding derivative by its square-free counterpart before continuing to differentiate.Without loss of generality, assume that this sequence is f, fx, fxx, and so on, and that nosquare-free reduction is ever needed. Denote the j-th element in this sequence as fj , forj = 0, 1, . . . (so f0 = f , f1 = fx, and so on). Assign each point p ∈ P to the first polynomialfj in the sequence for which p is non-singular; more precisely, we assign p to the first fj forwhich fj(p) = 0 but fj+1(p) 6= 0 (recall that f0(p) is always 0 by assumption). Similarly,assign each line ℓ to the first polynomial fj in the sequence for which ℓ is contained in Z(fj)but not contained in Z(fj+1) (again, by assumption, there always exists such a polynomialfj). If ℓ is assigned to fj then it can only contain points p that were assigned to some fkwith k ≥ j. Indeed, if ℓ contained a point p assigned to fk with k < j then fk+1(p) 6= 0 butℓ is contained in Z(fk+1), since k+1 ≤ j; this is a contradiction that establishes the claim.

Fix a line ℓ ∈ L, which is assigned to some fj . An incidence between ℓ and a pointp ∈ P , assigned to some fk, for k > j, can be charged to the intersection of ℓ with Z(fj+1)at p (by construction, p belongs to Z(fj+1)). The number of such intersections is at mostdeg(fj+1) ≤ D − j − 1 ≤ D, so the overall number of incidences of this sort, over all linesℓ ∈ L, is O(n0D) = O(D3). It therefore suffices to consider only incidences between pointsand lines that are assigned to the same zero set Z(fi).

The reductions so far have produced a finite collection of up to D polynomials, each ofdegree at most D, so that the points of P are partitioned among the polynomials and soare the lines of L, and we only need to bound the number of incidences between points andlines assigned to the same polynomial. Moreover, for each j, all the points assigned to fjare non-singular, by construction. For each j, let Pj and Lj denote the subsets of P and

26

of L0, respectively, that are assigned to fj , and put mj := |Pj | and nj := |Lj |. We have∑j mj ≤ m and

∑j nj ≤ n0.

We would like to apply the preceding analysis to Pj and Lj , but we face the technicalissue that fj might be reducible and have some linear factors. The theorem does not requirethe variety to be irreducible, but forbids it to have linear components.

We therefore proceed as follows. We first consider only those points and lines that arecontained in some nonlinear component of Z(fj). We apply the preceding analysis to thesesets, and obtain the incidence bound

O(m

2/3j n

1/3j s1/3 +mj + njD

).

Summing these bounds over all fj ’s, using Holder’s inequality, we get the overall bound

O(m2/3n

1/30 s1/3 +m+ n0D

).

To bound the number of incidences involving points and lines in the linear components ofZ(fj), for any fixed j, we order arbitrarily the planar components of Z(fj), assign eachpoint to first component that contains it, and assign each line to the first component thatcontains it. As Z(fj) has at most D such components, the number of “cross incidences,”between points and lines assigned to different components, is O(njD). For the number of“same component” incidences, we use the Toth-Zahl extension of the Szemeredi-Trotterbound [45, 47] in each plane, and sum them up, exactly as in the case of flat points andlines, and get a total bound of

O(m

2/3j n

1/3j s1/3 +mj + njD

),

and, summing these bounds over all fj ’s, as above, we get the overall bound

I(Ps, Ls) = O(m2/3n

1/30 s1/3 +m+ n0D

),

thereby completing the proof of part (b) (for the case where the ambient space is three-dimensional).

Reduction to three dimensions. To complete the analysis, we need to consider thecase where V is a two-dimensional variety embedded in Rd or in Cd, for d > 3.

Let H be a generic 3-flat, and denote by P ∗, L∗, and V ∗ the respective orthogonalprojections of P,L, and V onto H. Since H is generic, we may assume that all the projectedpoints in P ∗ are distinct, and so are all the projected lines in L∗. Clearly, every incidencebetween a point of P and a line of L corresponds to an incidence between the projectedpoint and line. Since no 2-flat contains more than s lines of L, and H is generic, repeatedapplications of Lemma 2.9 imply that no plane in H contains more than s lines of L∗.

One subtle point is that the set-theoretic projection V ∗ of V does not have to be a realalgebraic variety (in general, it is only a semi-algebraic set), but it is always contained in atwo-dimensional real algebraic variety V , which we call, as we did in an earlier work [36],the algebraic projection of V . The property that deg(V ) ≤ D is well known, and follows bystandard arguments in algebraic geometry; see, e.g., Beltrametti et al. [2, Proposition 3.4.8],

27

and also Harris [16]. That V does not contain a 2-flat follows by a suitable adaptation of theargument in [36, Lemma 2.1] (which is stated there for d = 4 over the reals), that appliesfor general d and over the complex field too.

In conclusion, we have I(P,L) ≤ I(P ∗, L∗), where P ∗ is a set of m points and L∗ is a setof n lines, all contained in the two-dimensional algebraic variety V , embedded in 3-space,which is of degree at most D and does not contain any plane, and no plane contains morethan s lines of L∗. The preceding analysis thus implies that the bound asserted in thetheorem applies in any dimension d ≥ 3.

4 Proof of Theorem 1.5

In most of this section we assume that the ambient space is four-dimensional, and work,with a few exceptions, over the complex field. The reduction from higher dimensions tofour dimensions is handled as the reduction to three dimensions at the end of the previoussection.

We exploit the following useful corollary of Theorem 1.3 (recall that we are now in fourdimensions). Note that in the bound given below, the term nD does not appear yet.

Corollary 4.1. Let f and g be two 4-variate polynomials, over R or C, of degree O(D),such that Z(f, g) is two-dimensional over C. Let P be a set of m points and L a set of nlines, such that all the points of P and all the lines of L are contained in the union of theirreducible components of Z(f, g) that are not 2-flats. Assume also that no 2-flat containsmore than s lines of L. Then we have(a) in the real case:

I(P,L) = O(m1/2n1/2D +m2/3D4/3s1/3 +m+ n

), (12)

(b) and in the complex case:

I(P,L) = O(m1/2n1/2D +m2/3D4/3s1/3 +D6 +m+ n

). (13)

Proof. Let Z(f, g) =⋃s

i=1 Vi be the decomposition of Z(f, g) into its irreducible com-ponents. By the generalized version of Bezout’s theorem [10], we have

∑si=1 deg(Vi) ≤

deg(f)deg(g) = O(D2). Assume that V1, . . . , Vk are the components that are not 2-flats,for some k ≤ s, and let W denote their union. As just observed, deg(W ) = O(D2). Apply-ing to W Theorem 1.3(a) (over the reals) or Theorem 1.3(b) (over the complex field) thuscompletes the proof.

The application of Theorem 1.3(b) goes smoothly in the complex case. In the realcase, it is possible that ZC(f, g) is two-dimensional while some components of its real partZ(f, g) = ZC(f, g) ∩ R2 are only one- (or zero-)dimensional. In the latter case, we applyTheorem 3.4 to ZC(f, g), viewed as a complex algebraic variety of degree O(D2), and deducethat there exists a subset L0 ⊆ L of at most O(D4) lines, such that the number of incidencesbetween P and L\L0 satisfies I(P,L\L0) = O

(m1/2n1/2D +m+ n

). To bound the number

of incidences between P and L0, we notice that L0 is a set of real lines, and proceed preciselyas in the final stretch for the real case in the proof of Theorem 1.3(a), thus completing theproof.

28

Remark. Corollary 4.1(a) is significant when D ≪ n1/4. For larger values of D, we canproject P , L, and Z(f, g) onto some generic 3-flat, and apply the incidence bound of Guthand Katz, as given in Theorem 1.1, within that 3-flat. When D ≥ n1/4, the resulting boundis better than the one in (12).

The proof of Theorem 1.5 revisits the proof of Theorem 1.2, as presented in [37], andapplies Corollary 4.1 as a major technical tool. The only difference between the real andthe complex cases is in the application of that corollary. Except for this application, we willwork over the complex domain, but the analysis carries over, in a straightforward manner,to the real case too.

Each line ℓ ∈ L that is not contained in V contributes at most D incidences, for a totalof O(nD) incidences. We thus assume, as we may, that all the lines of L are containedin V . Let V =

⋃ti=1 Vi be the decomposition of V into its irreducible components, and

assign each point p ∈ P , (resp., line ℓ ∈ L) to the first Vi (the one with the smallest indexi) that contains it (such a Vi always exists for a line ℓ). It is easy to verify that pointsand lines that are assigned to different Vi’s contribute at most nD incidences. Indeed, anysuch incidence (p, ℓ) can be charged to an intersection point of ℓ with the component Vi

that p is assigned to, and thus there are at most∑

i deg(Vi) = D such incidences for eachℓ, for an overall number of O(nD) such incidences. Therefore, it suffices to establish thebound in (5) or in (6) for the number of incidences between points and lines assigned to thesame component. Moreover, once we establish (5) or (6) for each irreducible component Vi

separately, the bound for the entire V follows by an easy application of Holder’s inequality,as detailed below. We thus assume that V is irreducible, and write V = Z(f), for some realor complex irreducible polynomial f of degree D.

We assume for now that P consists exclusively of non-singular points of the irreduciblevariety Z(f). The treatment of the singular points, similar to the handling of singularpoints in the proof of Theorem 1.3(b), will be given towards the end of the proof.

We recall the definition of the four-dimensional flecnode polynomial FL4f of f , as given

in Section 2. That is, FL4f vanishes at each flecnode p ∈ Z(f), namely, points p for whichthere exists a line that is incident to p and osculates to Z(f) up to the fourth order. Clearly,FL4f vanishes identically on every line of L, and thus also on P (assuming that each pointof P is incident to at least one line of L). As noted in Section 2, its degree is O(D).

If FL4f does not vanish identically on Z(f), then Z(f,FL4f ) := Z(f) ∩ Z(FL4f ) is a two-

dimensional variety that contains P and all the lines of L, and is of degree deg(f)·deg(FL4f ) =O(D2) (so we are in the setup assumed in Corollary 4.1). The other possibility is that FL4fvanishes identically on Z(f), and then Landsberg’s theorem [22], mentioned in Section 2(see also [37] for details), implies that Z(f) is ruled by (real or complex) lines.

First case: Z(f,FL4f ) is two-dimensional. Put g = FL4f and apply Corollary 4.1 to fand g. In the real case we obtain the bound

O(m1/2n1/2D +m2/3D4/3s1/3 +m+ n

),

and in the complex case we obtain the bound

O(m1/2n1/2D +m2/3D4/3s1/3 +D6 +m+ n

),

over all components of Z(f,FL4f ) that are not 2-flats.

29

Incidences within 2-flats contained in Z(f,FL4f ). From this point on, throughoutmost of the proof, there is no distinction between the real and complex cases, so we workstrictly over the complex domain; the differences between the real and complex cases willsurface again when we discuss incidences with singular points and the case where V isembedded in higher dimensions. The strategy here is to distribute the points of P andthe lines of L among the 2-flats that contain them (lines not contained in any 2-flat arecontained in some other component of Z(f,FL4f ) and are dealt with, as above, within thatcomponent). See also below for a more detailed account of this strategy. Points that belongto at most two such 2-flats get duplicated at most twice, and we bound the number ofincidences with these points by applying the planar bound (1) (which, as we recall, alsoholds in the complex plane [45, 47]) to each 2-flat separately, and sum up the bounds, toget O(m2/3n1/3s1/3+m+n), using Holder’s inequality, combined with the assumption thatno 2-flat contains more than s lines of L.

For the remaining points in P (points that belong to at least three 2-flats containedin Z(f,FL4f )), we proceed as follows. We first recall from Section 2 that a non-singularpoint p of Z(f) is linearly flat, if it is incident to at least three 2-flats that are contained inZ(f) (and thus also in the tangent hyperplane TpZ(f)). Linearly flat points are also flat,meaning that the second fundamental form of f vanishes at them (see Section 2 and [37]).This property, at a point p, is expressed by several polynomials of degree at most 3D − 4vanishing at p (see Section 2 and [37, Section 2.5]). We also recall that a line is flat if allits non-singular points are flat. Each line of L that is not flat contains at most O(D) flatpoints, and thus the non-flat lines contribute a total of at most O(nD) incidences with flatpoints, so we assume in what follows that the points of P and the lines of L are all flat.Since Z(f) is not a hyperplane, the second fundamental form does not vanish identicallyon Z(f) (this property holds in any dimension; see, e.g. [18, Exercise 3.2.12.2]), and it thenfollows from the characterization of flat points that there exists a certain polynomial Πsatisfying (i) deg(Π) ≤ 3D− 4, (ii) Z(f,Π) is two-dimensional, and (iii) the (flat) points ofP and the (flat) lines of L are contained in Z(f,Π).

Similar to Lemma 3.6, we have the property that all the (flat) points that lie on thesame flat line have the same tangent hyperplane to Z(f) (see [37, Lemma 2.15]). Using thisproperty, we partition the points and lines among tangent hyperplanes of Z(f), so that eachpoint p (assumed to be non-singular on Z(f)) is assigned to its tangent hyperplane to Z(f),and each line ℓ is assigned to the common hyperplane that is tangent to all non-singularpoints on ℓ. Let H denote the resulting set of tangent hyperplanes. Clearly, it suffices tobound the number of incidences within each hyperplane in H.

For each hyperplane h ∈ H, we now have a set Ph ⊆ P of mh points on h, a set Lh ⊆ Lof nh lines contained in h, and a set Fh of 2-flats, each of which is a 2-flat component ofZ(f,FL4f ) that is contained in h. Each point p ∈ Ph is contained in at least one 2-flatin Fh, and each line ℓ ∈ Lh is contained in at least one 2-flat in Fh. Note that we have∑

hmh ≤ m and∑

h nh ≤ n. Notice also that each 2-flat in Fh is also contained in thetwo-dimensional surface Z(f) ∩ h (it is indeed two-dimensional since Z(f) is assumed notto have any hyperplane component), which is of degree D, so, by the generalized version ofBezout’s theorem [10], it can contain at most D 2-flats, so we have |Fh| ≤ D. We assigneach point p ∈ Ph (resp., line ℓ ∈ Lh) to the first 2-flat in Fh (in some arbitrary fixedorder) that contains it. Similar to what has been observed above, the number of “cross-incidences”, between points and lines assigned to different 2-flats within h, is at most nhD,

30

for a total, over the hyperplanes h ∈ H, of at most nD incidences. Again, using the (complexversion of the) planar bound (1) and Holder’s inequality, the numbers of incidences within

the 2-flats of h sum up to O(m2/3h n

2/3h + mh + nhD), and, summing over the hyperplanes

h ∈ H, using Holder’s inequality once again, and the fact that nh ≤ s, we obtain a total ofO(m2/3n1/3s1/3 +m+ nD) incidences.

Remark. The novel feature of this step of the proof, as compared with the analogousargument used in [37], is that the number of 2-flats in Fh, for any fixed h, is at most D.This allows us to bound the number of incidences within each hyperplane h ∈ H separately,so that, within each such hyperplane, instead of using the Guth-Katz bound (in the realcase), we partition the points and lines among at mostD planes, and then use the Szemeredi-Trotter bound in the real case, or the Toth-Zahl bound in the complex case. The fact thatthere are at most D planes within each hyperplane h ∈ H guarantees that the number of“cross-incidences” (within h) is at most nhD, for a total of nD incidences (notice that, incontrast, the total number of planes (over all h ∈ H) can be arbitrarily large).

Second case: Z(f) is ruled by (complex) lines. We next consider the case wherethe four-dimensional flecnode polynomial FL4f vanishes identically on Z(f). By Landsberg’stheorem [22], this implies that Z(f) is ruled by (complex) lines.

Similar to the notations for the three-dimensional case treated in Section 2, we denoteby Σ3

p (resp., Σp), for p ∈ Z(f), the set of all lines that are incident to p and osculateto Z(f) to order 3 at p (resp., are contained in Z(f)). We put Σ3 :=

⋃p∈Z(f)Σ

3p, and

Σ :=⋃

p∈Z(f)Σp. Σ is the Fano variety of (lines contained in) Z(f), now represented in ahigher-dimensional projective space.

In [37], we proved that, for each p ∈ P , either |Σp| ≤ 6 or Σp is infinite. In the interestof completeness, we sketch here the outline of this argument. The analysis provides analgebraic characterization of points p of the latter kind, which uses an auxiliary polynomialU = U(p;u0, u1, u2, u3), called the u-resultant, defined in terms of f and its derivatives at p(see [37] and also [3] for details), where (u0, . . . , u3) denotes the direction of a line incidentto p (in homogeneous coordinates). The polynomial U is of degree O(D) in p and is ahomogeneous polynomial of degree15 six in u. The characterization is that Σ3

p is infinite ifand only if U(p;u0, u1, u2, u3) ≡ 0, as a polynomial of u, at p. In the complementary case,Bezout’s theorem [10] can be used to show that there are only six lines in Σ3

p, and thus atmost six lines in Σp. Pruning away points p ∈ P with |Σ3

p| ≤ 6 (the number of incidencesinvolving these points is at most 6m = O(m)), we may then assume that Σ3

p is infinite forevery p ∈ P .

If U(p;u0, u1, u2, u3) does not vanish identically (as a polynomial in u0, u1, u2, u3) atevery point p ∈ Z(f), then at least one of its coefficients, call it cU , which is a polynomialin p, of degree O(D), does not vanish identically on Z(f). In this case, as U vanishesidentically at every point of P (as a polynomial in u0, u1, u2, u3), we have P ⊂ Z(f, cU ),which is a two-dimensional variety. The machinery developed in the first case can then beapplied here (with g = cU ), and the bounds and properties derived for that case hold heretoo.

We may therefore assume that U(p;u0, u1, u2, u3) ≡ 0 at every non-singular point p ∈15The system of three homogeneous polynomial equations of degree 1, 2, 3, respectively, in the variable

(u0 : u1 : u2 : u3) have either at most 1 · 2 · 3 = 6 distinct solutions or infinitely many.

31

Z(f) (as a polynomial in u0, u1, u2, u3). By the aforementioned characterization via u-resultants, it follows that Σ3

p is then infinite at each such point.

We now use another theorem of Landsberg [18, Theorem 3.8.7]:

Theorem 4.2 (Landsberg). Let f be an irreducible polynomial over P4(C), such that thereexists an irreducible component Σ3

0 ⊂ Σ3 = Σ3(Z(f)) with the property that, for every pointp in a Zariski-open and dense set O ⊂ Z(f), dimΣ3

0,p ≥ 1, where Σ30,p is the set of lines in

Σ30 incident to p. Then, for every point p ∈ O, all lines in Σ3

0,p are contained in Z(f); that

is, for each p ∈ O, Σ30,p is equal to the set Σ0,p of lines incident to p and contained in Z(f).

Since Σ3p is infinite at each non-singular point p ∈ Z(f), its dimension is ≥ 1 at each such

point. As shown in [37], the main condition in Landsberg’s theorem, about the existenceof a component Σ3

0 of Σ3 with the required properties, is satisfied too. One can then arguethat the conclusion of Landsberg’s theorem holds at every point of Z(f); see Lemma 2.2 inSection 2 for a similar claim concerning two-dimensional surfaces. That is, Z(f) is infinitelyruled by (complex) lines, in the sense that each point p ∈ Z(f) is incident to infinitely many(complex) lines that are contained in Z(f), and, moreover, Σ3

0,p = Σ0,p at each p. That is,

Σ30 is contained in Σ. Denoting this set as Σ0, it is shown (in full detail) in [37] that the

union of the lines in Σ0 is equal to Z(f), and that dim(Σ0) ≥ 3.

Severi’s theorem. The following theorem was already used in [37], and we make a similaruse thereof here too. It has been obtained by Severi [33] in 1901. A variant of this resulthas also been obtained by Segre [31]; see also the more recent works [23, 27, 28]. The readercan find a (sketch of a) proof of this theorem in [37] (as suggested to us by A. J. de Jong).We state here a special case of the theorem that we need.

Theorem 4.3 (Severi’s Theorem [33]; special case). Let X ⊂ P4(C) be a three-dimensionalirreducible variety, and let Σ0 be a component of maximal dimension of the Fano varietyΣ = Σ(X) of X, such that the lines of Σ0 cover X. Then the following holds. (i) Ifdim(Σ0) = 4, then X is a hyperplane. (ii) If dim(Σ0) = 3, then either X is a quadric, orX is ruled by 2-flats.

Informally, dim(Σ0) = 3 corresponds to the case where X is infinitely ruled by linesof Σ0: There are four degrees of freedom to specify a line in Σ0, three to specify p ∈ X,and one to specify the line in Σ0,p. We can assume that Σ0,p is one-dimensional, becauseif it were two-dimensional, then X would have been a hyperplane. However, one degree offreedom has to be removed, to account for the fact that the same line (being contained inX) arises at each of its points. Severi’s theorem asserts, again informally, that in this casethe infinite family of lines of Σ0,p must form a 2-flat, unless X is a quadric or a hyperplane.Moreover, by [37, Theorem 3.9] (whose proof is based on Theorem 2.3, Σ0 has maximaldimension.

Applying the second case in Severi’s theorem to Z(f), which is justified by the precedingarguments, we conclude that either Z(f) is a quadric or it is ruled by 2-flats. The caseswhere Z(f) is a quadric or a hyperplane are ruled out by our assumption, so we only needto consider the case where Z(f) is ruled by (complex) 2-flats.

The case where Z(f) is ruled by 2-flats. Handling this last step is somewhat intricate;it resembles the analysis of flat points and lines in the first case, where here points and lines

32

are partitioned among the ruling 2-flats. In this case, every point p ∈ Z(f) is incident to atleast one 2-flat τp ⊂ Z(f). Let Dp denote the set of 2-flats that are incident to p and arecontained in Z(f).

For a non-singular point p ∈ Z(f), if |Dp| > 2, then p is a (linearly flat and thus) flatpoint of Z(f). Recall that we have bounded the number of incidences involving flat points(and lines) by partitioning them among a finite number of containing hyperplanes, and bybounding the incidences within each hyperplane. Lines incident to fewer than 3D−4 pointsof P have been pruned away, losing only O(nD) incidences, and the remaining lines are allflat. Repeating this argument here, noticing that here too, the number of 2-flats containedin a hyperplane is at most D, we obtain the bound

O(m2/3n1/3s1/3 +m+ nD

).

In what follows we therefore assume that all points of P are non-singular and non-flat (callthese points, as in [37], ordinary for short), and therefore |Dp| = 1 or 2, for each such p. PutH1(p) (resp., H1(p), H2(p)) for the 2-flat (resp., two 2-flats) in Dp, when |Dp| = 1 (resp.,|Dp| = 2).

Clearly, each line in L, containing at least one ordinary point p ∈ Z(f), is contained inat most two 2-flats contained in Z(f) (namely, the 2-flats of Dp).

Assign each ordinary point p ∈ P to each of the at most two 2-flats in Dp, and assigneach line ℓ ∈ L that is incident to at least one ordinary point to the at most two 2-flatsthat contain ℓ and are contained in Z(f) (it is possible that ℓ is not assigned to any 2-flat—see below). Changing the notation, enumerate these 2-flats, over all ordinary pointsp ∈ P , as U1, . . . , Uk, and, for each i = 1, . . . , k, let Pi and Li denote the respective subsetsof points and lines assigned to Ui, and let mi and ni denote their cardinalities. We thenhave

∑imi ≤ 2m and

∑i ni ≤ 2n, and the total number of incidences within the 2-flats

Ui (excluding lines not assigned to any 2-flat) is at most∑k

i=1 I(Pi, Li). This incidencecount can be obtained exactly as in the first case of the analysis, with the aid of Holder’sinequality, and yields the bound

k∑

i=1

I(Pi, Li) = O(m2/3n1/3s1/3 +m+ n

).

As noted, this bound does not take into account incidences involving lines which are notcontained in any of the 2-flats Ui (and are therefore not assigned to any such 2-flat). Itsuffices to consider only lines of this sort that are non-singular and non-flat, since singularor flat lines are only incident to singular or flat points, and we assumed above that all thepoints of P are ordinary points. If ℓ is a non-singular and non-flat line, and is not containedin any of the Ui, we call it a piercing line of Z(f). We need the following lemma from [37].

Lemma 4.4 ([37, Lemma 3.13]). Let p ∈ Z(f) be an ordinary point. Then p is incident toat most one piercing line.

Therefore, each ordinary point p ∈ P is incident to at most one piercing line, and thetotal contribution of incidences involving ordinary points and piercing lines is at most m.

In conclusion, combining the bounds that we have obtained for the various subcases ofthe second case, we get that the number of incidences accounted for so far, involving onlynon-singular points, satisfies the desired bound in (5).

33

Incidences involving singular points of Z(f). The forthcoming reasoning is verysimilar to the handling of singular points and lines in the proof of Theorem 1.3(b), althoughit is somewhat more involved because we need to ensure that the resulting polynomials thatwe construct be irreducible; we present the analysis in detail, for the sake of clarity.

In the analysis presented so far, we have assumed that the points of P are non-singularpoints of Z(f). To reduce the general setup to this situation we proceed as follows, similarto the way singularities were handled in Section 3; an identical reduction has also been usedin [37]. We only handle lines that are contained in Z(f), because the other lines contributeat most O(nD) incidences. We construct a sequence of partial derivatives of f that are notidentically zero on Z(f). For this we assume, as Section 3, that f , and each of its derivatives,are square-free. As before, assume that this sequence is obtained by always differentiatingwith respect to x, and denote the j-th element in this sequence as fj , for j = 0, 1, . . .. Assigneach point p ∈ P to the first polynomial fj in the sequence for which p is non-singular; moreprecisely, we assign p to the first fj for which fj(p) = 0 but fj+1(p) 6= 0 (recall that f0(p)is always 0 by assumption). Similarly, assign each line ℓ to the first polynomial fj in thesequence for which ℓ is contained in Z(fj) but not contained in Z(fj+1). If ℓ is assigned tofj then it can only contain points p that were assigned to some fk with k ≥ j.

Fix a line ℓ ∈ L, which is assigned to some fj . An incidence between ℓ and a pointp ∈ P , assigned to some fk, for k > j, can be charged to the intersection of ℓ with Z(fj+1)at p (by construction, p belongs to Z(fj+1)). The number of such intersections is at mostdeg(fj+1) ≤ D − j − 1 ≤ D, so the overall number of incidences of this sort, over all linesℓ ∈ L, is O(nD). It therefore suffices to consider only incidences between points and linesthat are assigned to the same zero set Z(fi).

The reductions so far have produced a finite collection of up to O(D) polynomials, eachof degree at most D, so that the points of P are partitioned among the polynomials andso are the lines of L, and each point p is non-singular with respect to the polynomial it isassigned to, and we only need to bound the number of incidences between points and linesassigned to the same polynomial. This is not the end yet, because the various (reducedforms of the) partial derivatives might be reducible, which we want to avoid. Thus, in afinal decomposition step, we split each derivative polynomial fj into its irreducible factors,and reassign the points and lines that were assigned to Z(fj) to the various factors, by thesame “first come first served” rule used above. The overall number of incidences that arelost in this process is again O(nD). The overall number of polynomials is O(D2), as caneasily be checked. Note also that the last decomposition step preserves non-singularity ofthe points in the special sense defined above; that is, as is easily verified, a point p ∈ Z(fj)with fj+1(p) 6= 0, continues to be a non-singular point of the irreducible component it isreassigned to.

We now fix one such final polynomial, call it fj , denote its degree by Dj (which is upperbounded by the original degree D), and denote by Pj and Lj the subsets of the original setsof points and lines that are assigned to fj , and by mj and nj their respective cardinalities.By construction, Z(fj) is irreducible, and Pj consists exclusively of non-singular points ofthe irreducible Z(fj).

The preceding analysis, concerning incidences with non-singular points, yields, for eachj, either the bound (5)

I(Pj , Lj) = O(m

1/2j n

1/2j Dj +m

2/3j n

1/3j s1/3 + njDj +mj

)

34

in the real case, or the bound (6)

I(Pj , Lj) = O(m

1/2j n

1/2j Dj +m

2/3j n

1/3j s1/3 +D6 + njDj +mj

)

in the complex case. Summing these bounds, upper bounding Dj by D, and using Holder’sinequality for the first two terms, we get the bound (5) or (6) for the entire sets P and L.This completes the proof for the case where the variety containing P and the lines of L areembedded in 4-space.

Reduction to the four-dimensional case. To complete the analysis, we need to con-sider the case where V is a three-dimensional variety embedded in d-dimensional space,for d > 4. The analysis follows closely the one at the end of the proof of Theorem 1.3, inSection 3.

Concretely, let H be a generic 4-flat, and denote by P ∗, L∗, and V ∗ the respectiveprojections of P,L, and V onto H. Since H is generic, we may assume that all the projectedpoints in P ∗ are distinct, and so are all the projected lines in L∗. Clearly, every incidencebetween a point of P and a line of L corresponds to an incidence between the projectedpoint and line. Since no 2-flat contains more than s lines of L, and H is generic, repeatedapplications of Lemma 2.9 imply that no 2-flat in H contains more than s lines of L∗.

As in Section 3, the set-theoretic projection V ∗ of V does not have to be a real algebraicvariety, so we use instead the algebraic projection V of V that contains V ∗. That V does notcontain a hyperplane or quadric is argued by a suitable adaptation of the preceding argument(see [36, Lemma 2.1]). Indeed, the case of a hyperplane is straightforward (reasoning as atthe end of the preceding section). For quadrics we have:

Claim. Let X be a three-dimensional algebraic variety in d dimensions, for d ≥ 5, suchthat a generic (real or complex) algebraic projection of V on 4-space is a quadric. Then Xis a quadratic variety in d dimensions.

Proof. Assume to the contrary that X is not a quadratic variety. This implies that, for ageneric (d− 2)-flat h, the curve Ch = X ∩ h is not a quadratic curve. For any 2-flat g ⊂ h,let Ch,g denote the projection of Ch onto g. This implies that for a generic choice of g anda (d − 2)-flat h satisfying g ⊂ h ⊂, the curve Ch,g is not a quadratic planar curve (thatis, a conic section) in g. Next, by taking a suitable rotation of the coordinate frame, wemay assume that g is the x1x2-flat, and h is the x1x2 . . . xd−2-flat. In these coordinates, itis easy to verify that Ch,g can be obtained by first projecting X onto the x1x2xd−1xd-flat,and then cutting it with the x1x2-plane. But the projection of X onto the x1x2xd−1xd-flat(which is actually a generic 4-flat) is a 3-quadric by assumption, and then cutting it withany 2-flat yields a quadratic planar curve, a contradiction that completes the proof.

In conclusion, we have I(P,L) ≤ I(P ∗, L∗), where P ∗ is a set of m points and L∗ is a setof n lines, all contained in the three-dimensional algebraic variety V , embedded in 4-space,which is of degree at most D and does not contain any hyperplane or quadric component,and no 2-flat contains more than s lines of L∗. The preceding analysis thus implies that thebound asserted in the theorem applies in any dimension d ≥ 4.

35

5 Discussion

(1) As already emphasized, most of the analysis in the proof of Theorem 1.3(a) is carriedout over the complex domain. The only place where the proofs of (a) and (b) bifurcateis in the final step. Over the reals we bound I(P,L0) using the bound of Guth and Katz(which only holds over the reals, because of the polynomial partitioning that it employs),as a “black box”. Over the complex field, we use a variant of the analysis to bypass thisstep, and obtain more or less the same bound, except for the additional term O(D3), whichbecomes insignificant for D ≪ m1/3, say.

For three-dimensional varieties, the proof of both parts of Theorem 1.5 are more or lessthe same, with the main difference being the application of the real or complex version ofCorollary 4.1 (which in itself is Theorem 1.3 “in disguise”). Another difference is in theapplication of the planar point-line incidence bound—the bound is the same in both cases,but the sources (Szemeredi-Trotter or Toth-Zahl) are different.

The derivation of fairly sharp point-line incidence bounds over the complex domain inhigher dimensions constitutes, in our opinion, significant progress in this theory.

(2) In view of the lower bound constructions in [15, 37, 40], the new bounds in Theorems 1.3and 1.5 do not hold without the assumption that the points lie on a variety of relativelysmall degree. We also note that, for a three-dimensional variety, we also get rid of the termm1/2n1/2q1/4; this term may arise only when we consider points on hyperplanes or quadrics,but in our case the variety does not contain any such components. Therefore, our theoremsindicate that these terms may only arise if the variety contains such components.

(3) As mentioned in the introduction, Corollary 1.4 can be extended to the case where V ,which is of constant degree D, also contains planes. Here too, we assume that no planecontains more than s lines of L, but this time it is not necessarily the case that s ≤ D.

Let π1, . . . , πk denote the planar components of V , where k ≤ D = O(1). For eachi = 1, . . . , k, the number of incidences within πi, namely, between the set Pi of pointscontained in πi and the set Li of lines contained in πi, in both real and complex cases, is

I(Pi, Li) = O(|Pi|2/3|Li|2/3 + |Pi|+ |Li|

)= O

(m2/3s2/3 +m+ s

).

Summing these bounds over the k = O(1) planes, we get the same asymptotic bound forthe overall number of the incidences within these planes. Any other incidence between apoint p lying in one of these planes πi and a line ℓ not contained in πi can be uniquelyidentified with the intersection of ℓ with πi. The overall number of such intersections is atmost nk = O(n). This leads to the following extension of Corollary 1.4.

Corollary 5.1. Let P be a set of m points and L a set of n lines in Rd or in Cd, for anyd ≥ 3, and let s ≤ n be a parameter, such that all the points and lines lie in a commontwo-dimensional algebraic surface of constant degree, and no 2-flat contains more than slines of L. Then

I(P,L) = O(m2/3s2/3 +m+ n

),

where the constant of proportionality depends on the degree of the surface.

36

(4) As already noted, one of the significant achievements of the analysis in Theorem 1.3 isthat the bound there does not include the term O(nD). Such a term arises naturally, whenone considers incidences between points lying in some irreducible component of V and linesnot contained in that component. These incidences can be bounded by nD, by chargingthem, as above, to line-component intersections. When D is large, eliminating the term nDcan be crucial for the analysis, as demonstrated in our earlier work [37].

(5) An interesting challenge is to establish a similar bound for I(P,L), for the case wherethe points of P lie on a two-dimensional variety V , but the lines need not be contained inV . A trivial extension of the proof adds the term O(nD) to the bound. The challenge is toavoid this term (if possible); see also Remark (4) above.

(6) Similar to item (2), Theorem 1.5(a) can be extended to the case where V also containshyperplane and quadric components, albeit only for the real case. Here, as in Theorem 1.2,we add the condition that no hyperplane or quadric contains more than q lines of L. Notethat here we do not assume that D is a constant.

Let H1, . . . , Hk denote the hyperplane and quadric components of V , where k ≤ D.Assign, whenever applicable, each point (resp., line) to the first Hi that contains it. Asobserved above, the number of “cross-incidences” is O(nD). By [37, Proposition 3.6], thetotal number of incidences within the hyperplanes and quadrics Hi, for i = 1, . . . , k, is

O(m1/2n1/2q1/4 +m2/3n1/3s1/3 +m+ n).

This leads to the following extension of Theorem 1.5(a).

Corollary 5.2. Let P be a set of m points and L a set of n lines in Rd, for any d ≥ 4,and let s ≤ q ≤ n be parameters, such that all the points and lines lie in a commonthree-dimensional algebraic surface of degree D, and assume that (i) no 3-flat or 3-quadriccontains more than q lines of L, and (ii) no 2-flat contains more than s lines of L. Then

I(P,L) = O(m1/2n1/2(D + q1/4) +m2/3n1/3s1/3 + nD +m

). (14)

(7) An interesting offshoot of Lemma 3.5 is the following result.

Proposition 5.3. Let V be a possibly reducible two-dimensional algebraic surface of degreeD > 1 in R3 or in C3, with no plane or regulus components, and let L be a set of n linescontained in V . Then the number of 2-rich points (points incident to at least two lines ofL) is O(nD).

Proof. Partition L into the subsets L1 and L0, as in the proof of Theorem 1.3. Recallthat L0 is the set of all lines that are either contained in non-ruled components of V , orcontained in more than one component, or are exceptional lines on ruled components. ByLemma 3.5, each line of L1 is non-conically incident to only O(D) other lines of L1, for atotal of O(nD) 2-rich points of this sort. Note that we now carry out the analysis withoutpruning any point (we do not want to do that), because V does not contain any plane orregulus component.

The number of 2-rich points that are exceptional points is at most the number of irre-ducible components of V , that is, at most D, so this number is negligible.

37

The number of lines in L0 is O(D2). Let h be a plane or a regulus. The number of linesof L0 contained in h is at most deg(V ∩ h) ≤ 2D = O(

√|L0|) (this holds if we assume, as

we may, that |L0| = Θ(D2)). It therefore follows from Guth and Katz [15] that the numberof 2-rich points involvoing the lines of L0 is O(|L0|3/2) = O(|L0|D).

It remains to consider 2-rich points that are intersection points of a line in L0 and a linein L1. By construction, each line ℓ ∈ L1 is contained in precisely one (ruled) component Wof V . If p ∈ ℓ is also incident to a line ℓ′ ∈ L0 then, again by construction, ℓ′ is containedin another component W ′ of V , which does not contain ℓ. Hence ℓ intersects W ′ in at mostdeg(W ′) points (one of which is p), for a total of at most deg(V ) = D points. Therefore,the number of 2-rich points involving one line in L1 and another in L0 is at most n1D.

As we have exhausted all cases, the assertion follows.

(8) Challenging directions for further research are (a) to bound the number of incidencesbetween points and lines on (d − 1)-dimensional varieties in Rd (or in higher dimensions),for d ≥ 5, (b) to bound the number of r-rich points, for any r ≥ 2, in a finite set of linescontained in such a variety, and (c) to bound the number of incidences between points ona variety and k-flats (under suitable restrictions) in three, four, or higher dimensions.

Acknowledgement. We are deeply grateful to Martin Sombra and to two anonymousreferees for their critical and thorough reading of the paper, and for many valuable com-ments.

References

[1] A. Beauville, Complex Algebraic Surfaces, Vol. 34, Cambridge University Press, Cam-bridge, 1996.

[2] M. Beltrametti, E. Carletti, D. Gallarati and G. M. Bragadin, Lectures on Curves,Surfaces and Projective Varieties: A Classical View of Algebraic Geometry, EuropeanMathematical Society, 2009.

[3] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction toComputational Algebraic Geometry and Commutative Algebra, Springer Verlag, Hei-delberg, 2007.

[4] D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Springer Verlag, Heidel-berg, 2005.

[5] W. L. Edge, The Theory of Ruled Surfaces, Cambridge University Press, Cambridge,2011.

[6] G. Elekes, Sums versus products in number theory, algebra and Erdos geometry–Asurvey, in Paul Erdos and his Mathematics II, Bolyai Math. Soc., Stud. 11, Budapest,2002, pp. 241–290.

[7] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimen-sions, J. Combinat. Theory, Ser. A 118 (2011), 962–977. Also in arXiv:0905.1583.

38

[8] P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

[9] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on ClassicMathematics, Amer. Math. Soc. Press, Providence, RI, 2007.

[10] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, ExpositoryLectures from the CBMS Regional Conference Held at George Mason University, June27–July 1, 1983, Vol. 54. AMS Bookstore, 1984.

[11] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Vol. 52, John Wiley &Sons, New York, 2011.

[12] L. Guth, Ruled surface theory and incidence geometry, in arXiv:1606.07682.

[13] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64,Amer. Math. Soc. Press, Providence, RI, 2016.

[14] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem,Advances Math. 225 (2010), 2828–2839. Also in arXiv:0812.1043.

[15] L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, AnnalsMath. 181 (2015), 155–190. Also in arXiv:1011.4105.

[16] J. Harris, Algebraic Geometry: A First Course, Vol. 133. Springer-Verlag, New York,1992.

[17] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1983.

[18] T. A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry viaMoving Frames and Exterior Differential Systems, Graduate Studies in Mathematics,volume 61, Amer. Math. Soc., Providence, RI, 2003.

[19] N. H. Katz, The flecnode polynomial: A central object in incidence geometry, inarXiv:1404.3412.

[20] J. Kollar, Szemeredi–Trotter-type theorems in dimension 3, Advances Math. 271 (2015),30–61. Also in arXiv:1405.2243.

[21] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer-Verlag, Heidelberg, 2012.

[22] J. M. Landsberg, Is a linear space contained in a submanifold? On the number ofderivatives needed to tell, J. Reine Angew. Math. 508 (1999), 53–60.

[23] E. Mezzetti and D. Portelli, On threefolds covered by lines, in Abhandlungen aus demMathematischen Seminar der Universitat Hamburg, Vol. 70, No. 1, Springer Verlag,Heidelberg, 2000.

[24] G. Monge, Application de l’Analyse a la Geometrie, Bernard, Paris, 1809.

[25] H. Pottmann and J. Wallner, Computational Line Geometry, Springer Science andBusiness Media, Berlin Heidelberg, 2009.

[26] A. Pressley, Elementary Differential Geometry, Springer Undergraduate MathematicsSeries, Springer Verlag, London, 2001.

39

[27] S. Richelson, Classifying Varieties With Many Lines, a senior thesis, Harvard univer-sity, 2008.

[28] E. Rogora, Varieties with many lines, Manuscripta Mathematica 82.1 (1994), 207–226.

[29] M. Rudnev, On the number of incidences between planes and points in three dimen-sions, in arXiv:1407.0426 (2014).

[30] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5thedition, Hodges, Figgis and co. Ltd., Dublin, 1915.

[31] B. Segre, Sulle Vn contenenti piu di ∞n−kSk, Nota I e II, Rend. Accad. Naz. Lincei 5(1948), 193–197, 275–280.

[32] J. M. Selig, Geometric Fundamentals of Robotics, Springer Science and Business Media,2004.

[33] F. Severi, Intorno ai punti doppi impropri etc., Rend. Cir. Math. Palermo 15 (10)(1901), 33–51.

[34] I. R. Shafarevich, Basic Algebraic Geometry, Vol. 197, Springer-Verlag, New York,1977.

[35] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions,in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st Sympos. Comput.Geom. (2015), 553–568, and in arXiv:1501.02544.

[36] M. Sharir and N. Solomon, Incidences between points and lines in R4, Proc. 30th Annu.Sympos. Comput. Geom., 2014, 189–197.

[37] M. Sharir and N. Solomon, Incidences between points and lines in R4, Discrete Comput.Geom. 57 (2017), 702–756. Also in Proc. 56th IEEE Sympos. Foundations of ComputerScience 2015, 1378–1394, and in arXiv:1411.0777.

[38] M. Sharir and N. Solomon, Incidences with curves and surfaces and applications todistinct and repeated distances, Proc. 28th ACM-SIAM Sympos. Discrete Algorithms,2017, 2456–2475. Also in arXiv:1610.01560.

[39] A. Sheffer, E. Szabo and J. Zahl, Point-curve incidences in the complex plane, Combi-natorica, to appear. Also in arXiv:1502.07003.

[40] N. Solomon and R. Zhang, Highly incidental patterns on a quadratic hypersurface inR4, Discrete Math. 340(4) (2017), 585–590. Also in arXiv:1601.01817.

[41] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput.Geom. 48 (2012), 255–280.

[42] J. Solymosi and F. de Zeeuw, Incidence bounds for complex algebraic curves on Carte-sian products, in arXiv:1502.05304.

[43] M. R. Spiegel, S. Lipschutz, and D. Spellman, Vector Analysis, 2nd edition, Schaum’soutlines, McGraw-Hill, 2009.

40

[44] E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combina-torica 3 (1983), 381–392.

[45] C.D. Toth, The Szemeredi-Trotter theorem in the complex plane, Combinatorica 35(2015), 95–126. Also in arXiv:0305283 (2003).

[46] B. L. van der Waerden, E. Artin, and E. Noether, Modern Algebra, Vol. 2, SpringerVerlag, Heidelberg, 1966.

[47] J. Zahl, A Szemeredi-Trotter type theorem in R4, Discrete Comput. Geom. 54 (2015),513–572. Also in arXiv:1203.4600.

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Part IV

Incidences between points andcurves and points and surfaces

191

7 Incidences with curves in Rd

193

Incidences with curves in Rd ∗

Micha Sharir† Adam Sheffer‡ Noam Solomon§

Submitted: Jan 4, 2016; Accepted: Oct 8, 2016 ; Published: Oct 28, 2016

Mathematics Subject Classifications: 52C10

Abstract

We prove that the number of incidences between m points and n bounded-degreecurves with k degrees of freedom in Rd is

O

m k

dk−d+1+εn

dk−ddk−d+1 +

d−1∑

j=2

mk

jk−j+1+εn

d(j−1)(k−1)(d−1)(jk−j+1) q

(d−j)(k−1)(d−1)(jk−j+1)

j +m+ n

,

for any ε > 0, where the constant of proportionality depends on k, ε and d, providedthat no j-dimensional surface of degree 6 cj(k, d, ε), a constant parameter depend-ing on k, d, j, and ε, contains more than qj input curves, and that the qj ’s satisfycertain mild conditions.

This bound generalizes the well-known planar incidence bound of Pach andSharir to Rd. It generalizes a recent result of Sharir and Solomon [21] concern-ing point-line incidences in four dimensions (where d = 4 and k = 2), and partlygeneralizes a recent result of Guth [9] (as well as the earlier bound of Guth andKatz [11]) in three dimensions (Guth’s three-dimensional bound has a better de-pendency on q2). It also improves a recent d-dimensional general incidence boundby Fox, Pach, Sheffer, Suk, and Zahl [8], in the special case of incidences with alge-braic curves. Our results are also related to recent works by Dvir and Gopi [5] andby Hablicsek and Scherr [13] concerning rich lines in high-dimensional spaces. Ourbound is not known to be tight in most cases.

∗Work on this paper was partially supported by Grant 2012/229 from the U.S.-Israel BinationalScience Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers forResearch Excellence (I-CORE) program (center no. 4/11), and by the Hermann Minkowski–MINERVACenter for Geometry at Tel Aviv University. Part of this research was performed while the authors werevisiting the Institute for Pure and Applied Mathematics (IPAM) at UCLA, which is supported by theNational Science Foundation. A preliminary version of this paper appeared in Proc. European Sympos.Algorithms, 2015, pages 977–988.†Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel; michas@post.tau.ac.il‡Dept. of Mathematics, California Institute of Technology, Pasadena, CA, USA; adamsh@gmail.com§Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel;

noamsolomon@post.tau.ac.il

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1 Introduction

Let C be a set of curves in Rd. We say that C has k degrees of freedom with multiplicitys if (i) for every k points in Rd there are at most s curves of C that are incident to all kpoints, and (ii) every pair of curves of C intersect in at most s points. The bounds thatwe derive depend more significantly on k than on s – see below.

In this paper we derive general upper bounds on the number of incidences between aset P of m points and a set C of n bounded-degree algebraic curves that have k degrees offreedom (with some constant multiplicity s). We denote the number of these incidencesby I(P , C).

Before stating our results, let us put them in context. The basic and most studiedcase involves incidences between points and lines. In two dimensions, writing L for thegiven set of n lines, the classical Szemeredi–Trotter theorem [28] yields the worst-casetight bound

I(P , L) = O(m2/3n2/3 +m+ n

). (1)

In three dimensions, in the 2010 groundbreaking paper of Guth and Katz [11], an improvedbound has been derived for I(P , L), for a set P of m points and a set L of n lines in R3,provided that not too many lines of L lie in a common plane. Specifically, they showed:

Theorem 1.1 (Guth and Katz [11]) Let P be a set of m distinct points and L a setof n distinct lines in R3, and let q2 6 n be a parameter, such that no plane contains morethan q2 lines of L. Then

I(P,L) = O(m1/2n3/4 +m2/3n1/3q

1/32 +m+ n

).

This bound was a major step in the derivation of the main result of [11], an almost-linearlower bound on the number of distinct distances determined by any set of n points in theplane, a classical problem posed by Erdos in 1946 [7]. Their proof uses several nontrivialtools from algebraic and differential geometry. This machinery comes on top of the maininnovation of Guth and Katz, the introduction of the polynomial partitioning technique;see below.

In four dimensions, Sharir and Solomon [22] have obtained the following sharp point-line incidence bound:

Theorem 1.2 (Sharir and Solomon [22]) Let P be a set of m distinct points and L aset of n distinct lines in R4, and let q2, q3 6 n be parameters, such that (i) no hyperplaneor quadric contains more than q3 lines of L, and (ii) no 2-flat contains more than q2 linesof L. Then

I(P , L) 6 2c√logm

(m2/5n4/5 +m

)+ A

(m1/2n1/2q

1/43 +m2/3n1/3q

1/32 + n

), (2)

where A and c are suitable absolute constants. When m 6 n6/7 or m > n5/3, we get thesharper bound

I(P , L) 6 A(m2/5n4/5 +m+m1/2n1/2q

1/43 +m2/3n1/3q

1/32 + n

). (3)

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In general, except for the factor 2c√logm, the bound is tight in the worst case, for any

values of m,n, with corresponding suitable ranges of q2 and q3.

This improves, in several aspects, an earlier treatment of this problem in Sharir andSolomon [21].

Another way to extend the Szemeredi–Trotter bound is for curves in the plane withk degrees of freedom (for lines, k = 2). This has been done by Pach and Sharir, whoshowed:1

Theorem 1.3 (Pach and Sharir [18]) Let P be a set of m points in R2 and let C be aset of bounded-degree algebraic curves in R2 with k degrees of freedom and with multiplicitys. Then

I(P , C) = O(m

k2k−1n

2k−22k−1 +m+ n

),

where the constant of proportionality depends on k and s.

Several special cases of this result, such as the cases of unit circles and of arbitrary circles,have been considered separately [4, 26]. Unlike the Szemeredi-Trotter result (which arisesas a special case of Theorem 1.3 with k = 2), the bound in Theorem 1.3 is not known tobe tight for any k > 3. In fact, it is known not to be tight for the case of arbitrary circles;see [1].

Here too one can consider the extension of these bounds to higher dimensions. Ex-cluding this paper, the following theorem states the current best bound for this case

Theorem 1.4 (Fox et al. [8]) Let P be a set of m points and let V be a set of nconstant-degree algebraic varieties, both in Rd, such that the incidence graph of P × Vdoes not contain a copy of Ks,t (here we think of s, t, and d as being fixed constants, andm and n as large). Then for every ε > 0, we have

I(P ,V) = O(m

(d−1)sds−1

+εnd(s−1)ds−1 +m+ n

),

where the constant of proportionality depends on ε, s, t, d, and the maximum degree ofthe varieties.

While the bound of Theorem 1.4 holds for varieties of any dimension, in this paperwe only consider the case of curves. Several better bounds are known for specific typesof curves. The case of lines is studied in several papers, such as [5, 13]. It is also worthmentioning here the work of Sharir, Sheffer and Zahl [20] on incidences between pointsand circles in three dimensions; an earlier study of this problem by Aronov et al. [2] givesa different, dimension-independent bound. A very recent result of Sharir and Zahl [24]gives an improved bound (over the one in Theorem 1.3) for curves in the plane.

The bounds given in Theorem 1.1 and Theorem 1.2 include a “leading term” thatdepends only on m and n (the terms m1/2n3/4 and 2c

√lognm2/5n4/5, respectively), and,

1Their result holds for more general families of curves, not necessarily algebraic, but, since algebraicitywill be assumed in higher dimensions, we assume it also in the plane.

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except for the two-dimensional case, a series of “lower-dimensional” terms (like the term

m2/3n1/3q1/32 in Theorem 1.1 and the terms m1/2n1/2q

1/43 and m2/3n1/3q

1/32 in Theorem 1.2).

The leading terms, in the case of lines, become smaller as d increases (when m is not toosmall and not too large with respect to n). Informally, by placing the lines in a higher-dimensional space, it should become harder to create many incidences on them.

Nevertheless, this is true only if the setup is “truly d-dimensional”. This means thatnot too many lines or curves are allowed to lie in a common lower-dimensional space. Thelower-dimensional terms handle incidences within such lower-dimensional spaces. Thereis such a term for every dimension j = 2, . . . , d−1, and the “j-dimensional” term handlesincidences within j-dimensional subspaces (which, as the quadrics in the case of lines infour dimensions in Theorem 1.2, are not necessarily linear and might be algebraic of lowconstant degree). Comparing the bounds for lines in two, three, and four dimensions, wesee that the j-dimensional term in d dimensions, for j < d, is a sharper variant of theleading term in j dimensions. More concretely, if that leading term in j dimensions ismanb then its counterpart in the d-dimensional bound, for d > j, is of the form mantqb−tj ,where qj is the maximum number of lines that can lie in a common j-dimensional flat orlow-degree variety, and t depends on j and d.

Our results. In this paper we consider a generalization of these results, to the case whereC is a family of bounded-degree algebraic curves with k degrees of freedom (and somemultiplicity s) in Rd. This is a very ambitious and difficult project, and the challengesthat it faces seem to be enormous. Here we make the first step in this direction, and obtainthe following bounds. As the exponents in the bounds are rather cumbersome expressionsin d, k, and j, we first state the special case of d = 3 (and prove it separately), and thengive the general bound in d dimensions.

Theorem 1.5 (Curves in R3) Let k > 2 be an integer, and let ε > 0. Then there existsa constant c(k, ε) that depends on k and ε, such that the following holds. Let P be a setof m points and C a set of n irreducible algebraic curves of constant degree with k degreesof freedom (and some multiplicity s) in R3, such that every algebraic surface of degree atmost c(k, ε) contains at most q2 curves of C. Then

I(P , C) = O

(m

k3k−2

+εn3k−33k−2 +m

k2k−1

+εn3k−34k−2 q

k−14k−2

2 +m+ n

),

where the constant of proportionality depends on k, s, and ε (and on the degree of thecurves).

The corresponding result in d dimensions is as follows.

Theorem 1.6 (Curves in Rd) Let d > 3 and k > 2 be integers, and let ε > 0. Thenthere exist constants cj(k, d, ε), for j = 2, . . . , d−1, that depend on k, d, j, and ε, such thatthe following holds. Let P be a set of m points and C a set of n irreducible algebraic curvesof constant degree with k degrees of freedom (and some multiplicity s) in Rd. Moreover,assume that, for j = 2, . . . , d− 1, every j-dimensional algebraic variety of degree at most

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cj(k, d, ε) contains at most qj curves of C, for given parameters q2 6 · · · 6 qd−1 6 n.Then we have

I(P , C) = O

(m

kdk−d+1

+εndk−d

dk−d+1 +d−1∑

j=2

mk

jk−j+1+εn

d(j−1)(k−1)(d−1)(jk−j+1) q

(d−j)(k−1)(d−1)(jk−j+1)

j +m+ n

),

where the constant of proportionality depends on k, s, d, and ε (and on the degree of thecurves), provided that, for any 2 6 j < l 6 d, we have (with the convention that qd = n)

qj >(ql−1ql

)l(l−2)ql−1. (4)

Note that the constraints (4) for l = j+1 simply require that the sequence q2, . . . , qd−1be (weakly) increasing, as already stipulated.

Discussion. The advantages of our results are obvious: They provide the first nontrivialbounds for the general case of curves with any number of degrees of freedom in anydimension (with the exception of one previous study of Fox et al. [8], in which weakerbounds are obtained, albeit for arbitrary varieties instead of algebraic curves). Apartfor the ε in the exponents, the leading term is “best possible,” in the sense that (i) thepolynomial partitioning technique [11] that our analysis employs (and that has been usedin essentially all recent works on incidences in higher dimensions) yields a recurrence thatsolves to this bound, and, moreover, (ii) it is (nearly) worst-case tight for lines in two,three, and four dimensions (as shown in the respective works cited above), and in factis likely to be tight for lines in higher dimensions too, using a suitable extension of aconstruction, due to Elekes and used in [11, 22].

Nevertheless, our bounds are not perfect, and tightening them further is a majorchallenge for future research. Specifically:

(i) While it seems likely that the leading terms in our bounds are tight for lines in Rd,they are probably not tight for most constant-degree algebraic curves. Sharir and Zahl[24] recently proved better bounds2 for the case of d = 2 and k > 3, and it seems likelythat better bounds also exist in higher dimensions. One common conjecture suggests thatin R2 the number of incidences between any n points and any n constant-degree curves(with no common components) should be O(n4/3); the conjecture does not always holdwhen the number of points and the number of curves are significantly different.

(ii) The bounds involve the factor mε. As the existing works indicate, getting rid of thisfactor is no small feat. Although the factor does not show up in the cases of lines in twoand three dimensions, it already shows up (sort of) in four dimensions (Theorem 1.2), aswell as in the case of circles in three dimensions [20]. (A recent study of Guth [9] alsopays this factor for the case of lines in three dimensions, in order to simplify the originalanalysis in the Guth–Katz paper [11]. Another recent simplified proof, due to Sharir and

2To be precise, it is assumed in [24] that the curves come from a “k-dimensional family of curves”,which is a similar constraint, albeit not quite the same, as having k degrees of freedom.

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Solomon [23], manages to get rid of this factor, except for some narrow range of m andn.) See the proofs and comments below for further elaboration of this issue.

(iii) The condition that no surface of degree cj(k, d, ε) contains too many curves of C, forj = 2, . . . , d − 1, is very restrictive, especially since the actual values of these constantsthat arise in the proofs can be quite large. Again, earlier works also “suffer” from thishandicap, such as Guth’s work [9] mentioned above, as well as an earlier version of Sharirand Solomon’s four-dimensional bound [21]. A recent interesting study of Guth andZahl [12] may offer some tools for better controlling these parameters.

(iv) Finally, the lower-dimensional terms that we obtain are not best possible. Forexample, the bound that we get in Theorem 1.5 for the case of lines in R3 (k = 2)

is O(m1/2+εn3/4 + m2/3+εn1/2q1/62 + m + n). When q2 n, the two-dimensional term

m2/3+εn1/2q1/62 in that bound is worse than the corresponding term m2/3n1/3q

1/32 in The-

orem 1.1 (even when ignoring the factor mε).Since the statement of Theorem 1.6 is rather involved, we also present two simplified

versions thereof. The first is a straightforward corollary as a simpler case.

Corollary 1.7 Let d > 3, k > 2 be integers, and let ε > 0. Then there exists a constantc(k, d, ε) that depends on k, d, and ε, such that the following holds. Let P be a set of mpoints and C a set of n irreducible algebraic curves of some constant maximum degree withk degrees of freedom (and some multiplicity s) in Rd, such that m = O(nd/(d−1)). More-over, assume that every algebraic variety of degree at most c(k, d, ε) contains a constantnumber of curves of C, where this constant may depend on d, k, and ε. Then we have

I(P , C) = O(m

kdk−d+1

+εndk−d

dk−d+1 + n),

where the constant of proportionality depends on k, s, d, ε, and the maximum degree ofthe curves.

The second simplification replaces the sequence of constraints on the number of curvesin lower-dimensional varieties of constant degrees by a single constraint involving only(d− 1)-dimensional varieties (hypersurfaces). Its proof is similar to that of Theorem 1.6,and will be briefly discussed later.

Theorem 1.8 Let d > 3, k > 2 be integers, and let ε > 0. Then there exists a constantc(k, d, ε) that depends on k, d, and ε, such that the following holds. Let P be a set of mpoints, let C be a set of n irreducible algebraic curves of some constant maximum degreeand with k degrees of freedom, both in Rd, and let q 6 n be another parameter, such thatevery hypersurface of degree at most c(k, d, ε) contains at most q curves of C. Then

I(P , C) = O(m

kdk−d+1

+εndk−d

dk−d+1 +mk

2k−1+εn

dk−d(d−1)(2k−1) q

(k−1)(d−2)(d−1)(2k−1) +m+ n

),

where the constant of proportionality depends on ε, k, d, and the maximum degree of thecurves.

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Our results are also related to recent works by Dvir and Gopi [5] and by Hablicsekand Scherr [13], that study rich lines in high dimensions. Specifically, let P be a set ofn points in Rd and let L be a set of r-rich lines (that is, each line of L contains at leastr points of P). If |L| = Ω(n2/rd+1) then there exists a hyperplane containing Ω(n/rd−1)points of P . Our bounds are relevant for extending this result to rich curves. Concretely,for a set P of n points in Rd and a collection C of r-rich constant-degree algebraic curveswith k degrees of freedom, if |C| is too large then the incidence bound becomes largerthan the “leading term” in Theorem 1.8, indicating that some hypersurface must containmany curves of C, which would then imply that such a surface has to also contain manypoints of P . We omit the rather routine, albeit fairly tedious, calculations.

As in the classical work of Guth and Katz [11], and in the numerous follow-up studiesof related problems, here too we use the polynomial partitioning method, as pioneeredin [11]. The reason why our bounds suffer from the aforementioned handicaps is that weuse a partitioning polynomial of (large but) constant degree. (The idea of using constant-degree partitioning polynomials for problems of this kind is due to Solymosi and Tao [25].)When using a polynomial of a larger, non-constant degree, we face the difficult task ofbounding incidences between points and curves that are fully contained in the zero setof the polynomial, where the number of curves of this kind can be large, because thepolynomial partitioning technique has no control over this value. We remark that forlines we have the classical Cayley–Salmon theorem (see, e.g., Guth and Katz [11]), whichessentially bounds the number of lines that can be fully contained in an algebraic surfaceof a given degree, unless the surface is ruled by lines. However, such a property hasnot been known for more general curves. Nevertheless, Nilov and Skopenkov [17] haverecently established such a result involving lines and circles in R3, and, very recently,Guth and Zahl [12] have done the same for general algebraic curves in three dimensions.Handling these incidences requires heavy-duty machinery from algebraic geometry, andleads to profound new problems in that domain that need to be tackled.

In contrast, using a polynomial of constant degree makes this part of the analysis muchsimpler, as can be seen below, but then handling incidences within the cells of the partitionbecomes non-trivial, and a naive approach yields a bound that is too large. To handlethis part, one uses induction within each cell of the partitioning, and it is this inductionprocess that is responsible for the weaker aspects of the lower-dimensional terms in theresulting bound, as well as the extra mε factor in the leading term. Nevertheless, withthese “sacrifices” we are able to obtain a “general purpose” bound that holds for a broadspectrum of instances. It is our hope that this study will motivate further research on thisproblem that would improve our results along the “handicaps” mentioned above. Recallinghow inaccessible were these kinds of problems prior to Guth and Katz’s breakthroughseight and six years ago, it is quite gratifying that so much new ground can be gained inthis area, including the progress made in this paper.

Background. Incidence problems have been a major topic in combinatorial and compu-tational geometry for the past thirty years, starting with the aforementioned Szemeredi-Trotter bound [28] back in 1983 (and even earlier). Several techniques, interesting in theirown right, have been developed, or adapted, for the analysis of incidences, including the

the electronic journal of combinatorics 23(4) (2016), #P4.16 7

crossing-lemma technique of Szekely [27], and the use of cuttings as a divide-and-conquermechanism (e.g., see [4]). Connections with range searching and related algorithmic prob-lems in computational geometry have also been noted and exploited, and studies of theKakeya problem (see, e.g., [29]) indicate the connection between this problem and inci-dence problems. See Pach and Sharir [19] for a comprehensive (albeit a bit outdated)survey of the topic.

The landscape of incidence geometry has dramatically changed in the past eight years,due to the infusion, in two groundbreaking papers by Guth and Katz [10, 11], of new toolsand techniques drawn from algebraic geometry. Although their two direct goals have beento obtain a tight upper bound on the number of joints in a set of lines in three dimensions[10], and a near-linear lower bound for the classical distinct distances problem of Erdos[11], the new tools have quickly been recognized as useful for incidence bounds. See[6, 14, 15, 20, 25, 30, 31] for a sample of recent works on incidence problems that use thenew algebraic machinery.

The present paper continues this line of research, and aims at extending the collectionof instances where nontrivial incidence bounds in higher dimensions can be obtained.

2 The three-dimensional case

Proof of Theorem 1.5. We fix ε > 0, and prove by induction on m+ n that

I(P , C) 6 α1

(m

k3k−2

+εn3k−33k−2 +m

k2k−1

+εn3k−34k−2 q

k−14k−2

)+ α2(m+ n), (5)

where α1, α2 are sufficiently large constants, α1 depends on ε and k (and s), and α2

depends on k (and s).For the induction basis, the case where m,n are sufficiently small constants can be

handled by choosing sufficiently large values of α1, α2.Another base case is m = O(n1/k). Since the incidence graph, as a subgraph of

P×C, does not contain Kk,s+1 as a subgraph, the Kovari-Sos-Turan theorem (e.g., see [16,Section 4.5]) implies that I(P , C) = O(mn1−1/k+n), where the constant of proportionalitydepends on k (and s). When m = O(n1/k), this implies the bound I(P , C) = O(n), whichis subsumed in (5) if we choose α2 sufficiently large. We may thus assume that n 6 cmk, forsome absolute constant c, and that m and n are at least some sufficiently large constants.

Applying the polynomial partitioning technique. We construct an r-partitioningpolynomial f for P , for a sufficiently large constant r (depending on ε). That is, asestablished in Guth and Katz [11], f is of degree O(r1/3) (the constant in the O notationis an absolute constant), and the complement of its zero set Z(f) is partitioned intou = O(r) open connected cells, each containing at most m/r points of P . Denote the(open) cells of the partition as τ1, . . . , τu. For each i = 1, . . . , u, let Ci denote the setof curves of C that intersect τi and let Pi denote the set of points that are contained inτi. We set mi = |Pi| and ni = |Ci|, for i = 1, . . . , u, and m′ =

∑imi, and notice that

mi 6 m/r for each i (and m′ 6 m). An obvious property (which is a consequence of

the electronic journal of combinatorics 23(4) (2016), #P4.16 8

Bezout’s theorem, see, e.g., [25, Theorem A.2]) is that every curve of C intersects O(r1/3)cells of R3 \Z(f). Therefore,

∑i ni 6 bnr1/3, for a suitable constant b > 1 (that depends

on the degree of the curves in C). Using Holder’s inequality, we have

∑

i

n3k−33k−2

i 6(∑

i

ni

) 3k−33k−2

(∑

i

1

) 13k−2

6 b′(nr

13

) 3k−33k−2

r1

3k−2 = b′n3k−33k−2 r

k3k−2 ,

∑

i

n3k−34k−2

i 6(∑

i

ni

) 3k−34k−2

(∑

i

1

) k+14k−2

6 b′(nr

13

) 3k−34k−2

rk+14k−2 = b′n

3k−34k−2 r

k2k−1 ,

for another absolute constant b′. Combining the above with the induction hypothesis,applied within each cell of the partition, implies

∑

i

I(Pi, Ci) 6∑

i

(α1

(m

k3k−2

+ε

i n3k−33k−2

i +mk

2k−1+ε

i n3k−34k−2

i qk−14k−2

2

)+ α2(mi + ni)

)

6 α1

m

k3k−2

+ε

rk

3k−2+ε

∑

i

n3k−33k−2

i +m

k2k−1

+εqk−14k−2

2

rk

2k−1+ε

∑

i

n3k−34k−2

i

+

∑

i

α2(mi + ni)

6 α1b′

m

k3k−2

+εn3k−33k−2

rε+m

k2k−1

+εn3k−34k−2 q

k−14k−2

2

rε

+ α2

(m′ + bnr1/3

).

Our assumption that n = O(mk) implies that n = O(m

k3k−2n

3k−33k−2

)(with an absolute

constant of proportionality). Thus, when α1 is sufficiently large with respect to r, k, andα2, we have

∑

i

I(Pi, Ci) 6 2α1b′

m

k3k−2

+εn3k−33k−2

rε+m

k2k−1

+εn3k−34k−2 q

k−14k−2

2

rε

+ α2m

′.

When r is sufficiently large, such that rε > 6b′, we have

∑

i

I(Pi, Ci) 6α1

3

(m

k3k−2

+εn3k−33k−2 +m

k2k−1

+εn3k−34k−2 q

k−14k−2

2

)+ α2m

′. (6)

Incidences on the zero set Z(f). It remains to bound incidences with points that lieon Z(f). Set P0 := P∩Z(f) and m0 = |P0| = m−m′. Let C0 denote the set of curves thatare fully contained in Z(f), and set C ′ := C \ C0, n0 := |C0|, and n′ := |C ′| = n−n0. Sinceevery curve of C ′ intersects Z(f) in O(r1/3) points, we have, taking α1 to be sufficientlylarge, and arguing as above,

I(P0, C ′) = O(nr1/3) 6 α1

3m

k3k−2

+εn3k−33k−2 . (7)

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Finally, we consider the number of incidences between points of P0 and curves of C0.For this, we set c(k, ε) to be the degree of f , which is O(r1/3), and can be taken to beO((6b′)1/(3ε)). Then, by the assumption of the theorem, we have |C0| 6 q2. We considera generic plane π ⊂ R3 and project P0 and C0 onto two respective sets P∗ and C∗ onπ. Since π is chosen generically, we may assume that no two points of P0 project to thesame point in π, and that no pair of distinct curves in C0 have overlapping projectionsin π. Moreover, the projected curves still have k degrees of freedom, in the sense that,given any k points on the projection γ∗ of a curve γ ∈ C0, there are at most s − 1 otherprojected curves that go through all these points. This is argued by lifting each pointp back to the point p on γ in R3, and by exploiting the facts that the original curveshave k degrees of freedom, and that, for a sufficiently generic projection, any curve thatdoes not pass through p does not contain any point that projects to p. The number ofintersection points between a pair of projected curves may increase but it must remain aconstant since these are intersection points between constant-degree algebraic curves withno common components. By applying Theorem 1.3, we obtain

I(P0, C0) = I(P∗, C∗) = O(mk

2k−1

0 q2k−22k−1

2 +m0 + q2),

where the constant of proportionality depends on k (and s). Since q2 6 n and m0 6 m,

we have mk

2k−1

0 q2k−22k−1

2 6 mk

2k−1n3k−34k−2 q

k−14k−2

2 . We thus get that I(P0, C0) is at most

O

(m

k2k−1n

3k−34k−2 q

k−14k−2

2 + n+m0

)6 α1

3m

k2k−1n

3k−34k−2 q

k−14k−2

2 + b2n+ α2m0, (8)

for sufficiently large α1 and α2; the constant b2 comes from Theorem 1.3, and is indepen-dent of ε and of the choices for α1, α2 made so far.

By combining (6), (7), and (8), including the case m = O(n1/k), and choosing α2

sufficiently large, we obtain

I(P , C) 6 α1

(m

k3k−2

+εn3k−33k−2 +m

k2k−1

+εn3k−34k−2 q

k−14k−2

2

)+ α2(m+ n).

This completes the induction step and thus the proof of the theorem. 2

Example 1: The case of lines. Lines in R3 have k = 2 degrees of freedom, and wealmost get the bound of Guth and Katz in Theorem 1.1. There are three differences thatmake this derivation somewhat inferior to that in Guth and Katz [11], as detailed in items(i)–(iii) in the discussion in the introduction. We also recall the two follow-up studies ofpoint-line incidences in R3, of Guth [9] and of Sharir and Solomon [23]. Guth’s boundsuffers from weaknesses (i) and (ii), but avoids (iii), using a fairly sophisticated inductiveargument. Sharir and Solomon’s bound avoids (i) and (iii), and almost avoids (ii), in asense that we do not make explicit here. In both cases, considerably more sophisticatedmachinery is needed to achieve these improvements.

Example 2: The case of circles. Circles in R3 have k = 3 degrees of freedom, and weget the bound

I(P , C) = O(m3/7+εn6/7 +m3/5+εn3/5q

1/52 +m+ n

).

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The leading term is the same as in Sharir et al. [20], but the second term is weaker, becauseit relies on the general bound of Pach and Sharir (Theorem 1.3), whereas the bound in[20] exploits an improved bound for point-circle incidences, due to Aronov et al. [2], whichholds in any dimension. If we plug that bound into the above scheme, we obtain an exactreconstruction of the bound in [20]. In addition, considering the items (i)–(iii) discussedearlier, we note: (i) The requirements in [20] about the maximum number of circles on asurface are weaker, and are only for planes and spheres. (ii) The mε factors are presentin both bounds. (iii) Even after the improvement noted above, the bounds still seem tobe weak in terms of their dependence on q2, and improving this aspect, both here and in[20], is a challenging open problem.

Theorem 1.5 can easily be restated as bounding the number of rich points.

Corollary 2.1 For each ε > 0 there exists a parameter c(k, ε) that depends on k and ε,such that the following holds. Let C be a set of n irreducible algebraic curves of constantdegree and with k degrees of freedom (with some multiplicity s) in R3. Moreover, assumethat every surface of degree at most c(k, ε) contains at most q2 curves of C. Then thereexists some constant r0(k, ε) depending on ε, k (and s), such that for any r > r0(k, ε), thenumber of points that are incident to at least r curves of C (so-called r-rich points), is

O

(n3/2+ε

r3k−22k−2

+ε+n3/2+εq

1/2+ε2

r2k−1k−1

+ε+n

r

), where the constant of proportionality depends on k, s

and ε.

Proof. Denoting by mr the number of r-rich points, the corollary is obtained by com-bining the upper bound in Theorem 1.5 with the lower bound rmr. 2

3 Incidences in higher dimensions

Proof of Theorem 1.6. Again, we fix ε > 0, and prove, by double induction, where theouter induction is on the dimension d and the inner induction is on m + n, that I(P , C)is at most

α1,d

(m

kdk−d+1

+εndk−d

dk−d+1 +d−1∑

j=2

mk

jk−j+1+εn

d(j−1)(k−1)(d−1)(jk−j+1) q

(d−j)(k−1)(d−1)(jk−j+1)

j

)+ α2,d(m+ n), (9)

where α1,d, α2,d are sufficiently large constants, α1,d depends on k (and s), ε, d, and themaximum degree of the curves, and α2,d depends only on k (and s), d, and the maximumdegree of the curves.

For the outer induction basis, the case d = 2 follows by Theorem 1.3, and the cased = 3 is just Theorem 1.5, proved in the previous section. We assume therefore that theclaim holds up to dimension d−1, and prove it in dimension d > 4. The base cases of theinner induction (that is, when d is fixed, we induct over m+n) is when m,n are sufficientlysmall constants, and when m = O(n1/k). The bound in (9) can be enforced in the formercase by choosing sufficiently large values of α1,d, α2,d, and in the latter case exactly as ford = 3, so we may assume, as before, that n 6 cmk for some absolute constant c.

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Applying the polynomial partitioning technique. The analysis is somewhat repet-itive and resembles the one in the previous section, although many details are different;it is given in detail for the convenience of the reader, and in the interest of completeness.

Let f be an r-partitioning polynomial, for a sufficiently large constant r. Accordingto the polynomial partitioning theorem [11], we have degf = O(r1/d). Denote the (open)cells of the partition as τ1, . . . , τu, where u = O(r). For each i = 1, . . . , u, let Ci denote theset of curves of C that intersect τi and let Pi denote the set of points that are containedin τi. We set mi = |Pi|, and ni = |Ci|, for i = 1, . . . , u, and m′ =

∑imi, and notice that

mi 6 m/r for each i (and m′ 6 m). Arguing as before, every curve of C intersects at mostdeg(f) = O(r1/d) cells of Rd \ Z(f). Therefore,

∑i ni 6 bdnr

1/d, for a suitable constantbd > 1 that depends on d and the degree of the curves. Using Holder’s inequality, we have

∑

i

ndk−d

dk−d+1

i 6 b′d

(nr

1d

) dk−ddk−d+1

r1

dk−d+1 6 b′dndk−d

dk−d+1 rk

dk−d+1 , and

∑

i

nd(j−1)(k−1)

(d−1)(jk−j+1)

i 6 b′d

(nr

1d

) d(j−1)(k−1)(d−1)(jk−j+1)

rdk−jk+j−1

(d−1)(jk−j+1) 6 b′dnd(j−1)(k−1)

(d−1)(jk−j+1) rk

jk−j+1 ,

for each j = 2, . . . , d − 1, where b′d is another constant parameter that depends on d.Combining the above with the induction hypothesis implies that

∑i I(Pi, Ci) is at most

∑

i

(α1,d

(m

kdk−d+1

+ε

i ndk−d

dk−d+1

i +d−1∑

j=2

mk

jk−j+1+ε

i nd(j−1)(k−1)

(d−1)(jk−j+1)

i q(d−j)(k−1)

(d−1)(jk−j+1)

j

)+ α2,d(mi + ni)

)

6 α1,d

m

kdk−d+1

+ε

rk

dk−d+1+ε

∑

i

ndk−d

dk−d+1

i +d−1∑

j=2

mk

jk−j+1+εq

(d−j)(k−1)(d−1)(jk−j+1)

j

rk

jk−j+1+ε

∑

i

nd(j−1)(k−1)

(d−1)(jk−j+1)

i

+∑

i

α2,d(mi + ni)

6 α1,db′d

m

kdk−d+1

+εndk−d

dk−d+1

rε+

∑d−1j=2 m

kjk−j+1

+εnd(j−1)(k−1)

(d−1)(jk−j+1) q(d−j)(k−1)

(d−1)(jk−j+1)

j

rε

+α2,d

(m′ + bdnr

1/d).

Since we assume that n = O(mk), we have n = O(m

kdk−d+1n

dk−ddk−d+1

), with a constant of

proportionality that depends only on d. Thus, when α1,d is sufficiently large with respectto r, d, and α2,d, we have

∑

i

I(Pi, Ci) 6 2α1,db

(m

kdk−d+1

+εndk−d

dk−d+1

rε+m

k2k−1

+εndk−d

(d−1)(2k−1) q(k−1)(d−2)(d−1)(2k−1)

rε

)+ α2,dm

′.

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When r is sufficiently large, such that rε > 6b′, the bound is at most

α1,d

3

(m

kdk−d+1

+εndk−d

dk−d+1 +d−1∑

j=2

mk

jk−j+1+εn

d(j−1)(k−1)(d−1)(jk−j+1) q

(d−j)(k−1)(d−1)(jk−j+1)

j

)+ α2,dm

′. (10)

Incidences on the zero set Z(f). It remains to bound incidences with points that lieon Z(f). Set P0 = P ∩ Z(f) and m0 = |P0| = m −m′. Let C0 denote the set of curvesthat are fully contained in Z(f), and set C ′ = C \ C0, n0 = |C0|, and n′ = |C ′| = n − n0.Since every curve of C ′ intersects Z(f) in O(r1/d) points, we have, arguing as above,

I(P0, C ′) 6 bdn′r1/d = O(nr1/d) 6 α1,d

3m

kdk−d+1

+εndk−d

dk−d+1 , (11)

provided that α1,d is chosen sufficiently large.Finally, we consider the number of incidences between points of P0 and curves of C0.

For this, we set cd−1(k, d, ε) to be the degree of f , which is O(r1/d) = O((6b′)1/(εd)). Then,by the assumption of the theorem, we have |C0| 6 qd−1. We consider a generic hyperplaneH ⊂ Rd and project P0 and C0 onto two respective sets P∗ and C∗ on H. Arguing as inthe three-dimensional case, we can enforce that I(P0, C0) = I(P∗, C∗), that the projectedcurves have k degrees of freedom, and that, for j < d − 1, the pairs (qj, cj) remainunchanged for P∗ and C∗ within H. Applying the induction hypothesis for dimensiond− 1, and recalling that |C0| 6 qd−1, we obtain

I(P0, C0) = I(P∗, C∗) 6 α1,d−1

(d−1∑

j=2

mk

jk−j+1+εq

(d−1)(j−1)(k−1)(d−2)(jk−j+1)

d−1 q(d−j−1)(k−1)(d−2)(jk−j+1)

j

)+ α2,d−1(m+ n).

As is easily verified, Equation (4) with l = d (and qd = n) implies that, for each j,

q(d−1)(j−1)(k−1)(d−2)(jk−j+1)

d−1 q(d−j−1)(k−1)(d−2)(jk−j+1)

j 6 nd(j−1)(k−1)

(d−1)(jk−j+1) q(d−j)(k−1)

(d−1)(jk−j+1)

j .

By choosing α1,d > 3α1,d−1 and α2,d > α2,d−1, we have that I(P0, C0) is at most

α1,d

3

(d−1∑

j=2

mk

jk−j+1+εn

d(j−1)(k−1)(d−1)(jk−j+1) q

(d−j)(k−1)(d−1)(jk−j+1)

j

)+ α2,d(m+ n). (12)

By combining (10), (11), and (12), including the case m = O(n1/k), and choosing α2,d

sufficiently large, we obtain

I(P , C) 6 α1,d

(m

kdk−d+1

+εndk−d

dk−d+1 +mk

2k−1+εn

dk−d(d−1)(2k−1) q

(k−1)(d−2)(d−1)(2k−1)

)+ α2,d(m+ n).

This completes the induction step and thus the proof of the theorem. 2

Proof of Theorem 1.8. The proof is similar to that of Theorem 1.6, except that, whenhandling incidences between points and curves on Z(f), we simply project the points and

the electronic journal of combinatorics 23(4) (2016), #P4.16 13

curves onto some generic 2-plane, argue that the projected curves also have k degreesof freedom (and degree at most D), and apply the Pach–Sharir planar bound, given inTheorem 1.3 to the projected points and curves. Both terms in the bound “go through”the induction controlled by the polynomial partitioning. This is clear for the leading term,and follows for the second term in much the same way as in the preceding proof.

As a consequence of Theorem 1.6, we have:

Example: incidences between points and lines in R4. In the earlier version [21] ofSharir and Solomon’s study of point-line incidences in four dimensions, we have obtainedthe following weaker version of Theorem 1.2.

Theorem 3.1 For each ε > 0, there exists an integer cε, so that the following holds. LetP be a set of m distinct points and L a set of n distinct lines in R4, and let q, s 6 n beparameters, such that (i) for any polynomial f ∈ R[x, y, z, w] of degree 6 cε, its zero setZ(f) does not contain more than q lines of L, and (ii) no 2-plane contains more than slines of L. Then,

I(P,L) 6 Aε

(m2/5+εn4/5 +m1/2+εn2/3q1/12 +m2/3+εn4/9s2/9

)+ A(m+ n),

where Aε depends on ε, and A is an absolute constant.

This result follows from our main Theorem 1.6, if we impose Equation (4) on q2 = s,

q3 = q, and n, which in this case is equivalent to s 6 q 6 n and q9

n8 < s. This illustrateshow the general theory developed in this paper extends similar results obtained earlierfor “isolated” instances. Nevertheless, as already mentioned earlier, the bound for linesin R4 has been improved in Theorem 1.2 of [22], in its lower-dimensional terms.

Discussion. We first notice that similarly to the three-dimensional case, Theorem 1.6implies an upper bound on the number of k-rich points in d dimensions (see Corollary 2.1in three dimensions), and the proof thereof applies verbatim, with the appropriate mod-ifications of the various exponents that now depend also on d. We leave it to the readerto work out the precise (and, admittedly, somewhat cumbersome) statement.

Second, we note that Theorems 1.5 and 1.6 have several weaknesses. The obvious onesare the items (i)–(iii) discussed in the introduction. Another, less obvious weakness, has todo with the way in which the qj-dependent terms in the bounds are derived. Specifically,these terms facilitate the induction step, when the constraining parameters qj are passedunchanged to the inductive subproblems. Informally, since the overall number of lines ina subproblem goes down, one would expect the various parameters qj to decrease too, butso far we do not have a clean mechanism for doing so. This weakness is manifested, e.g.,in Corollary 2.1, where one would like to replace the second term by one with a smallerexponent of n and a larger one of q = q2. Specifically, for lines in R3, one would like toget a term close to O(nq2/k

3). This would yield O(n3/2/k3) for the important special caseq2 = O(n1/2) considered in [11]; the present bound is weaker.

A final remark concerns the relationships between the parameters qj, as set forthin Equation (4). These conditions are forced upon us by the induction process. Asnoted above, for incidences between points and lines in R4, the bound derived in our

the electronic journal of combinatorics 23(4) (2016), #P4.16 14

main Theorem 1.6 is (asymptotically) the same as that of the main result of Sharir andSolomon in [21]. The difference is that there, no restrictions on the qj are imposed. Theproof in [21] is facilitated by the so called “second partitioning polynomial” (see [14, 21]).Recently, Basu and Sombra [3] proved the existence of a third partitioning polynomial(see [3, Theorem 3.1]), and conjectured the existence of a k-th partitioning polynomial forgeneral k > 3 (see [3, Conjecture 3.4] for an exact formulation); for completeness we referalso to [8, Theorem 4.1], where a weaker version of this conjecture is proved. Buildingupon the work of [3], the proof of Sharir and Solomon [22] is likely to extend and yield thesame bound as in our main Theorem 1.6, for the more general case of incidences betweenpoints and bounded degree algebraic curves in dimensions at most five, and, if Conjecture3.4 of [3] holds, in every dimension, without any conditions on the qj.

References

[1] P. K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lensesin arrangements of pseudocircles and their applications, J. ACM 51 (2004), 139–186.

[2] B. Aronov, V. Koltun, and M. Sharir, Incidences between points and circles in threeand higher dimensions, Discrete Comput. Geom. 33 (2005), 185–206.

[3] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension twoand point-hypersurface incidences in four dimensions, Discrete Comput. Geom. 55(2016), 158–184.

[4] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorialcomplexity bounds for arrangements of curves and spheres, Discrete Comput. Geom.5 (1990), 99–160.

[5] Z. Dvir and S. Gopi, On the number of rich lines in truly high dimensional sets, Proc.31st Annu. Sympos. Comput. Geom., 2015, 584–598.

[6] G. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimen-sions, J. Combinat. Theory, Ser. A 118 (2011), 962–977. Also in arXiv:0905.1583.

[7] P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

[8] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version ofZarankiewicz’s problem, J. European Math. Soc., to appear.

[9] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, DiscreteComput. Geom. 53 (2015), 428–444. Also arXiv:1404.2321.

[10] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya prob-lem, Advances Math. 225 (2010), 2828–2839. Also arXiv:0812.1043.

[11] L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane,Annals Math. 181 (2015), 155–190. Also arXiv:1011.4105.

[12] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly ruled surfaces,arXiv:1503.02173.

[13] M. Hablicsek and Z. Scherr, On the number of rich lines in high dimensional realvector spaces, Discrete Comput. Geom. 55 (2016), 955–962.

the electronic journal of combinatorics 23(4) (2016), #P4.16 15

[14] H. Kaplan, J. Matousek, Z. Safernova and M. Sharir, Unit distances in three dimen-sions, Combinat. Probab. Comput. 21 (2012), 597–610. Also arXiv:1107.1077.

[15] H. Kaplan, J. Matousek and M. Sharir, Simple proofs of classical theorems in discretegeometry via the Guth–Katz polynomial partitioning technique, Discrete Comput.Geom. 48 (2012), 499–517. Also arXiv:1102.5391.

[16] J. Matousek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, 2002.

[17] F. Nilov and M. Skopenkov, A surface containing a line and a circle through eachpoint is a quadric, Geometria Dedicata 163.1 (2013), 301–310.

[18] J. Pach and M. Sharir, On the number of incidences between points and curves,Combinat. Probab. Comput. 7 (1998), 121–127.

[19] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of GeometricGraphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Prov-idence, RI, 2004, pp. 185–223.

[20] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between pointsand circles, Combinat. Probab. Comput. 24 (2015), 490–520. Also in Proc. 29th ACMSymp. on Computational Geometry (2013), 97–106, and arXiv:1208.0053.

[21] M. Sharir and N. Solomon, Incidences between points and lines in four dimensions,Proc. 30th ACM Sympos. on Computational Geometry (2014), 189–197.

[22] M. Sharir and N. Solomon, Incidences between points and lines in R4, Proc.56th IEEE Symp. on Foundations of Computer Science (2015), 1378–1394. AlsoarXiv:1411.0777.

[23] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions,in Intuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st ACM Sympos. onComputational Geometry (2015), 553–568, and arXiv:1501.02544.

[24] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applica-tions, arXiv:1604.07877.

[25] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput.Geom. 48 (2012), 255–280.

[26] J. Spencer, E. Szemeredi and W.T. Trotter, Unit distances in the Euclidean plane,In: Graph Theory and Combinatorics (B. Bollobas, ed.), Academic Press, New York,1984, 293–303.

[27] L. Szekely, Crossing numbers and hard Erdos problems in discrete geometry, Com-binat. Probab. Comput. 6 (1997), 353–358.

[28] E. Szemeredi and W.T. Trotter, Extremal problems in discrete geometry, Combina-torica 3 (1983), 381–392.

[29] T. Tao, From rotating needles to stability of waves: Emerging connections betweencombinatorics, analysis, and PDE, Notices AMS 48(3) (2001), 294–303.

[30] J. Zahl, An improved bound on the number of point-surface incidences in threedimensions, Contrib. Discrete Math. 8(1) (2013), 100–121. Also arXiv:1104.4987.

[31] J. Zahl, A Szemeredi-Trotter type theorem in R4, Discrete Comput. Geom. 54 (2015),513–572. Also arXiv:1203.4600.

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210

8 Incidences with curves and surfacesin three dimensions,with applications to distinct andrepeated distances

211

Incidences with curves and surfaces in three dimensions,

with applications to distinct and repeated distances∗

Micha Sharir† Noam Solomon‡

August 13, 2017

Abstract

We study a wide spectrum of incidence problems involving points and curves or points and surfacesin R3. The current (and in fact the only viable) approach to such problems, pioneered by Guth andKatz [38, 39], requires a variety of tools from algebraic geometry, most notably (i) the polynomialpartitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recentstudies [40], by algebraic curves of some constant degree. By exploiting and refining these tools, weobtain new and improved bounds for numerous incidence problems in R3.

In broad terms, we consider two kinds of problems, those involving points and constant-degreealgebraic curves, and those involving points and constant-degree algebraic surfaces. In some variantswe assume that the points lie on some fixed constant-degree algebraic variety, and in others we considerarbitrary sets of points in 3-space.

The case of points and curves has been considered in several previous studies, starting with Guthand Katz’s work on points and lines [39]. Our results, which are based on a recent work of Guth andZahl [40] concerning surfaces that are doubly ruled by curves, provide a grand generalization of allprevious results. We reconstruct the bound for points and lines, and improve, in certain signifcant ways,recent bounds involving points and circles (in [54]), and points and arbitrary constant-degree algebraiccurves (in [53]). While in these latter instances the bounds are not known (and are strongly suspectednot) to be tight, our bounds are, in a certain sense, the best that can be obtained with this approach,given the current state of knowledge.

In the case of points and surfaces, the incidence graph between them can contain large completebipartite graphs, each involving points on some curve and surfaces containing this curve (unlike earlierstudies, we do not rule out this possibility, which makes our approach more general). Our boundsestimate the total size of the vertex sets in such a complete bipartite graph decomposition of the incidencegraph. In favorable cases, our bounds translate into actual incidence bounds. Overall, here too our resultsprovide a “grand generalization” of most of the previous studies of (special instances of) this problem.

As an application of our point-curve incidence bound, we consider the problem of bounding thenumber of similar triangles spanned by a set of n points in R3. We obtain the bound O(n15/7), whichimproves the bound of Agarwal et al. [1].

As applications of our point-surface incidence bounds, we consider the problems of distinct andrepeated distances determined by a set of n points in R3, two of the most celebrated open problems incombinatorial geometry. We obtain new and improved bounds for two special cases, one in which thepoints lie on some algebraic variety of constant degree, and one involving incidences between pairs inP1 × P2, where P1 is contained in a variety and P2 is arbitrary.

∗Work on this paper by Noam Solomon and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation.Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by the IsraeliCenters of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Scienceat Tel Aviv University and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. A preliminaryversion of the paper has appeared in Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (2017), 2456–2475.†School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. michas@tau.ac.il‡School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. noam.solom@gmail.com

1

Keywords. Combinatorial geometry, incidences, the polynomial method, algebraic geometry, distinctdistances, repeated distances.

1 Introduction

1.1 The setups

Incidences between points and curves in three dimensions. Let P be a set of m points and Ca set of n irreducible algebraic curves of constant degree in R3. We consider the problem of obtainingsharp incidence bounds between the points of P and the curves of C. This is a major topic in incidencegeometry since the groundbreaking work of Guth and Katz [39] on point-line incidences in R3, with manyfollow-up studies, some of which are reviewed below. Building on the recent work of Guth and Zahl [40],which bounds the number of 2-rich points determined by a set of bounded-degree algebraic curves in R3

(i.e., points incident to at least two of the given curves), we are able to generalize Guth and Katz’s point-line incidence bound to a general bound on the number of incidences between points and bounded-degreeirreducible algebraic curves that satisfy certain natural assumptions, discussed in detail below.

Incidences between points and surfaces in three dimensions. Let P be a set of m points, andS a set of n two-dimensonal algebraic varieties of constant maximum degree in R3. Here too we imposecertain natural assumptions on the surfaces in S, discussed in detail below.

Let G(P, S) ⊆ P × S denote the incidence graph of P and S; its edges connect all pairs (p, σ) ∈ P × Ssuch that p is incident to σ. In general, I(P, S) := |G(P, S)| might be as large as the maximum possiblevalue mn, by placing all the points of P on a suitable curve, and make all the surfaces of S containthat curve,1 in which case G(P, S) = P × S. The bound that we are going to obtain will of courseacknowledge this possibility, and will in fact bypass it altogether. Concretely, rather than bounding I(P, S),

our basic approach will represent G(P, S) as a union of complete bipartite subgraphs⋃

γ∈Γ0

(Pγ × Sγ

), and

of a “leftover” subgraph G0(P, S) (which, in certain cases, might be empty), and derive an upper boundfor the overall size of their vertex sets, namely, a bound on

J(P, S) :=∑

γ∈Γ0

(|Pγ |+ |Sγ |

),

where the decomposition is over a set Γ0 of constant-degree algebraic curves γ so that, for each γ ∈ Γ0,Pγ = P ∩ γ and Sγ is the set of the surfaces of S that contain γ. (In some cases we will derive differentbounds on

∑γ |Pγ | and on

∑γ |Sγ |.) For the residual subgraph G0(P, S), we derive a sharp bound on the

actual number of incidences that it encodes (namely, the number of its edges). This generalizes previousresults in which one had to require that G(P, S) does not contain some fixed-size complete bipartite graph,or (only for spheres or planes) that the surfaces in S be “non-degenerate” ([5, 26]; see below).

Incidences between points on a variety and surfaces. An interesting special case is where P iscontained in some two-dimensional algebraic variety (surface) V of constant degree. Besides being, as webelieve, a problem of independent interest, it arises as a key subproblem in our analysis of the general casediscussed above.

1This situation can arise in many instances, for example in the case of planes (where many of them can intersect in a commonline), or spheres (where many can intersect in a common circle), but there are also many cases where this is impossible. Inthis latter situation, which we do not yet know how to characterize in a simple and general manner, our analysis becomessharper—see below.

2

We assume that the surfaces of S are taken from an s-dimensional family of surfaces, meaning thateach of them can be represented by a constant number of real parameters (e.g., by the coefficients of themonomials of the polynomial whose zero set is the surface), so that, in this parametric space, the pointsrepresenting the surfaces of S lie in an s-dimensional algebraic variety F of some constant degree (to whichwe refer as the “complexity” of F). This assumption, which holds in practically all applications, extends,in an obvious manner, to lower-dimensional varieties (e.g., curves) and to higher dimensions; see Sharirand Zahl [60] for a more thorough study of this notion.

1.2 Background

Points and curves, the planar case. The case of incidences between points and curves has a richhistory, starting with the aforementioned case of points and lines in the plane [23, 65, 66], where the worst-case tight bound on the number of incidences is Θ(m2/3n2/3 + m + n), where m is the number of pointsand n is the number of lines. Still in the plane, Pach and Sharir [49] extended this bound to incidencebounds between points and curves with k degrees of freedom. These are curves with the property that, foreach set of k points, there are only µ = O(1) curves that pass through all of them, and each pair of curvesintersect in at most µ points; µ is called the multiplicity (of the degrees of freedom).

Theorem 1.1 (Pach and Sharir [49]). Let P be a set of m points in R2 and let C be a set of n bounded-degreealgebraic curves in R2 with k degrees of freedom and with multiplicity µ. Then

I(P, C) = O(m

k2k−1n

2k−22k−1 +m+ n

),

where the constant of proportionality depends on k and µ.

Remark. The result of Pach and Sharir holds for more general families of curves, not necessarily algebraic,but, since algebraicity will be assumed in higher dimensions, we assume it also in the plane.

Except for the case k = 2 (lines have two degrees of freedom), the bound is not known, and stronglysuspected not to be tight in the worst case. Indeed, in a series of papers during the 2000’s [2, 9, 46], animproved bound has been obtained for incidences with circles, parabolas, or other families of curves withcertain properties (see [2] for the precise formulation). Specifically, for a set P of m points and a set C ofn circles, or parabolas, or similar curves [2], we have

I(P, C) = O(m2/3n2/3 +m6/11n9/11 log2/11(m3/n) +m+ n). (1)

Some further (slightly) improved bounds, over the bound in Theorem 1.1, for more general families ofcurves in the plane have been obtained by Chan [18, 19] and by Bien [12]. They are, however, considerablyweaker than the bound in (1).

Recently, Sharir and Zahl [60] have considered general families of constant-degree algebraic curves in theplane that belong to an s-dimensional family of curves. Similarly to the case of surfaces, discussed above,this means that each curve in that family can be represented by a constant number of real parameters, sothat, in this parametric space, the points representing the curves lie in an s-dimensional algebraic varietyF of some constant degree (to which we refer, as above, as the “complexity” of F). See [60] for moredetails.

Theorem 1.2 (Sharir and Zahl [60]). Let C be a set of n algebraic plane curves that belong to an s-dimensional family F of curves of maximum constant degree E, no two of which share a common irreduciblecomponent, and let P be a set of m points in the plane. Then, for any ε > 0, the number I(P, C) ofincidences between the points of P and the curves of C satisfies

I(P, C) = O(m

2s5s−4n

5s−65s−4

+ε +m2/3n2/3 +m+ n),

where the constant of proportionality depends on ε, s, E, and the complexity of the family F .

3

Except for the factor O(nε), this is a significant improvement over the bound in Theorem 1.1 (fors ≥ 3), in cases where the assumptions in Theorem 1.2 imply (as they often do) that C has k = s degreesof freedom. Concretely, when k = s, we obtain an improvement, except for the factor nε, for the entire“meaningful” range n1/s ≤ m ≤ n2, in which the bound is superlinear. The factor nε makes the boundin [60] slightly weaker only when m is close to the lower end n1/s of that range. Note also that for circles(where s = 3), the bound in Theorem 1.2 nearly coincides with the slightly more refined bound (1).

Incidences with curves in three dimensions. The seminal work of Guth and Katz [39] establishesthe sharper bound O(m1/2n3/4 + m2/3n1/3q1/3 + m + n) on the number of incidences between m pointsand n lines in R3, provided that no plane contains more than q of the given lines. This has lead tomany recent works on incidences between points and lines or other curves in three and higher dimensions;see [17, 40, 53, 54, 55, 59] for a sample of these results. Most relevant to our present study are the works ofSharir, Sheffer, and Solomon [53] on incidences between points and curves in any dimension, the work ofSharir, Sheffer, and Zahl [54] on incidences between points and circles in three dimensions, and the workof Sharir and Solomon [55] on incidences between points and lines in four dimensions, as well as severalother studies of point-line incidences by the authors [56, 59].

Of particular significance is the recent work of Guth and Zahl [40] on the number of 2-rich points in acollection of curves, namely, points incident to at least two of the given curves. For the case of lines, Guthand Katz [39] have shown that the number of such points is O(n3/2), when no plane or regulus containsmore than O(n1/2) lines. Guth and Zahl obtain the same asymptotic bound for general algebraic curves,under analogous (but stricter) restrictive assumptions.

The new bounds that we will derive require the extension to three dimensions of the notions of havingk degrees of freedom and of being an s-dimensional family of curves. The definitions of these concepts, asgiven above for the planar case, extend, more or less verbatim, to three (or higher) dimensions, but, even intypical situations, these two concepts do not coincide anymore. For example, lines in three dimensions havetwo degrees of freedom, but they form a 4-dimensional family of curves (this is the number of parametersneeded to specify a line in R3).

Points and surfaces. Many of the earlier works on point-surface incidences have only considered specialclasses of surfaces, most notably planes and spheres (see below). The case of more general surfaces hasbarely been considered, till the recent work of Zahl [69], who has studied the general case of incidencesbetween m points and n bounded-degree algebraic surfaces in R3 that have k degrees of freedom. Moreprecisely, in analogy with the case of curves, one needs to assume that for any k points there are at most

µ = O(1) of the given surfaces that pass through all of them. Zahl’s bound is O(m2k

3k−1n3k−33k−1 + m + n),

with the constant of proportionality depending on k, µ, and the maximum degree of the surfaces.

By Bezout’s theorem, if we require every triple of the given surfaces to have finite intersection, thenumber of intersection points would be at most E3, where E is the degree of the surfaces. In particular,E3 + 1 points would then have at most two of the given surfaces passing through all of them. In manyinstances, though, the actual number of degrees of freedom can be shown to be much smaller.

Zahl’s bound was later generalized by Basu and Sombra [11] to incidences between points and bounded-degree hypersurfaces in R4 satisfying certain analogous conditions.

Points and planes. Although we will not specifically address this special case, we refer the reader tothe earlier works on this problem, going back to Edelsbrunner, Guibas and Sharir [25]. More recently,Apfelbaum and Sharir [4] (see also Brass and Knauer [14] and Elekes and Toth [26]) have shown that ifthe incidence graph, for a set P of m points and a set H of n planes, does not contain a copy of Kr,s, forconstant parameters r and s, then I(P,H) = O(m3/4n3/4 + m + n). In more generality, Apfelbaum and

4

Sharir [4] have shown that if I = I(P,H) is significantly larger than this bound, then G(P,H) must containa large complete bipartite subgraph P ′×H ′, such that |P ′| · |H ′| = Ω(I2/(mn))−O(m+n). Moreover, asalso shown in [4] (slightly improving a similar result of Brass and Knauer [14]), G(P,H) can be expressedas the union of complete bipartite graphs Pi×Hi so that

∑i(|Pi|+ |Hi|) = O(m3/4n3/4 +m+n). (This is

a specialization to the case d = 3 of a similar result of [4, 14] in any dimension d, and it concurs with theapproach followed in this paper for more general scenarios.) Recently, Solomon and Sharir [57] improvedthis bound substantially when all the points of P lie on a constant-degree variety V .

Points and spheres. Earlier works on the special case of point-sphere incidences have considered thegeneral setup, where the points of P are arbitrarily placed in R3. Initial partial results go back to Chung [22]and to Clarkson et al. [23], and continue with the work of Aronov et al. [7]. Later, Agarwal et al. [1]have bounded the number of non-degenerate spheres with respect to a given point set; their bound wassubsequently improved by Apfelbaum and Sharir [5].2

The aforementioned recent work of Zahl [69] can be applied in the case of spheres if one assumesthat no three, or any larger but constant number, of the spheres intersect in a common circle. In thiscase the family has k = 3 degrees of freedom — any three points determine a unique circle that passesthrough all of them, and, by assumption, only O(1) spheres contain that circle. Zahl’s bound then becomesO(m3/4n3/4+m+n). In particular, this bound holds for congruent (unit) spheres (where three such spherescan never contain a common circle). The case of incidences with unit spheres have also been studied inKaplan et al. [43], with the same upper bound; see also [58].

If many spheres of the family can intersect in a common circle, the bound does no longer hold. Theonly earlier work that handled this situation is by Apfelbaum and Sharir [4], where it was assumed thatthe given spheres are non-degenerate. In this case the bound obtained in [4] is O(m8/11n9/11 + m + n).Interestingly, this is also the bound that Zahl’s result would have yielded if the sphere had k = 4 degreesof freedom, which however they only “almost have”: four generic points determine a unique sphere thatpasses through all of them, but four co-circular points determine an infinity of such spheres.

Distinct and repeated distances in three dimensions. The case of spheres is of particular interest,because it arises, in a standard and natural manner, in the analysis of distinct and repeated distancesdetermined by n points in three dimensions (see Section 6, where we use these well-known reductions inour analysis). After Guth and Katz’s almost complete solution of the number of distinct distances inthe plane [39], the three-dimensional case has moved to the research forefront. The prevailing conjectureis that the lower bound is Ω(n2/3) (the best possible in the worst case), but the current record, due toSolymosi and Vu [63], is still far smaller3 (close to Ω(n3/5)), and the problem seems much harder than itstwo-dimensional counterpart. Obtaining lower bounds for distinct distances using circles or spheres has ingeneral been suboptimal when compared with more effective methods (such as in [39]), but here we useit effectively to obtain new lower bounds (larger than Ω(n2/3)) when the points lie on a variety of fixeddegree.

The status of the case of repeated distances is also far from being satisfactory. The planar case is“stuck” with the upper bound O(n4/3) of Spencer et al. [64] from the 1980’s. This bound also holds forpoints on the 2-sphere, and there it is tight in the worst case (when the repeated distance is 1, say, and theradius of the sphere is 1/

√2) [29], but it is strongly believed that in the plane the correct bound is close

to linear. In three dimensions, the aforementioned bound of [43, 69] immediately implies the upper boundO(n3/2) on the number of repreated distances (a slight improvement over the earlier bound of Clarkson et

2Given a finite point set P ⊂ R3 and a constant 0 < η < 1, a sphere σ ⊂ R3 is called η-degenerate (with respect to P ), ifthere exists a circle c ⊂ σ such that |c ∩ P | ≥ η|σ ∩ P |.

3This follows by substituting the new lower bound Ω(n/ logn) of Guth and Katz for distinct distances in the plane, in therecursive analysis of [63].

5

al. [23]), but the best known lower bound is only Ω(n4/3 log log n) [28].

1.3 Our results

Incidences with curves. We first consider the problem of incidences between points and algebraiccurves. Before we state our results, we discuss three notions that are used in these statements. These arethe notions of k degrees of freedom (already mentioned above), of constructibility, and of surfaces infinitelyruled by curves.

k degrees of freedom. Let C0 be an infinite family of irreducible algebraic curves of constant degree Ein R3. Formally, in complete analogy with the planar case, we say that C0 has k degrees of freedom withmultiplicity µ, where k and µ are constants, if (i) for every tuple of k points in R3 there are at most µcurves of C0 that are incident to all k points, and (ii) every pair of curves of C0 intersect in at most µpoints. As in [49], the bounds that we derive depend more significantly on k than on µ—see below.

We remark that the notion of k degrees of freedom gets more involved for surfaces, and raises severalannoying technical issues. For example, how many points does it take to define, say, a sphere (up to afixed multiplicity)? As already observed earlier, four generic points do the job (they define a unique spherepassing through all four of them), but four co-circular points do not.

While it seems possible to come up with some sort of working definition, we bypass this issue in thispaper, by defining this notion, for a family F of surfaces, only with respect to a given surface V , by sayingthat F has k degrees of freedom with respect to V if the family of the irreducible components of the curvesσ ∩ V | σ ∈ F, counted without multiplicity, has k degrees of freedom, in the sense just defined. In thecase of spheres, for example, this definition gives, as is easily checked, four degrees of freedom when V isneither a plane nor a sphere, but only three when V is a plane or a sphere.

Constructibility. In the statements of the following theorems, we also assume that C0 is a constructiblefamily of curves. This notion generalizes the notion of being algebraic, and is discussed in detail in Guthand Zahl [40]. Informally, a set Y ⊂ Cd is constructible if it is a Boolean combination of algebraic sets.The formal definition goes as follows (see, e.g., Harris [42, Lecture 3]). For z ∈ C, define v(0) = 0 andv(z) = 1 for z 6= 0. Then Y ⊆ Cd, for some fixed d, is a constructible set if there exist a finite set ofpolynomials fj : Cd → C, for j = 1, . . . , JY , and a subset BY ⊂ 0, 1JY , so that x ∈ Y if and only if(v(f1(x)), . . . , v(fJY (x))) ∈ BY .

When we apply this definition to a set of curves, we think of them as points in some parametric(complex) d-space, where d is the number of parameters needed to specify a curve. When JY = 1 we get allthe algebraic hypersurfaces (that admit the implied d-dimensional representation) and their complements.An s-dimensional family of curves, for s < d, is obtained by taking JY = d− s and BY = 0JY . In doingso, the curves that we obtain are complete intersections. Following Guth and Zahl (see also a commentto that effect in the appendix), this involves no loss of generality, because every curve is contained in acurve that is a complete intersection. In what follows, when we talk about constructible sets, we implicitlyassume that the ambient dimension d is constant.

The constructible sets form a Boolean algebra. This means that finite unions and intersections ofconstructible sets are constructible, and the complement of a constructible set is constructible. Anotherfundamental property of constructible sets is that, over C, the projection of a constructible set is con-structible; this is known as Chevalley’s theorem (see Harris [42, Theorem 3.16] and Guth and Zahl [40,Theorem 2.3]). If Y is a constructible set, we define the complexity of Y to be min(deg f1 + · · ·+ deg fJY ),where the minimum is taken over all representations of Y , as described above. As just observed, con-structibility of a family C0 of curves extends the notion of C0 being s-dimensional. One of the main

6

motivations for using the notion of constructible sets (rather than just s-dimensionality) is the fact, estab-lished by Guth and Zahl [40, Proposition 3.3], that the set C3,E of irreducible curves of degree at most Ein complex 3-dimensional space (either affine or projective) is a constructible set of constant complexitythat depends only on E. Moreover, Theorem 1.13, one of the central technical tools that we use in ouranalysis (see below for its statement and proof), holds for constructible families of curves.

The connection between degrees of freedom and constructibility/dimensionality. Looselyspeaking, in the plane the number of degrees of freedom and the dimensionality of a family of curvestend to be equal. In three dimensions the situation is different. This is because the constraint that a curveγ passes through a point p imposes two equations on the parameters defining γ. We therefore expect thenumber of degrees of freedom to be half the dimensionality. A few instances where this is indeed the caseare: (i) Lines in three dimensions have two degrees of freedom, and they form a 4-dimensional family ofcurves (this is the number of parameters needed to specify a line in R3). (ii) Circles in three dimensionshave three degrees of freedom, and they form a 6-dimensional family of curves (e.g., one needs three param-eters to specify the plane containing the circle, two additional parameters to specify its center, and a sixthparameter for its radius). (iii) Ellipses have five degrees of freedom, but they form an 8-dimensional familyof curves, as is easily checked. (This discrepancy (for ellipses) is explained by noting that four points arenot sufficient to define the ellipse because the first three determine the plane containing it, so the fourthpoint, if at all coplanar with the first three, only imposes one constraint on the parameters of the ellipse.)

Remark. The definition of constructibility is given over the complex field C. This is in accordance withmost of the basic algebraic geometry tools, which have been developed over the complex field. Some carehas to be exercised when applying them over the reals. For example, Theorem 1.13, one of the centraltechnical tools that we use in our analysis, as well as the results of Guth and Zahl [40], apply over thecomplex field, but not over the reals. On the other hand, when we apply the partitioning method of [39](as in the proofs of Theorems 1.4) and 1.12 or when we use Theorem 1.2, we (have to) work over the reals.

It is a fairly standard practice in algebraic geometry that handles a real algebraic variety V , defined byreal polynomials, by considering its complex counterpart VC, namely the set of complex points at whichthe polynomials defining V vanish. The rich toolbox that complex algebraic geometry has developed allowsone to derive various properties of VC, which, with some care, can usually be transported back to the realvariety V .

This issue arises time and again in this paper. Roughly speaking, we approach it as follows. We applythe polynomial partitioning technique to the given sets of points and of curves or surfaces, in the originalreal (affine) space, as we should. Within the cells of the partitioning we then apply some field-independentargument, based either on induction or on some ad-hoc combinatorial argument. Then we need to treatpoints that lie on the zero set of the partitioning polynomial. We can then switch to the complex field,when it suits our purpose, noting that this step preserves all the real incidences; at worst, it might addadditional incidences involving the non-real portions of the variety and of the curves or surfaces. Hence,the bounds that we obtain for this case transport, more or less verbatim, to the real case too.

Surfaces infinitely ruled by curves. Back in three dimensions, a surface V is (singly, doubly, orinfinitely) ruled by some family Γ of curves of degree at most E, if each point p ∈ V is incident to (at leastone, at least two, or infinitely many) curves of Γ that are fully contained in V . The connection betweenruled surface theory and incidence geometry goes back to the pioneering work of Guth and Katz [39] andshows up in many subsequent works. See Guth’s recent survey [36] and recent book [37], and Kollar [44]for details.

In most of the previous works, only singly-ruled and doubly-ruled surfaces have been considered. Look-ing at infinitely-ruled surfaces adds a powerful ingredient to the toolbox, as will be demonstrated in this

7

paper.

We recall that the only surfaces that are infinitely ruled by lines are planes (see, e.g., Fuchs andTabachnikov [31, Corollary 16.2]), and that the only surfaces that are infinitely ruled by circles are spheresand planes (see, e.g., Lubbes [45, Theorem 3] and Schicho [52]; see also Skopenkov and Krasauskas [62] forrecent work on celestials, namely surfaces doubly ruled by circles, and Nilov and Skopenkov [48], provingthat a surface that is ruled by a line and a circle through each point is a quadric). It should be noted that,in general, for this definition to make sense, it is important to require that the degree E of the ruling curvesbe much smaller than deg(V ). Otherwise, every variety V is infinitely ruled by, say, the curves V ∩ h, forhyperplanes h, having the same degree as V . A challenging open problem is to characterize all the surfacesthat are infinitely ruled by algebraic curves of degree at most E (or by certain special classes thereof).However, the following result of Guth and Zahl provides a useful sufficient condition for this property tohold.

Theorem 1.3 (Guth and Zahl [40]). Let V be an irreducible surface, and suppose that it is doubly ruledby curves of degree at most E. Then deg(V ) ≤ 100E2.

In particular, an irreducible surface that is infinitely ruled by curves of degree at most E is doublyruled by these curves, so its degree is at most 100E2. Therefore, if V is irreducible of degree D larger thanthis bound, V cannot be infinitely ruled by curves of degree at most E. This leaves a gray zone, in whichthe degree of V is between E and 100E2. We would like to conjecture that in fact no irreducible varietywith degree in this range is infinitely ruled by degree-E curves. Being unable to establish this conjecture,we leave it as a challenging open problem for further research.

Finally, we remark that the notion of surfaces infinitely ruled by curves also plays a crucial role in oneof our results on point-surface incidences (see Theorem 1.8).

Our results: points and curves. We can now state our main results on point-curve incidences.

Theorem 1.4 (Curves in R3). Let P be a set of m points and C a set of n irreducible algebraic curves ofconstant degree E, taken from a constructible family C0, of constant complexity, with k degrees of freedom(and some multiplicity µ) in R3, such that no surface that is infinitely ruled by curves of C0 contains morethan q curves of C, for a parameter q < n. Then

I(P, C) = O(m

k3k−2n

3k−33k−2 +m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

), (2)

where the constant of proportionality depends on k, µ, E, and the complexity of the family C0.

Remarks. (1) In certain favorable situations, such as in the cases of lines or circles, discussed above,the surfaces that are infinitely ruled by curves of C0 have a simple characterization. In such cases thetheorem has a stronger flavor, as its assumption on the maximum number of curves on a surface has tobe made only for this concrete kind of surfaces. For example, as already noted, for lines (resp., circles)we only need to require that no plane (resp., no plane or sphere) contains more than q of the curves. Ingeneral, as mentioned, characterizing infinitely-ruled surfaces by a specific family of curves is a difficulttask. Nevertheless, we can overcome this issue by replacing the assumption in the theorem by a morerestrictive one, requiring that no surface that is infinitely ruled by curves of degree at most E contain morethan q curves of C. By Theorem 1.3, any infinitely ruled surface of this kind must be of degree at most100E2. Hence, an even simpler (albeit weaker) formulation of the theorem is to require that no surface ofdegree at most 100E2 contains more than q curves of C. This can indeed be much weaker: In the case ofcircles, say, instead of making this requirement only for planes and spheres, we now have to make it forevery surface of degree at most 400.

8

(2) In several recent works (see [34, 53, 54]), the assumption in the theorem is replaced by a much morerestrictive assumption, that no surface of degree at most cε contains more than q given curves, where cεis a constant that depends on another prespecified parameter ε > 0 (where ε appears in the exponents inthe resulting incidence bound), and is typically very large (and increases as ε becomes smaller). Gettingrid of such an ε-dependent constant (and of the ε in the exponent) is a significant feature of Theorem 1.4.

(3) Theorem 1.4 generalizes the incidence bound of Guth and Katz [39], obtained for the case of lines. Inthis case, lines have k = 2 degrees of freedom, they certainly form a constructible (in fact, a 4-dimensional)family of curves, and, as just noted, planes are the only surfaces in R3 that are infinitely ruled by lines.Thus, in this special case, both the assumptions and the bound in Theorem 1.4 are identical to those inGuth and Katz [39]. That is, if no plane contains more than q input lines, the number of incidences isO(m1/2n3/4 +m2/3n1/3q1/3 +m+ n).

Improving the bound. The bound in Theorem 1.4 can be further improved, if we also throw into theanalysis the dimensionality s of the family C0. Actually, as will follow from the proof, the dimensionalitythat will be used is only that of any subset of C0 whose members are fully contained in some variety that isinfinitely ruled by curves of C0. As just noted, such a variety must be of constant degree (at most 100E2,or smaller as in the cases of lines and circles), and the additional constraint that the curves be containedin the variety can typically be expected to reduce the dimensionality of the family.

For example, if C0 is the collection of all circles in R3, then, since the only surfaces that are infinitelyruled by circles are spheres and planes, the subfamily of all circles that are contained in some sphere orplane is only 3-dimensional (as opposed to the entire C0, which is 6-dimensional).

We capture this setup by saying that C0 is a family of reduced dimension s if, for each surface Vthat is infinitely ruled by curves of C0, the subfamily of the curves of C0 that are fully contained in V iss-dimensional. In this case we obtain the following variant of Theorem 1.4.

Theorem 1.5 (Curves in R3). Let P be a set of m points and C a set of n irreducible algebraic curves ofconstant degree E, taken from a constructible family C0 with k degrees of freedom (and some multiplicityµ) in R3, such that no surface that is infinitely ruled by curves of C0 contains more than q of the curves ofC, and assume further that C0 is of reduced dimension s. Then

I(P, C) = O(m

k3k−2n

3k−33k−2

)+Oε

(m2/3n1/3q1/3 +m

2s5s−4n

3s−45s−4 q

2s−25s−4

+ε +m+ n), (3)

for any ε > 0, where the first constant of proportionality depends on k, µ, s, E, and the maximumcomplexity of any subfamily of C0 consisting of curves that are fully contained in some surface that isinfinitely ruled by curves of C0, and the second constant also depends on ε.

Remarks. (1) Theorem 1.5 is an improvement of Theorem 1.4 when s ≤ k and m > n1/k, in cases whereq is sufficiently large so as to make the second term in (2) dominate the first term; for smaller values of mthe bound is always linear. This is true except for the term qε, which affects the bound only when m is veryclose to n1/k (when s = k). When s > k we get a threshold exponent β = 5s−4k−2

ks−4k+2s (which becomes 1/kwhen s = k), so that the bound in Theorem 1.5 is stronger (resp., weaker) than the bound in Theorem 1.4when m > nβ (resp., m < nβ), again, up to the extra factor qε.

(2) The bounds in Theorems 1.4 and 1.5 improve, in three dimensions, the recent result of Sharir, Sheffer,and Solomon [53], in three significant ways: (i) The leading terms in both bounds are essentially thesame, but our bound is sharper, in that it does not include the factor O(nε) appearing in [53]. (ii) Theassumption here, concerning the number of curves on a low-degree surface, is much weaker than the onemade in [53], where it was required that no surface of some (constant but potentially very large) degreecε, that depends on ε, contains more than q curves of C. (See also Remark (2) following Theorem 1.4.)

9

(iii) The two variants of the non-leading terms here are significantly smaller than those in [53], and, in acertain sense (that will be elaborated following the proof of Theorem 1.5) are best possible.

Point-circle incidences in R3. Theorem 1.5 yields a new bound for the case of incidences between pointsand circles in R3, which improves over the previous bound of Sharir, Sheffer, and Zahl [54]. Specifically,we have:

Theorem 1.6. Let P be a set of m points and C a set of n circles in R3, so that no plane or spherecontains more than q circles of C. Then

I(P, C) = O(m3/7n6/7 +m2/3n1/3q1/3 +m6/11n5/11q4/11 log2/11(m3/q) +m+ n

).

Here too we have the three improvements noted in Remark (2) above. In particular, in the sense ofpart (iii) of that remark, the new bound is “best possible” with respect to the best known bound (1) forthe planar or spherical cases. See Section 3 for details. Theorem 1.6 has an interesting application to theproblem of bounding the number of similar triangles spanned by a set of n points in R3. It yields thebound O(n15/7), which improves the bound of Agarwal et al. [1]. See Section 3 for details.

Incidence graph decomposition, for points on a variety and surfaces. Our first main result onpoint-surface incidences deals with the special case where the points of P lie on some algebraic variety Vof constant degree. Besides being of independent interest, this is a major ingredient of the analysis for thegeneral case of an arbitrary set of points in R3 and surfaces.

In the statements of the following theorems we assume that the set S of the given surfaces is takenfrom some infinite family F that either has k degrees of freedom with respect to V (with some multiplicityµ), as defined earlier, for suitable constant parameters k (and µ), or is of reduced dimension s with respectto V , for some constant parameter s, meaning that the family Γ := σ ∩ V | σ ∈ F is an s-dimensionalfamily of curves (this is reminiscent of the notion of reduced dimension defined above for curves).

Theorem 1.7. Let P be a set of m points on some algebraic surface V of constant degree D in R3, andlet S be a set of n algebraic surfaces in R3 of maximum constant degree E, taken from some family Fof surfaces, which either has k degrees of freedom with respect to V (with some multiplicity µ), or is ofreduced dimension s with respect to V , for some constant parameters k (and µ) or s. We also assume thatthe surfaces in S do not share any common irreducible component (which certainly holds when they areirreducible). Then the incidence graph G(P, S) can be decomposed as

G(P, S) =⋃

γ

(Pγ × Sγ), (4)

where the union is over all irreducible components of curves γ of the form σ ∩ V , for σ ∈ S, and, for eachsuch γ, Pγ = P ∩ γ and Sγ is the set of surfaces in S that contain γ.

If F has k degrees of freedom then

∑

γ

|Pγ | = O(m

k2k−1n

2k−22k−1 +m+ n

), (5)

and if F is s-dimensional then we have, for any ε > 0,

∑

γ

|Pγ | = O(m

2s5s−4n

5s−65s−4

+ε +m2/3n2/3 +m+ n), (6)

where the constants of proportionality depends on D, E, and the complexity of the family F , and either onk and µ in the former case, or on ε and s in the latter case.

10

Moreover, in both cases we have∑

γ |Sγ | = O(n), where the constant of proportionality depends on Dand E.

Remark. A major feature of this result is that it does not impose any restrictions on the incidence graph,such as requiring it not to contain some fixed complete bipartite graph Kr,r, for r a constant, as is done inthe preceding studies [11, 43, 69]. We re-iterate that, to allow for the existence of large complete bipartitegraphs, the bounds in (5) and (6), as well as the bound

∑γ |Sγ | = O(n), are not on the number of incidences

(that is, on the number of edges in G(P, S), which could be as high as mn) but on the overall size of thevertex sets of the subgraphs in the complete bipartite graph decomposition of G(P, S). This would lead tothe same asymptotic bound on |G(P, S)| itself, if one assumes that this graph does not contain Kr,r as asubgraph, for a constant r.

This kind of compact representation of incidences has already been used in the previous studies of Brassand Knauer [14], Apfelbaum and Sharir [4], and our recent works [57, 58], albeit only for the special casesof planes or spheres.

Remark. Another way of bypassing the possible presence of large complete bipartite graphs in G(P, S),used in several earlier works [1, 4, 26], is to assume that the surfaces in S are non-degenerate. Thesestudies, already mentioned earlier, only considered the cases of planes and spheres (or of hyperplanes andspheres in higher dimensions) [1, 26]. For spheres, for example, this means that no more than some fixedfraction of the points of P on any given sphere can be cocircular. Although large complete bipartite graphscan exist in G(P, S) in this case, the non-degeneracy assumption allows us to control, in a sharp form,the number of incidences (and shows that the resulting complete bipartite graphs are not so large afterall). It would be interesting (and, as we believe, doable) to extend our analysis to the case of (suitablydefined) more general non-degenerate surfaces. These remarks also apply to the general case (involvingpoints anywhere in R3), given in Theorem 1.12 below.

A mixed incidence bound (for points on most varieties and general surfaces). Our secondresult is an improvement of Theorem 1.7, still for the case where the points of P lie on some algebraicvariety V of constant degree, where we now also assume that V is not infinitely ruled by the (irreduciblecomponents of the) intersection curves of pairs of members of the given family F of surfaces. In thiscase we obtain an improved, “mixed” bound, in which G(P, S) can be split into two subgraphs, G0(P, S)and G1(P, S), where the bound in (5) or in (6) now holds for |G0(P, S)|, i.e., for the actual number ofincidences that it represents, and where G1(P, S) admits a complete bipartite graph decomposition, asabove, for which the sum of the vertex sets is only4 O(m+ n). The actual bound is slightly sharper—seebelow.

Specializing the theorem to the case of spheres, as is done later on (in Section 6), leads to interestingimplications to distinct and repeated distances in three dimensions.

Theorem 1.8. Let P be a set of m points on some irreducible algebraic surface V of constant degree D inR3, and let S be a set of n algebraic surfaces in R3 of constant degree E, which do not share any commonirreducible component, taken from some infinite constructible family F of surfaces that either has k degreesof freedom with respect to V (with some multiplicity µ) or is s-dimensional with respect to V , for someconstant parameters k (and µ) or s. Assume further that V is not infinitely ruled by the family C0 ofthe irreducible components of the intersection curves of pairs of surfaces5 in F . Then the incidence graphG(P, S) can be decomposed as

G(P, S) = G0(P, S) ∪⋃

γ

(Pγ × Sγ), (7)

4In fact, many “bad” things must happen for G1(P, S) to be nontrivial, and in many situations one would expect G1(P, S)to be empty; see below.

5A stricter assumption is that V is not infinitely ruled by algebraic curves of degree at most E2, which will hold if weassume that each irreducible component of V has degree larger than 100E4.

11

where the union is over all irreducible curves γ contained in (one-dimensional) intersections of the formσ ∩ σ′ ∩ V , for σ 6= σ′ ∈ S, and, for each such γ, Pγ ⊆ P ∩ γ (for some points on some curves, theirincident pairs are moved to, and counted in G0(P, S)), and Sγ is the set (of size at least two) of surfacesin S that contain γ.

Moreover, if F has k degrees of freedom with respect to V (with some multiplicity µ) then

|G0(P, S)| = O(m

k2k−1n

2k−22k−1 +m+ n

), (8)

and if F is s-dimensional with respect to V then, for any ε > 0,

|G0(P, S)| = O(m

2s5s−4n

5s−65s−4

+ε +m2/3n2/3 +m+ n), (9)

where the constants of proportionality depends on D, E, and the complexity of the family F , and either onk and µ in the former case, or on ε and s in the latter case. In either case we also have

∑

γ

|Pγ | = O(m), and∑

γ

|Sγ | = O(n),

where the constants of proportionality depend on D, E, and the complexity of the family F , and either onk (and µ) in the former case, or on ε and s in the latter case.

Remarks. (1) As already alluded to, we note that, typically, one would expect the complete bipartitedecomposition part of (7) to be empty or trivial. To really be significant, (a) many surfaces of S wouldhave to intersect in a common curve, and, in cases where the multiplicity of these curves is not that large,(b) many curves of this kind would have to be fully contained in V . Thus, in many cases, in which (a)and (b) do not hold, the bounds in (8) or in (9) in Theorem 1.8 are for the overall number of incidences.Note also that both Theorem 1.7 and Theorem 1.8 yield a decomposition of (the whole or a portion of)G(P, S) into complete bipartite subgraphs. The major difference is that the bound

∑γ |Pγ | on the overall

P -vertex sets size of these graphs is (relatively) large in Theorem 1.7, but it is only linear in m and n (ifat all nonzero) in Theorem 1.8. (The bound on

∑γ |Sγ | remains O(n) in both cases.)

(2) We note that if V is infinitely ruled by our curves the results break down. For a simple example, take mpoints and N lines in the plane which form Θ(m2/3N2/3) incidences between them. Now pick any surfaceV in R3, say the paraboloid z = x2 +y2 for specificity, and lift up each of the N lines to a vertical parabolaon V . Clearly, V is infinitely ruled by such parabolas, and we get a system of m points and n parabolaswith Θ(m2/3N2/3) incidences between them. It is also easy to turn this construction into a point-surfaceincidence structure, in which

∑γ |Pγ | is equal to this bound, which is larger than the lower bound O(m+N)

asserted in the theorem. The line y = ax+b in the plane is lifted to the parabola γa,b = (x, y, z) ∈ R3 : y =ax+ b, z = x2 + y2 contained in the paraboloid V . Define a family S of quadratic surfaces parameterizedby a, b, c0, c1, c2 ∈ R by Sa,b,c0,c1,c2 := (x, y, z) ∈ R3 | (z−x2− y2) + (y− ax− b)(c0 + c1x+ c2y) = 0. Forany c0, c1, c2 ∈ R, the quadric Sa,b,c0,c1,c2 contains the parabola γa,b, i.e., many surfaces in S intersect in acommon parabola.

Incidences between points on a variety and spheres. A particular case of interest is when S is aset of spheres. The intersection curves of spheres are circles, and, as already noted, the only surfaces thatare infinitely ruled by circles are spheres and planes. Hence, to apply Theorem 1.8, we need to assumethat the constant-degree surface V that contains the points of P has no planar or spherical components,thereby ensuring that V is not infinitely ruled by circles. Clearly, as already noted, spheres in R3 have fourdegrees of freedom with respect to any constant-degree variety with no planar or spherical components,and they form a four-dimensional family of surfaces, with respect to any such variety (and also in general).We can therefore apply Theorem 1.8, with s = 4, and conclude:

12

Theorem 1.9. Let P be a set of m points on some algebraic surface V of constant degree D in R3, whichhas no linear or spherical components, and let S be a set of n spheres, of arbitrary radii, in R3. Theincidence graph G(P, S) can be decomposed as

G(P, S) = G0(P, S) ∪⋃

γ∈Γ

(Pγ × Sγ), (10)

where Γ is the set of circles that are contained in V and in at least two spheres of S, and such that, foreach γ ∈ Γ, Pγ = P ∩ γ and Sγ is the set of all spheres in S that contain γ. We have

|G0(P, S)| = O(m1/2n7/8+ε +m2/3n2/3 +m+ n

), (11)

∑

γ

|Pγ | = O(m), and∑

γ

|Sγ | = O(n), ,

for any ε > 0, where the constant of proportionality depends on D and ε.

Remark. Since V does not contain a planar or spherical component, the number of circles in Γ is O(D2),as follows by Guth and Zahl [40]. That is, the union in (11) is only over a constant number of circles. Onthe other hand, there might also be incidence edges contained in complete bipartite graphs correspondingto circles that are not contained in V , whose number might be quite large. These incidences are recordedin G0(P, S) and their number is bounded in (11).

Zahl’s assumption that G(P, S) does not contain Kr,3, for some (arbitrary) constant r (that is, byassuming that every triple of spheres intersect in at most r points of P ), leads to the bound I(P, S) =O(m3/4n3/4 +m+ n); our bound is better for m > n1/2 (ignoring the nε factor in our bound). Except forthis rather restrictive assumption, Zahl’s result is more general, as it does not require the points to lie ona constant-degree variety.

We also note that if we assume that G(P, S) does not contain any Kr,r, for r > 3 a constant, the boundin the second part of (11) becomes a bound on the number of incidences, so, under this somewhat weakerassumption (than that of Zahl), we improve Zahl’s bound for points on a variety and for m > n1/2.

The bound in (11) further improves when either (i) the centers of the spheres of S lie on V (or onsome other constant-degree variety), or (ii) the spheres of S have the same radius. In both cases, S is onlythree-dimensional, so the bound improves to

|G0(P, S)| = O(m6/11n9/11+ε +m2/3n2/3 +m+ n

), (12)

for any ε > 0. When both conditions hold—the spheres are congruent and their centers lie on V—S isonly two-dimensional with respect to V , and the bound improves still further to

|G0(P, S)| = O(m2/3n2/3+ε +m+ n

).

Using a slightly refined machinery, developed in a companion paper [58], the latter bound can be actuallyimproved further to

O(m2/3n2/3 +m+ n). (13)

Applications of Theorem 1.9 and (12), (13): Distinct distances. As already mentioned, and aswill be detailed in the proofs of the following results, the new bounds on point-sphere incidences haveimmediate applications to the study of distinct and repeated distances determined by a set of n points inR3, when the points (or a subset thereof—see below) lie on some fixed-degree algebraic variety. Specifically,for distinct distances, we have the following results.

13

Theorem 1.10. (a) Let P be a set of n points on an algebraic surface V of constant degree D in R3, withno linear or spherical components. Then the number of distinct distances determined by P is Ω(n7/9−ε),for any ε > 0, where the constant of proportionality depends on D and ε.

(b) Let P1 be a set of m points on a surface V as in (a), and let P2 be a set of n arbitrary points in R3.Then the number of distinct distances determined by pairs of points in P1 × P2 is

Ω(

minm4/7−εn1/7−ε, m1/2n1/2, m

),

for any ε > 0, where the constant of proportionality depends on D and ε.

Remark. In a recent work [58], we have obtained slightly improved bounds, without the ε in the exponents,using a more refined space decomposition technique, which can be applied for arrangements of spheres.

While we believe that the bounds in the theorem are not tight, we note that the bounds in both (a)and (b) (with, say, m = n) are significantly larger than the conjectured best-possible lower bound Ω(n2/3)for arbitrary point sets in R3.

Repeated distances. As another application, we bound the number of unit (or repeated) distancesinvolving points on a surface V , as above.

Theorem 1.11. (a) Let P be a set of n points on some algebraic surface V of constant degree D in R3,which does not contain any planar or spherical components. Then P determines O(n4/3) unit distances,where the constant of proportionality depends on D.

(b) Let P1 be a set of m points on a surface V as in (a), and let P2 be a set of n arbitrary points in R3.Then the number of unit distances determined by pairs of points in P1 × P2 is

O(m6/11n9/11+ε +m2/3n2/3 +m+ n

),

for any ε > 0, where the constant of proportionality depends on D and ε.

In part (a) we extend, to the case of general constant-degree algebraic surfaces, the known boundO(n4/3), which is worst-case tight when V is a sphere [29]. Part (b) gives (say, for the case m = n) anintermediate bound between O(n4/3) and the best known upper bound O(n3/2) for a arbitrary set of pointsin R3 [43, 69].

Another thing to notice is that, for distinct distances, the situation is quite different when V is (orcontains) a plane or a sphere, in which case the bound goes up to Ω(n/ log n) [39, 67] (see also Sheffer’ssurvey [61] for details).

Incidence graph decomposition (for arbitrary points and surfaces). Our final main result onpoint-surface incidences deals with the general setup involving a set S of constant-degree algebraic surfacesand an arbitrary set of points in R3. The analysis in this general setup proceeds by a recursive argument,based on the polynomial partitioning technique of Guth and Katz [39], in which Theorem 1.7 plays acentral role6. This result extends a recent result in preliminary work by the authors [58, Theorem 1.4]from spheres to general surfaces, and extends the aforementioned result of Zahl [69], for general algebraicsurfaces, to the case where no constraints are imposed on G(P, S).

6Ideally, applying Theorem 1.8 would yield a better estimate, but, unfortunately, we cannot control the polynomial gener-ated by the polynomial partitioning technique.

14

Theorem 1.12. Let P be a set of m points in R3, and let S be a set of n surfaces from some s-dimensionalfamily7 F of surfaces, of constant maximum degree E in R3. Then the incidence graph G(P, S) can bedecomposed as

G(P, S) = G0(P, S) ∪⋃

γ

(Pγ × Sγ), (14)

where the union is now over all curves γ of intersection of at least two of the surfaces of S, and, for eachsuch γ, Pγ = P ∩ γ and Sγ is the set (of size at least two) of surfaces in S that contain γ. Moreover, wehave, for any ε > 0,

J(P, S) :=∑

γ

(|Pγ |+ |Sγ |

)= O

(m

2s3s−1n

3s−33s−1

+ε +m+ n), and |G0(P, S)| = O(m+ n), (15)

where the constants of proportionality depend on ε, s, D, E, and the complexity of the family F .

As already noted, this result extends Zahl’s bound [69] to the case where no restrictions are imposedon the incidence graph (see the remark following Theorem 1.7). Zahl’s bound is the same as ours, exceptfor the extra factor nε in our bound.

We also note that Theorem 1.12 only applies to s-dimensional families F , and not to families withk degrees of freedom. The main issue here is that in Theorems 1.7 and 1.8, the notion of k degrees offreedom (and that of s-dimensionality) is applied to the intersection curves of the surfaces from F withsome constant-degree variety, whereas here it has to hold for the surfaces themselves in the entire three-dimensional space. So far we are lacking a good definition of this notion that will facilitate certain stepsin the proof. See a discussion of this issue following the proof, in Section 7.

1.4 The main techniques

There are three main ingredients used in our approach. The first ingredient, already mentioned in thecontext of planar point-curve incidences, is the techniques of Pach and Sharir [49] (given in Theorem 1.1),and of Sharir and Zahl [60] (Theorem 1.2) concerning incidences between points and algebraic curves inthe plane. The latter bound will be used in the analysis of incidences both between points and curves, andbetween points and surfaces.

The second ingredient, relevant to the proofs of Theorems 1.4 and 1.12, is the polynomial partitioningtechnique of Guth and Katz [39], and its more recent extension by Guth [35], which yields a divide-and-conquer mechanism via space decomposition by the zero set of a suitable polynomial. This will producesubproblems that will be handled recursively, and will leave us with the overhead of analyzing the incidencepattern involving the points that lie on the zero set itself. The latter step will be accomplished by astraightforward application of Theorem 1.7. We assume familiarity of the reader with these results; moredetails will be given in the applications of this technique in the proofs of the aforementioned theorems.

The third ingredient arises in the proof of Theorems 1.4 and 1.5, where we argue that a “generic”point on a variety V , that is not infinitely ruled by constant-degree curves of some given family, as in thestatement of the theorems, is incident to at most a constant number of the given curves that are fullycontained in V . Moreover, we can also control the number and structural properties of “non-generic”points.

Before formally stating, in detail, the technical properties that we need, we review a few notations.

Fix a constructible set C0 ⊂ C3,E of irreducible curves of degree at most E in 3-dimensional space, anda trivariate polynomial f . Following Guth and Zahl [40, Section 9], we call a point p ∈ Z(f) a (t, C0, r)-flecnode, if there are at least t curves γ1, . . . , γt ∈ C0, such that, for each i = 1, . . . , t, (i) γi is incident to p,

7Here we use the general notion of s-dimensionality, not confined to points on a variety.

15

(ii) p is a non-singular point of γi, and (iii) γi osculates to Z(f) to order r at p. This is a generalizationof the notion of a flecnodal point, due to Salmon [51, Chapter XVII, Section III] (see also [39, 55] formore details). Our analysis requires the following theorem. It is a consequence of the analysis of Guthand Zahl [40, Corollary 10.2], which itself is a generalization of the Cayley–Salmon theorem on surfacesruled by lines (see, e.g., Guth and Katz [39]), and is closely related to Theorem 1.3 (also due to Guth andZahl [40]). The novelty in this theorem is that it addresses surfaces that are infinitely ruled by certainfamilies of curves, where the analysis in [40] only handles surfaces that are doubly ruled by such curves.

Theorem 1.13. (a) For given integer parameters c and E, there are constants c1 = c1(c, E), r = r(c, E),and t = t(c, E), such that the following holds. Let f be a complex irreducible polynomial of degree D E,and let C0 ⊂ C3,E be a constructible set of complexity at most c. If there exist at least c1D

2 curves of C0, suchthat each of them is contained in Z(f) and contains at least c1D points on Z(f) that are (t, C0, r)-flecnodes,then Z(f) is infinitely ruled by curves from C0.

(b) In particular, if Z(f) is not infinitely ruled by curves from C0 then, except for at most c1D2 exceptional

curves, every curve in C0 that is fully contained in Z(f) is incident to at most c1D points that are incidentto at least t curves in C0 that are also fully contained in Z(f).

Note that, by making c1 sufficiently large (specifically, choosing c1 > E), the assumption that each ofthe c1D

2 curves in the premises of the theorem is fully contained in Z(f) follows (by Bezout’s theorem)from the fact that each of them contains at least c1D points on Z(f). Although the theorem is a corollaryof the work of Guth and Zahl in [40], we review (in the appendix) the machinery needed for its proof, andsketch a brief version of the proof itself, for the convenience of the reader and in the interest of completeness.

2 Proofs of Theorems 1.4 and 1.5 (points and curves)

The proofs of both theorems are almost identical, and they differ in only one step in the analysis. We willgive a full proof of Theorem 1.4, and then comment on the few modifications that are needed to establishTheorem 1.5.

Proof of Theorem 1.4. Since the family C has k degrees of freedom with multiplicity µ, the incidencegraph G(P, C), as a subgraph of P × C, does not contain Kk,µ+1 as a subgraph. The Kovari-Sos-Turantheorem (e.g., see [47, Section 4.5]) then implies that I(P, C) = O(mn1−1/k + n), where the constant ofproportionality depends on k (and µ). We refer to this as the naive bound on I(P, C). In particular, whenm = O(n1/k), we get I(P, C) = O(n). We may thus assume that m ≥ a′n1/k, for some absolute constanta′.

The proof proceeds by double induction on n and m, and establishes the bound

I(P, C) ≤ A(m

k3k−2n

3k−33k−2 +m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

), (16)

for a suitable constant A that depends on k, µ, E, and the complexity of C0.

The base case for the outer induction on n is n ≤ n0, for a suitable sufficiently large constant n0 thatwill be set later. The bound (16) clearly holds in this case if we choose A ≥ n0.

The base case for the inner induction on m is m ≤ a′n1/k, in which case the naive bound implies thatI(P, C) = O(n), so (16) holds with a sufficiently large choice of A. Assume then that the bound (16) holdsfor all sets P ′, C′ with |C′| < n or with |C′| = n and |P ′| < m, and let P and C be sets of sizes |P | = m,|C| = n, such that n > n0, and m > a′n1/k.

16

It is instructive to notice that the two terms mk

3k−2n3k−33k−2 and m in (16) compete for dominance; the

former (resp., latter) dominates when m ≤ n3/2 (resp., m ≥ n3/2). One therefore has to treat these twocases somewhat differently; see below and also in earlier works [39, 56].

Applying the polynomial partitioning technique. We construct a partitioning polynomial f forthe set C of curves, as in the recent variant of the polynomial partitioning technique, due to Guth [35].Specifically, we choose a degree

D =

cm

k3k−2 /n

13k−2 , for a′n1/k ≤ m ≤ an3/2,

cn1/2, for m > an3/2,(17)

for suitable constants c, a, and a′ (whose values will be set later), and obtain a polynomial f of degreeat most D, such that each of the O(D3) (open) connected components of R3 \ Z(f) is crossed by at mostO(n/D2) curves of C, where the former constant of proportionality is absolute, and the latter one dependson E. Note that in both cases 1 ≤ D n1/2, if a, a′, and c are chosen appropriately. Denote the cells ofthe partition as τ1, . . . , τu, for u = O(D3). For each i = 1, . . . , u, let Ci denote the set of curves of C thatintersect τi, and let Pi denote the set of points that are contained in τi. We set mi = |Pi| and ni = |Ci|, fori = 1, . . . , u, put m′ =

∑imi ≤ m, and notice that ni = O(n/D2), for each i. An obvious property (which

is a consequence of the generalized version of Bezout’s theorem [32]) is that every curve of C intersects atmost ED + 1 cells of R3 \ Z(f).

When a′n1/k ≤ m ≤ an3/2, within each cell τi of the partition, for i = 1, . . . , u, we use the naive bound

I(Pi, Ci) = O(min1−1/ki + ni) = O

(mi(n/D

2)1−1/k + n/D2),

and, summing over the O(D3) cells, we get a total of

O

(mn1−1/k

D2(1−1/k)+ nD

).

With the above choice of D, we deduce that the total number of incidences within the cells is

O(m

k3k−2n

3k−33k−2

).

When m > an3/2, within each cell τi of the partition we have ni = O(n/D2) = O(1), so the numberof incidences within τi is at most O(mini) = O(mi), for a total of O(m) incidences. Putting these twoalternative bounds together, we get a total of

O(m

k3k−2n

3k−33k−2 +m

)(18)

incidences within the cells.

Incidences within the zero set Z(f). It remains to bound incidences with points that lie on Z(f). SetP ∗ := P ∩ Z(f) and m∗ := |P ∗| = m −m′. Let C∗ denote the set of curves that are fully contained inZ(f), and set C′ := C \ C∗, n∗ := |C∗|, and n′ := |C′| = n− n∗. Since every curve of C′ intersects Z(f) in atmost ED = O(D) points, we have (for either choice of D)

I(P ∗, C′) = O(nD) = O(m

k3k−2n

3k−33k−2 +m

). (19)

Finally, we consider the number of incidences between points of P ∗ and curves of C∗. Decompose finto (complex) irreducible components f1, . . . , ft, for t ≤ D, and assign each point p ∈ P ∗ (resp., curveγ ∈ C∗) to the first irreducible component fi, such that Z(fi) contains p (resp., fully contains γ; such

17

a component always exists). The number of “cross-incidences”, between points and curves assigned todifferent components, is easily seen, arguing as above, to be O(nD), which satisfies our bound. In whatfollows, we recycle the symbols mi (resp., ni), to denote the number of points (resp., curves) assigned tofi, and put Di = deg(fi), for i = 1, . . . , t. We clearly have

∑imi = |P ∗| = m∗,

∑i ni = |C∗| = n∗, and∑

iDi = deg(f) = D.

For each i = 1, . . . , t, there are two cases to consider.

Case 1: Z(fi) is infinitely ruled by curves of C0. By assumption, there are at most q curves of Con Z(fi), implying that ni ≤ q. We project the points of Pi and the curves of Ci onto some generic planeπ0. A suitable choice of π0 guarantees that (i) no pair of intersection points or points of Pi project to thesame point, (ii) if p is not incident to γ then the projections of p and of γ remain non-incident, (iii) no pairof curves in Ci have overlapping projections, and (iv) no curve of Ci contains any segment orthogonal toπ0. Moreover, the number of degrees of freedom does not change in the projection (see Sharir et al. [53]).The number of incidences for the points and curves assigned to Z(fi) is equal to the number of incidencesbetween the projected points and curves, which, by Theorem 1.1, is

O

(m

k2k−1

i n2k−22k−1

i +mi + ni

)= O

(m

k2k−1

i nk−12k−1

i qk−12k−1 +mi + ni

).

Summing over i = 1, . . . , t, and using Holder’s inequality, we get the bound

O(m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

),

which, by making A sufficiently large, is at most

A4

(m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

). (20)

Remark. This is the only step in the proof where being of reduced dimension s, for s sufficiently small,might yield an improved bound (over the one in (20)); see below, in the follow-up proof of Theorem 1.5,for details.

Case 2: Z(fi) is not infinitely ruled by curves of C0. In this case, Theorem 1.13(b) implies thatthere exist suitable constants c1, t that depend on E and on the complexity of C0, such that there are atmost c1D

2i exceptional curves, namely, curves that contain at least c1Di points that are incident to at least

t curves from C∗. Therefore, by choosing c (in the definition of D) sufficiently small, we can ensure that,in both cases (of small m and large m),

∑iD

2i ≤ (

∑iDi)

2 = D2 n. This allows us to apply inductionon the number of curves, to handle the exceptional curves. Concretely, we have an inductive instance ofthe problem involving mi points and at most c1D

2i n curves of C. By the induction hypothesis, the

corresponding incidence bound is at most

A

(m

k3k−2

i (c1D2i )

3k−33k−2 +m

k2k−1

i (c1D2i )

k−12k−1 q

k−12k−1 +mi + c1D

2i

).

We now sum over i. For the first and fourth terms, we bound each mi by m, and use the fact that∑iD

αi ≤ Dα for any α ≥ 1. For the second terms, we use Holder’s inequality. Overall, we get the

incidence bound

A

(c3k−33k−2

1 mk

3k−2 (D2)3k−33k−2 + c

k−12k−1

1 mk

2k−1 (D2)k−12k−1 q

k−12k−1 +m+ c1D

2

),

which, with a proper choice of c (in (17)), can be upper bounded by

A4

(m

k3k−2n

3k−33k−2 +m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

). (21)

18

Except for these incidences, for each fi, each non-exceptional curve in C∗ that is assigned to Z(fi) is incidentto at most c1Di points that are incident to at least t curves from C∗; the total number of incidences ofthis kind involving the ni curves assigned to Z(fi) and their incident points is O(niDi). Other incidencesinvolving the non-exceptional curves in C that are assigned to Z(fi) only involve points assigned to Z(fi)that are incident to at most t = O(1) curves from C∗; the number of such point-curve incidences is thusO(mit) = O(mi). Therefore, when Z(fi) is not infinitely ruled by curves of C∗, the number of incidencesassigned to Z(fi) is O(mi+niDi), plus terms that are accounted for by the induction. Summing over thesecomponents Z(fi), we get the bound O(m + nD), plus the inductive bounds in (21), and, choosing A tobe sufficiently large, these bounds will collectively be at most

A2

(m

k3k−2n

3k−33k−2 +m

k2k−1n

k−12k−1 q

k−12k−1 +m+ n

). (22)

In summary, by choosing A sufficiently large, the number of incidences is well within the bound of (16),thus establishing the induction step, and thereby completing the proof. 2

Proof of Theorem 1.5. The proof proceeds by the same double induction on n and m, and establishesthe bound, for any prespecified ε > 0,

I(P, C) ≤ Am k3k−2n

3k−33k−2 +Aε

(m

2s5s−4n

3s−45s−4 q

2s−25s−4

+ε +m2/3n1/3q1/3 +m+ n), (23)

for a suitable constant A that depends on k, µ, s, E, and the complexity of C0, and another constant Aεthat also depends on ε. The flow of the proof is very similar to that of the preceding proof. The maindifference is in the case where some component Z(fi) of Z(f) is infinitely ruled by curves from C0. Again,in this case it contains at most q curves of C∗.

We take the points of P ∗ and the curves of C∗ that are assigned to Z(fi), and project them onto somegeneric plane π0 (the same plane can be used for all such components), as in the proof of Theorem 1.4 andget the same properties (i)–(iv) of the projected points and curves. Let Pi and Ci denote, respectively, theset of projected points and the set of projected curves; the latter is a set of ni plane irreducible algebraiccurves of constant maximum degree8 DE. Moreover, as in the preceding proof, the contribution of Z(fi) toI(P ∗, C∗) is equal to the number I(Pi, Ci) of incidences between Pi and Ci. We can now apply Theorem 1.2to Pi and Ci. To do so, we first note:

Lemma 2.1. Ci is contained in an s-dimensional family of curves.

Proof. Here it is more convenient to work over the complex field C (see the general remark in theintroduction). Let Π0 denote the projection of C3 onto π0. Let C0(fi) denote the family of the curves ofC0 that are contained in Z(fi), and let C0(fi) denote the family of their projections onto π0 (under π0).Define the mapping ψ : C0(fi) → C0(fi), by ψ(γ) = Π0(γ), for γ ∈ C0(fi). By Green and Morrison [33](see also [42, Lecture 21] and Ellenberg et al. [27, Section 2]), C0(fi) and C0(fi) are algebraic varieties andψ is a (surjective) morphism from C0(fi) to C0(fi). In general, if ψ : X 7→ Y is a surjective morphism ofalgebraic varieties, then the dimension of X is at least as large as the dimension of Y . Indeed, Definition11.1 in [42] defines the dimension via such a morphism, provided that it is finite-to-one. A complete proofof the general case is given in [70]. Therefore, C0(fi) is of dimension at most dim(C0(fi)) = s, and the proofof the lemma is complete. 2

Applying Theorem 1.2 to the projected points and curves, we conclude that the number of incidencesfor the points and curves assigned to Z(fi) is at most

Bε

(m

2s5s−4

i n5s−65s−4

+ε

i +m2/3i n

2/3i +mi + ni

)≤ Bε

(m

2s5s−4

i n3s−45s−4

i q2s−25s−4

+ε +m2/3i n

1/3i q1/3 +mi + ni

),

8A projection preserves irreducibility and does not increase the degree; see, e.g., Harris [42] for a reference to these facts.

19

with a suitable constant of proportionality Bε that depends on s and on ε. By Holder’s inequality, summingthis bound over all such components Z(fi), we get the bound

B′ε(m

2s5s−4n

3s−45s−4 q

2s−25s−4

+ε +m2/3n1/3q1/3 +m+ n),

for another constant B′ε proportional to Bε. By making Aε sufficiently large, this bound is at most

Aε4

(m

2s5s−4n

3s−45s−4 q

2s−25s−4

+ε +m2/3n1/3q1/3 +m+ n).

The rest of the proof proceeds as the previous proof, more or less verbatim, except that we need a morecareful (albeit straightforward) separate handling of the leading term, multiplied by A, and the otherterms, multiplied by Aε. The induction step then establishes the bound in (23) in much the same way asabove. 2

Remarks. (1) As already mentioned in the introduction, the “lower-order” terms

O(m

2s5s−4n

3s−45s−4 q

2s−25s−4

+ε +m2/3n1/3q1/3 +m+ n)

in the bound are “best possible” in the following sense. If the bound in Theorem 1.2 were optimal, ornearly optimal, in the worst case, for points and curves of C0 that lie in a constant-degree surface V thatis infinitely ruled by such curves, the same would also hold for the lower-order terms in the bound inTheorem 1.5.9 This is shown by a simple packing argument, in which we take n/q generic copies of V , andplace on each of them mq/n points and q curves, so as to obtain

Ω(

(mq/n)2s

5s−4 q5s−65s−4 + (mq/n)2/3q2/3 +mq/n+ q

)

incidences on each copy, for a total of

(n/q)·Ω(

(mq/n)2s

5s−4 q5s−65s−4 + (mq/n)2/3q2/3 +mq/n+ q

)= Ω

(m

2s5s−4n

3s−45s−4 q

2s−25s−4 +m2/3n1/3q1/3 +m+ n

)

incidences. (This construction works when m > n/q. Otherwise, the bound is linear, and clearly bestpossible. Also, we assume that the lower bound does not involve the factor qε, to simplify the reasoning.)In particular, this remark applies to the case of points and circles, as discussed in Theorem 1.6.

(2) There is an additional step in the proof in which the fact that C0 is of some constant (not necessarilyreduced) dimension s′ could lead to an improved bound. This is the base case m = O(n1/k), where we usethe Kovari-Sos-Turan theorem to obtain a linear bound on I(P, C). Instead, we can use the result of Foxet al. [30, Corollary 2.3], and the fact that the incidence graph does not contain Kk,µ+1 as a subgraph, toshow that, when m = O(n1/s′), the number of incidences is linear. The problem is that here we need to usethe dimension s′ of the entire C0, rather than the reduced dimension s (which, as we recall, applies only tosubsets of C0 on a variety that is infinitely ruled by curves of C0). Typically, as already noted, s is largerthan k (generally twice as large as k), making this bootstrapping bound inferior to what we have. Still, incases where s′ happens to be smaller than k, this would lead to a further improved incidence bounds, inwhich the leading term is also smaller.

Rich points. Theorems 1.4 and 1.5 can easily be restated as bounding the number of r-rich points fora set C of curves with k degrees of freedom (and or reduced dimension s) in R3, when r is at least somesufficiently large constant. The case r = 2 is treated in Guth and Zahl [40], and the same bound that theyobtain holds for larger values of r (albeit without an explicit dependence on r), smaller than the thresholdin the following corollary.

9Theorem 1.2 is formulated, and proved in [60], only for plane curves. Nevertheless, it also holds for curves contained in avariety V of constant degree, simply by projecting the points and curves onto some generic plane, as done in the proofs.

20

Corollary 2.2. (a) Let C be a set of n irreducible algebraic curves, taken from some constructible familyC0 of irreducible curves of degree at most E and with k degrees of freedom (with some multiplicity µ) inR3, and assume that no surface that is infinitely ruled by curves of C0, or, alternatively, by curves of degreeat most E, contains more than q curves of C (e.g., make this assumption for all surfaces of degree at most100E2). Then there exists some constant r0, depending on k (and µ) and on C0, or, more generally, on E,such that, for any r ≥ r0, the number of points that are incident to at least r curves of C (so-called r-richpoints) is

O

(n3/2

r3k−22k−2

+nq

r2k−1k−1

+n

r

),

where the constant of proportionality depends on k and E (and on µ).

(b) If C0 is also of reduced dimension s, the bound on the number of r-rich points becomes

O

(n3/2

r3k−22k−2

+nq

2s−23s−4

+ε

r5s−43s−4

+n

r

),

where the constant of proportionality now also depends on s and ε. (Actually, the first term comes with aconstant that is independent of ε.)

Proof. Denoting by mr the number of r-rich points, the corollary is obtained by combining the upperbound in Theorem 1.4 or Theorem 1.5 with the lower bound rmr. 2

The bound in (b) is an improvement, for s = k, when q > rk+ε′ , for another arbitrarily small parameterε′, which is linear in the prespecified ε. (To be more precise, this is an improvement at all only when thesecond term dominates the bound.)

It would be interesting to close the gap, by obtaining an r-dependent bound also for values of r between3 and r0. It does not seem that the technique in Guth and Zahl [40] extends to this setup.

3 Incidences between points and circles and similar triangles in R3

We first briefly discuss the fairly straightforward proof of Theorem 1.6. As already discussed in theintroduction, we have k = s = 3, for the case of circles, so we can apply Theorem 1.5 in the context ofcircles, and obtain the bound

I(P, C) = O(m3/7n6/7 +m2/3n1/3q1/3 +m6/11n5/11q4/11+ε +m+ n

),

for any ε > 0, where q is the maximum number of the given circles that are coplanar or cospherical. Infact, the extension of the planar bound (1) to higher dimensions, due to Aronov et al. [6], asserts that, forany set C of circles in any dimension, we have

I(P, C) = O(m2/3n2/3 +m6/11n9/11 log2/11(m3/n) +m+ n

), (24)

which is slightly better than the general bound of Sharir and Zahl [60] (given in Theorem 1.2). If we usethis bound, instead of that in Theorem 1.2, in the proof of Theorem 1.5 (specialized for the case of circles),we get the slight improvement (in which the two constants of proportionality are now absolute)

I(P, C) = O(m3/7n6/7 +m2/3n1/3q1/3 +m6/11n5/11q4/11 log2/11(m3/q) +m+ n

),

which establishes Theorem 1.6.

21

The number of similar triangles. Theorem 1.6 has the following interesting application. Let P be aset of n points in R3, and let ∆ = abc be a fixed given triangle. The goal is to bound the number, denotedas S∆(P ), of triangles spanned by P and similar to ∆. The best known upper bound for S∆(P ), obtainedby Agarwal et al. [1], is O(n13/6), and the proof that establishes this bound in [1] is fairly involved. UsingTheorem 1.6, we obtain the following simple and fairly straightforward improvement.

Theorem 3.1. S∆(P ) = O(n15/7).

Proof. Following a standard strategy, fix a pair p, q of points in P , and consider the locus γpq of all pointsr such that the triangle pqr is similar to ∆ (when p, q, r are mapped to a, b, c, respectively). Clearly, γpq isa circle whose axis (line passing through the center of γpq and perpendicular to its supporting plane) passesthrough p and q. Moreover, there exist at most two (ordered) pairs p, q and p′, q′ for which γpq = γp′q′ .Let C denote the set of all these circles (counted without multiplicity). Then S∆(P ) is at most two thirdsof the number I(P, C) of incidences between the n points of P and the N = O(n2) circles of C.

By Theorem 1.6 we thus have

S∆(P ) = O(n3/7(n2)6/7 + n2/3(n2)1/3q1/3 + n6/11(n2)5/11q4/11 log2/11 n+ n2

),

where q is the maximum number of circles in C that are either coplanar or cospherical. That is, we have

S∆(P ) = O(n15/7 + n4/3q1/3 + n16/11q4/11 log2/11 n+ n2

). (25)

We claim that q = O(n). This is easy for coplanarity, because, for any fixed plane π, each point p ∈ P cangenerate at most one circle γpq in C that is contained in π. Indeed, the axis of such a circle is perpendicularto π and passes through p. This fixes the center of γpq, and it is easily checked that the radius is also fixed.A similar argument holds for cospherical circles. Here too, for a fixed sphere σ, each point p ∈ P that isnot the center o of σ can generate at most one circle γpq in C that is contained in σ. This is because theaxis of such a circle must pass through o, which fixes the center of the circle, and the radius is also fixed,as an easy calculation shows. For p = o there are at most n− 1 additional such circles.

Hence, plugging q = O(n) into (25), we get S∆(P ) = O(n15/7), as asserted. 2

4 Proof of Theorem 1.7 (points on a variety and surfaces)

Let P , V , S, F , m, and n be as in the statement of the theorem. We first restrict the analysis to thecase where V is irreducible. This involves no loss of generality, because, when V is reducible, we candecompose it into its irreducible components, assign each point of P to each component that containsit, and assign the surfaces of S to all the components. This decomposes the problem into at most Dsubproblems, each involving an irreducible surface, and it thus follows that the original vertex set countis at most D = O(1) times the bound for the irreducible case. In the remainder of this section we thusassume that V is irreducible. To obtain the bound in (5) or in (6) on

∑γ |Pγ |, we reduce this problem to

the case of incidences between points and algebraic curves in the plane, and then apply either Theorem 1.1or Theorem 1.2, as appropriate.

Surfaces with k degrees of freedom. Recall that a family F of surfaces is said to have k degrees offreedom with respect to a constant-degree variety V , if the family of the irreducible components of theintersection curves σ ∩ V | σ ∈ F, counted without multiplicity, has k degrees of freedom, with someconstant multiplicity µ, as defined for curves in R3 in Section 1.3.

Note that this definition means that, for any k points on V there are at most µ curves of the formσ ∩ V , for σ ∈ F , that pass through all the points; the number of surfaces that pass through all the points

22

could be much larger, even infinite. For example, spheres in R3 have four degrees of freedom with respectto any variety that is neither a sphere nor a plane, because four non-cocircular points determine a uniquesphere that passes through all four, whereas four cocircular points (over-)determine a unique circle thatpasses through all of them, but an infinity of spheres with this property. Interestingly, when V is a sphereor a plane, the number of degrees of freedoms goes down to three.

The reduction. Consider the intersection curves γσ := σ ∩ V , for σ ∈ S. These are algebraic curvesof degree O(DE) = O(1). These curves are not necessarily distinct, and pairs of distinct curves can havecommon irreducible components. We denote by Γ the multiset of the irreducible components of thesecurves, where each component appears with multiplicity equal to the number of surfaces that contain it;we also denote by Γ0 the underlying set of distinct irreducible components of the curves, obtained byremoving duplications from Γ.

We may assume that V does not fully contain any surface of S. Indeed, since V is irreducible, it cancontain (that is, coincide with) at most one such surface, which contributes at most m to

∑γ |Pγ |. Ignoring

this surface, we have that each γσ is at most one-dimensional; it can be empty, and it may have isolatedpoints. To treat these points, we note that each such curve has only O(1) such points,10 so the isolatedpoints contribute a total of at most O(n) incidences with their corresponding surfaces. For uniformity, wesimply record these incidences, as well as those involving the surface fully contained in (coinciding with)V , if any, as trivial (complete) bipartite graphs, with total vertex set size O(m + n) on the P -side, andO(n) on the S-side.

We represent (the remainder of) G(P, S) simply as the union⋃γ∈Γ0

(Pγ × Sγ), where, for each γ ∈ Γ0,Pγ = P ∩ γ and Sγ is the set of all surfaces in S that contain γ. This representation is not necessarilyedge disjoint, but a pair (p, σ) can appear in this union at most O(DE) times, because σ ∩ V can have atmost O(DE) irreducible components, and (p, σ) appears in the union once for each of these componentsthat contains p. The argument just offered also shows that

∑γ |Sγ | = O(n). The corresponding sum∑

γ |Pγ | (excluding the special cases treated above, which only add O(m+ n) to the count) is the numberof incidences I(P,Γ0) between the points of P and the curves of Γ0 (counted without multiplicity). Wetherefore proceed to estimate I(P,Γ0).

We follow an argument very similar to the one in the proofs of Theorems 1.4 and 1.5; due to certaindifferences, some of which are rather nontrivial, we spell it out for clarity. We take a generic plane π0 andproject the points of P and the curves of Γ0 onto π0. As before, a suitable choice of π0 guarantees that (i)no pair of intersection points or points of P project to the same point, (ii) if p is not incident to γ then theprojections of p and of γ remain non-incident, (iii) no pair of curves in Γ0 have overlapping projections,and (iv) no curve of Γ0 contains any segment orthogonal to π0. Let P ∗ and Γ∗0 denote, respectively, theset of projected points and the set of projected curves; the latter is a set of n plane irreducible algebraiccurves of constant maximum degree O(DE) (see a previous footnote). Moreover, I(P,Γ0) is equal to thenumber I(P ∗,Γ∗0) of incidences between P ∗ and Γ∗0.

We now bifurcate according to which property F is assumed to satisfy. Consider first the case whereF is of reduced dimension s with respect to V . We have the following lemma, which is an extension of thesimpler variant given in Lemma 2.1 above.

Lemma 4.1. Γ∗0 is contained in an s-dimensional family of curves.

Proof. As in the preceding proof, we work over the complex field C. Let Π0 denote the projection of C3

onto π0. Define the family of curves ΓF = σ∩V | σ ∈ F, and let Γ∗F denote the family of the projectionsof the curves in ΓF under Π0. Define mappings ϕ : F → ΓF and ψ : ΓF → Γ∗F , by ϕ(σ) = σ ∩ V , and

10The number of isolated points on a curve is easily seen to be quadratic in its degree. In our case, this degree is O(DE) =O(1), and the claim follows.

23

ψ(γ) = Π0(γ). As above, by [27, 33, 42], ΓF and Γ∗F are algebraic varieties and ψ is a morphism from ΓFto Γ∗f . By Fulton [32, Section 3.4], ϕ is a morphism from F to ΓF , and since both ϕ and ψ are surjective,it follows that their composition Φ = ψ ϕ is a surjective morphism from F to Γ∗F . As in the proof ofLemma 2.1, the definition in Harris [42, Definition 11.1], and its extension in [70], imply that the dimensionof F is at least as large as the dimension of Γ∗F . Therefore, Γ∗F is of dimension at most dim(F) = s, andthe proof of the lemma is complete. 2

Remark. It might be the case that Γ∗0 is of smaller dimension than s. As will follow from the proof, theincidence bound depends on the dimension of Γ∗0 and not on the dimension of F , so the bound will improveif Γ∗0 is indeed of smaller dimension.

In addition, the curves of Γ∗0 are of constant maximum degree. Applying Theorem 1.2 to our setup, weget the bound in (6). Adding the counts obtained separately for the preceding special cases completes theproof of Theorem 1.7 when F is of reduced dimension s with respect to V .

Consider next the case where F has k degrees of freedom with respect to V . Then Theorem 1.1 isapplicable to P ∗ and Γ∗0, and we have

I(P ∗,Γ∗0) = O(m

k2k−1n

2k−22k−1 +m+ n

),

and adding to this the bounds obtained in the other cases yields the bound asserted in (5). 2

5 Proof of Theorem 1.8 (points on a variety and general surfaces)

Let P , V , S, F , k, µ, s, m, and n be as in the statement of the theorem. We first restrict the analysisto the case where V is irreducible. The general case can be handled, as in the case of Theorem 1.7, byrepeating the analysis to each of the O(1) irreducible components of V , and summing up the resultingbounds, to obtain the same asymptotic bound (multiplied by an extra factor of D = O(1)).

Let then f be an irreducible complex polynomial such that V = Z(f), and let C denote the set ofirreducible curves that (i) are fully contained in V , (ii) are contained in at least two surfaces of S, and(iii) contain at least one point of P . By Bezout’s theorem [32] and condition (ii), we have deg(γ) ≤ E2 foreach γ ∈ C. For each curve γ ∈ C we form the bipartite subgraph Pγ × Sγ of G(P, S), where Pγ = P ∩ γ(the actual sets Pγ for some of the curves will be smaller—see below), and Sγ is the set of the surfacesof S that contain γ. To estimate

∑γ |Pγ | and

∑γ |Sγ |, we argue as follows. First,

∑γ |Pγ | is the number

of incidences between the points of P and the curves of C, counted without multiplicity. By assumption,V is not infinitely ruled by the irreducible components of the intersection curves of pairs of surfaces fromF , and C is contained in this family of curves. To apply Theorem 1.13, it remains to argue that C isconstructible. Although the proof of this property is not too hard, it is rather technical, and we willpresent it in Lemma A.4 in the Appendix. We then conclude that, for a suitable constant t = t(E, c)(where c is the complexity of the family C), except for possibly O(D2) exceptional curves, every curve inC contains only O(D) points that are incident to at least t curves of C.

Consider first incidences between the non-exceptional curves in C and the “rich” points (those incidentto at least t curves of C). Each surface σ ∈ S intersect V in a curve of degree at most DE, and cantherefore contain at most O(DE) curves from C. Each of these curves contains at most O(D) t-rich points,for a total of O(D2E) incidences for each surface σ ∈ S, so the overall number of incidences of this kindis O(nD2E) = O(n). Note that this is a bound on the actual number of point-surface incidences. Weinclude the incident pairs of this kind in G0(P, S), and the resulting bound O(n) is clearly subsumed bythe asserted bound in (8) or (9).

Removing these edges from the complete bipartite decomposition, the remaining incidences counted inI(P, C) are estimated as follows. The number of incidences with the t-poor points (each lying on at most

24

t curves of C) is at most mt = O(m), and the number of incidences between the O(D2) exceptional curvesand the points of P is at most O(mD2) = O(m), for a total of O(m) incidences.

In summary, the complete bipartite decomposition that we end up with is of the form⋃γ P′γ×Sγ , where

γ ranges over the curves of C, and (i) for each of the O(D2) exceptional curves γ we have P ′γ = Pγ , and (ii)for each of the non-exceptional curves γ, P ′γ is the set of the t-poor points of P that lie on γ. We thus obtainthat

∑γ∈C |P ′γ | = O(m). As V is irreducible and does not contain any of the surfaces σ ∈ S (except possibly

for at most one, which then coincides with V and which we may ignore, as before), the preceding argumentimplies again that each σ ∈ S generates at most DE curves of C, so

∑γ |Sγ | ≤ nDE = O(n). This gives

us the complete bipartite graph decomposition portion of the representation in (7), which satisfies all theproperties asserted in the theorem.

Let G0(P, S) denote the remainder of the incidence graph. For the moment, ignore the pairs involvingthe t-rich points on the curves of C, which are also part of the final G0(P, S). For each σ ∈ S, putγσ := (σ ∩ V ) \⋃ C. As just noted, each γσ is at most one-dimensional (i.e., a curve). By construction, itdoes not contain any curve in C, and it might also be empty (for this or for other reasons). Note that ifσ ∩ V does contain a curve γ in C, then the incidences between σ and the points of P on γ are all alreadyrecorded in P ′γ × Sγ , or are the O(n) special incidences with t-rich points on the curves in C, so ignoringthem is “safe”. Finally, we may ignore the isolated points of γσ, because, as already argued, each curveγσ can contain at most O(1) such points, which contribute a total of at most O(n) incidences with theircorresponding surfaces. Let G0(P, S) continue to denote the remaining portion of G(P, S), after pruningaway all the incidences already accounted for. Put I0(P, S) = |G0(P, S)|.

Let Γ denote the set of the n curves γσ, for σ ∈ S (and notice that this time it is an actual set, not amultiset). The curves of Γ are algebraic curves of degree at most DE, and, as is easily checked, any pairof curves γσ, γσ′ ∈ Γ intersect in at most minDE2, E4 = O(1) points.

Note that I0(P, S) is equal to the number I(P,Γ) of incidences between the points of P and the curvesof Γ. To bound the latter quantity, we proceed exactly as in the preceding proof, bifurcating accordingto whether F is of reduced dimension s with respect to V or has k degrees of freedom with respect toV . In both cases we project the points and curves onto some generic plane π0, and bound the number ofincidences between the projected points and curves, using either Theorem 1.1 (for families with k degreesof freedom) or Theorem 1.2 (for s-dimensional families), obtaining the respective bounds asserted in (8)or in (9). 2

6 Distinct and repeated distances in three dimensions

In this section we prove Theorems 1.10 and 1.11, the applications of Theorems 1.7 and 1.8 to distinct andrepeated distances in three dimensions; see also our earlier work [58] that handles these problems in asomewhat different manner. The theorems present four results, in each of which the problem is reduced toone involving incidences between spheres and points on a surface V . However, except for Theorem 1.10(b),the spheres that arise in the other three cases are restricted, by requiring their centers to lie on V and /or to have a fixed radius. This makes the number of degrees of freedom (with respect to the variety) andthe dimensionality of the corresponding families of spheres go down to 3 or 2. The case of two degrees offreedom (in Theorem 1.11(a)) is the simplest, and requires very little of the machinery developed here (seebelow). The cases of three degrees of freedom (in Theorem 1.10(a) and Theorem 1.11(b)) yield improved“in-between” bounds.

Proof of Theorem 1.10 (distinct distances). We will first establish the more general bound in (b);handling (a) will be done later, in a similar, somewhat simpler manner.

(b) Let t denote the number of distinct distances in P1 × P2. For each q ∈ P2, draw t spheres centered at

25

q and having as radii the t distinct distances. We get a collection S of nt spheres, a set P1 of m pointson V , which we relabel as P , to simplify the notation, and exactly mn incidences between the points of Pand the spheres of S.

Let C denote the set of intersection circles of pairs of spheres from S that are contained in V , countedwithout multiplicity; we keep in C only circles that contain points of P1. For each γ ∈ C, let µ(γ) ≥ 2 denoteits multiplicity, namely the number of spheres containing γ; note that µ(γ) is equal to the number of pointsof P2 that lie on the axis of γ, namely the line that passes through the center of γ and is orthogonal tothe plane containing γ. The maximum possible multiplicity of a circle is at most 2t, because the distancesof the corresponding centers in P2 to the points of P1 ∩ γ 6= ∅ are all distinct, up to a possible multiplicityof 2. For each k, let Ck (resp., C≥k) denote the subset of circles in C of multiplicity exactly (resp., at least)k, and put Nk := |Ck|, N≥k := |C≥k|.

In order to effectively apply the bound in Theorem 1.9, we first have to control the term∑

γ |Pγ | · |Sγ |,which arises since we deal here with the actual number of incidences. Specifically, we claim that most ofthe mn incidences do not come from this part of the incidence graph, unless t = Ω(n). Indeed, write thissum as ∑

γ

|Pγ | · |Sγ | =∑

k≥2

k∑

γ∈Ck|Pγ |.

Putting Ek :=∑

γ∈Ck |Pγ |, and E≥k :=∑

γ∈C≥k|Pγ |, we then have

∑

γ

|Pγ | · |Sγ | =∑

k≥2

kEk = 2E≥2 +∑

k≥3

E≥k.

By Theorem 1.8, we have E≥k = O(m), so we have

∑

γ

|Pγ | · |Sγ | = 2E≥2 +∑

k≥3

E≥k = O

(m+

2t∑

k=3

m

)= O(mt).

If this would have accounted for more than, say, half the incidences, we would get mn = O(mt), or t = Ω(n),as claimed, and then (a much larger lower bound than) the bound in the theorem would follow. We maythus ignore this term, and write

mn = O(m1/2(nt)7/8+ε +m2/3(nt)2/3 +m+ nt

),

for any ε > 0, or

t = Ω(

minm4/(7+8ε)n(1−8ε)/(7+8ε), m1/2n1/2, m

),

which, by replacing ε by another, still arbitrarily small ε′, becomes the bound asserted in the theorem (andis also smaller than the bound for the complementary situation treated above).

(a) Here we are in a more favorable situation, because the spheres in S belong to a three-dimensionalfamily of surfaces—a family that can be represented simply as V ×R. We can therefore apply Theorem 1.8with dimensionality s = 3 (which is the actual dimensionality of the family, not the reduced one withrespect to V ), arguing first that, as in the proof of (b), we may ignore the term

∑γ |Pγ | · |Sγ | in the bound

on I(P, S), which is negligible unless t = Ω(n). We thus get the inequality

n2 = O(n6/11(nt)9/11+ε + n2/3(nt)2/3 + nt

),

for any ε > 0, which yields t = Ω(n7/(9+11ε)), which, by replacing ε, as in the proof of (b), can easily bemassaged into the bound asserted in the theorem. 2

26

Proof of Theorem 1.11 (repeated distances). Consider (a) first. Following the standard approachto problems involving repeated distances, we draw a unit sphere sp around each point p ∈ P , and seek anupper bound on the number of incidences between these spheres and the points of P ; this latter numberis exactly twice the number of unit distances determined by P .

This instance of the problem has several major advantages over the general analysis in Theorem 1.9.First, in this case the incidence graph G(P, S) cannot contain K3,3 as a subgraph, eliminating altogetherthe complete bipartite graph decomposition in (10) (or, rather, bounding the overall number of edges inthese subgraphs by O(n)).

More importantly, the family of the fixed-radius spheres whose centers lie on V is 2-dimensionaland has two degrees of freedom, which leads to the standard Szemeredi-Trotter-like bound I(P, S) =O(|P |2/3|S|2/3 + |P | + |S|) = O(n4/3), so the number of repeated distances in this case is O(n4/3), asclaimed. (The last bound can be obtained by applying Theorem 1.7, but it can also be obtained moredirectly, e.g., via Szekely’s technique [65].)

We remark that the last bound does not use much of the machinery developed in this paper. Still, weare not aware of any previous claim of the bound for the general case of points on a surface; see Brass etal. [15, Section 5.2] and Brass [13] for a discussion of closely-related problems.

We now consider (b). Again, we reduce the problem to that of bounding the number of incidences betweenthe m points of P1, which lie on V , and the n unit spheres centered at the points of P2. Here too theoverall number of edges in the complete bipartite graph decomposition is O(m+ n), so we can ignore thispart of the bound.

In this case, the family of unit spheres is 3-dimensional. Applying the same reasoning as in the proofof Theorem 1.7, we conclude that the number of unit distances in this case is

O(m6/11n9/11+ε +m2/3n2/3 +m+ n

),

for any ε > 0, as claimed. 2

7 Proof of Theorem 1.12 (surfaces and arbitrary points)

The proof establishes the bound in (15), via induction on m, with a prespecified fixed parameter ε > 0.Concretely, we claim that, for any such choice of ε > 0, we can write

G(P, S) = G0(P, S) ∪⋃

γ∈Γ0

(Pγ × Sγ),

where Γ0 is a collection of distinct constant-degree irreducible algebraic curves in R3, and, for each γ ∈ Γ0,Pγ = P ∩ γ and Sγ is the set of surfaces in S that contain γ. We only include in Γ0 curves γ with |Sγ | ≥ 2(as the cases where |Sγ | = 1 will be “swallowed” in G0(P, S)), so each γ ∈ Γ0 is an irreducible componentof an intersection curve of at least two surfaces from S, and its degree is therefore at most E2. We thenclaim that

J(P, S) :=∑

γ∈Γ0

(|Pγ |+ |Sγ |

)≤ A

(m

2s3s−1n

3s−33s−1

+ε +m+ n), (26)

and |G0(P, S)| ≤ A(m+ n),

for a suitable constant A that depends on ε, s, E, D, and the complexity of the family F .

The base cases are when m ≤ n1/s or when m ≤ m0, for some sufficiently large constant m0 that will beset later. Consider first the case m ≤ n1/s. Note that in this case the right-hand side of (26) is O(n1+ε). We

27

will actually establish the bound O(n) on both J(P, S) and |G0(P, S)|, as follows. Since the surfaces of Scome from an s-dimensional family, a suitable extension of the analysis in Sharir and Zahl [60, Lemma 3.2]shows that there exists an s-dimensional real parametric space Rs, and a duality mapping that sends eachsurface σ ∈ S to a point σ∗ ∈ Rs, and sends each point p ∈ P to a constant-degree algebraic hypersurfacep∗ in that space, so that if p is incident to σ then σ∗ is incident to p∗. This holds with the exception of atmost O(1) ‘bad’ points in P and at most O(1) ‘bad’ surfaces in S, and the constants depend on s, E, andthe complexity of F . The contribution to J(P, S), or rather to I(P, S), of the bad points and surfaces isonly O(m+ n), so we can ignore it, or, rather, place these incidences in the remainder subgraph G0(P, S).

Construct the arrangement of the dual surfaces p∗, for p ∈ P . Its complexity is O(ms) = O(n), andthis bound also holds if we count each face of the arrangement, of any dimension, with multiplicity equalto the number of surfaces that contain it. For each such (relatively open) face11 f , we form the completebipartite graph Pf × Sf , where Sf is the set of all surfaces σ such that σ∗ ∈ f , and Pf is the set of allpoints p whose dual surface p∗ contains f . We have

∑f |Sf | ≤ n, and

∑f |Pf | = O(ms) = O(n). Back in

primal space, if |Sf | ≥ 2, then all the points of Pf lie in the intersection γf :=⋂σ∈Sf

σ, which, by Bezout’s

theorem, is either one-dimensional, i.e., a curve in Γ0 as in the theorem, or a discrete set of at most E3

points. In the latter case |Pf | ≤ E3 = O(1), implying that∑

f |Pf ||Sf | = O(∑

f |Sf |) = O(n). Similarly, if|Sf | = 1, then we also have

∑f |Pf ||Sf | = O(

∑f |Pf |) = O(ms) = O(n). Clearly,

∑f |Pf ||Sf |, over faces

f for which either |Sf | ≤ 1 or γf is discrete, counts the number of incidences between P and S that fallinto these special cases, so the number of these incidences is only O(n). We are left with a portion of theincidence graph that can be written as the union of complete bipartite graphs

⋃f Pf × Sf , over faces f

for which |Sf | ≥ 2 and γf is a curve in Γ0. This union is of the form asserted in the theorem, and thecorresponding J(P, S) is O(n). The asserted bound thus holds by choosing A sufficiently large.

The case m ≤ m0 follows easily (since in this case we have I(P, S) ≤ m0n) if we choose A sufficientlylarge. This holds for any choice of m0 (and a corresponding choice of A); the value that we choose isspecified later.

Suppose then that (26) holds for all sets P ′, S′, with |P ′| < m, and consider the case where the setsP, S are of respective sizes m,n, and we have m > n1/s and m > m0.

Before continuing, we also dispose of the case m ≥ n3. In this case we consider the arrangement A(S)(in R3) of the surfaces in S. The complexity of A(S) is O(n3) = O(m). More precisely, this bound holds,and is asymptotically tight, for surfaces in general position. In our case, S is likely not to be in generalposition, and then the complexity of A(S) might be smaller, because vertices and edges might be incidentto many surfaces. Nevertheless, if we count each vertex and edge of A(S) with its multiplicity, we still getthe complexity upper bound O(n3). (Here we reason in complete analogy with the dual s-dimensional casetreated above.) This means that the number of incidences with points that are either vertices or lie on the(relatively open) 2-faces of A(S) is O(n3 + m) = O(m). Incidences with points that lie on the (relativelyopen) edges of A(S) (note that each such edge is a portion of some curve of intersection between at leasttwo surfaces of S) are recorded, as usual, by a complete bipartite graph decomposition

⋃γ(Pγ×Sγ), where

the curves γ are as stipulated in the theorem, and where, as just argued, we have∑

γ |Pγ | ≤ m and∑γ |Sγ | = O(n3) = O(m). This implies that (26) holds in this case. Thus, in what follows, we assume

that m ≤ n3. Since we also assume that m > m0, we have n ≥ m1/3 > n0 := m1/30 .

Applying the polynomial partitioning technique. We fix a sufficiently large constant parameterD m1/3, whose concrete choice will be specified later, and apply the polynomial partitioning technique

11Technically, rather than considering individual faces f , we should consider the full varieties that contain these faces andare obtained by intersecting subsets of the surfaces p∗, where each such intersection might contain many faces f . However, indoing so, we want to exclude faces that lie on such an intersection and are of dimension smaller than that of the intersection(because they lie on other surfaces too), and treat them separately. To simplify the presentation, we ignore this modification,which does not affect the asymptotic bound that is derived here.

28

of Guth and Katz [39]. We obtain a polynomial f ∈ R[x, y, z] of degree at most D, whose zero setZ(f) partitions 3-space into O(D3) (open) connected components (cells), and each cell contains at mostO(m/D3) points. By duplicating cells if necessary12, we may also assume that each cell is crossed byat most O(n/D) surfaces of S; this duplication keeps the number of cells O(D3) (because each surfacecrosses only O(D2) cells, a well known property that follows, e.g., from Warren’s theorem [68]).13 That is,we obtain at most aD3 subproblems, for some absolute constant a, each associated with some cell of thepartition, so that, for each i ≤ aD3, the i-th subproblem involves a subset Pi ⊂ P and a subset Si ⊂ S,such that mi := |Pi| ≤ b0m/D

3 and ni := |Si| ≤ bn/D, for another absolute constant b0 and a constant bthat depends on E. Set P0 := P ∩ Z(f) and P ′ = P \ P0. We have

J(P, S) ≤ J(P0, S) + J(P ′, S). (27)

We first bound J(P0, S). Similarly to what was done earlier, decompose Z(f) into its O(D) irreduciblecomponents, assign each point of P0 to every component that contains it, and assign the surfaces of S toall components. We now fix a component, and bound the vertex count in the complete bipartite graphdecomposition involving incidences between the points and surfaces assigned to that component; J(P0, S)is at most D times the bound that we get. We may therefore assume that Z(f) is irreducible. Sincedeg(Z(f)) ≤ D is a constant, Theorem 1.7 implies that we can write G(P0, S) as

⋃γ P0γSγ , over a suitable

set of curves γ ⊂ V , with P0γ = P0 ∩ γ and Sγ is the set of surfaces in S containing γ, and we have

J(P0, S) = O(m

2s5s−4n

5s−65s−4

+ε +m2/3n2/3 +m+ n),

where the constant of proportionality depends on ε, s, D, E, and the complexity of F . (Here, as inTheorem 1.7, the contribution of S to this bound, namely to

∑γ |Sγ |, is only O(n). Unfortunately, this

no longer holds in the estimation of J(P ′, S), given, so we ignore this improvement). As is easily checked,this bound is subsumed in (26) for m ≥ n1/s (and s ≥ 3), if we choose A sufficiently large (so here, as inall the other steps, A depends on all the parameters just listed). Finally, we estimate

J(P ′, S) ≤aD3∑

i=1

J(Pi, Si).

By the induction hypothesis, we get, for each i,

J(Pi, Si) ≤ A(m

2s3s−1

i n3s−33s−1

+ε

i +mi + ni

).

Summing this over i, we get

J(P ′, S) ≤ A · aD3(

(b0m/D3)

2s3s−1 (bn/D)

3s−33s−1

+ε + (b0m/D3) + (bn/D)

)

=Aab

2s3s−1

0 b3s−33s−1

+ε

Dεm

2s3s−1n

3s−33s−1

+ε +Aab0m+AabD2n.

We note that m2s

3s−1n3s−33s−1

+ε ≥ nε ·m and m2s

3s−1n3s−33s−1

+ε ≥ nε ·n for n1/s ≤ m ≤ n3. We choose D sufficiently

large so that Dε ≥ 4Aab2s

3s−1

0 b3s−33s−1

+ε, and then the bound is at most

(A

4+Aab0nε

+AabD2

nε

)m

2s3s−1n

3s−33s−1

+ε.

12By using Guth’s recent technique for partitioning sets of varieties [35], already mentioned earlier, we could do withoutthis cell duplication step.

13In actuality, the bound is O(D2E2), because the intersection curve σ ∩ Z(f), for any σ ∈ S, is of degree at most DE.Here we treat E as a much smaller quantity than D, and bear in mind that the relevant constants may depend on E.

29

Choosing n0 (that is, m0) sufficiently large, so that nε0 ≥ 4a ·maxb0, bD2, we ensure that, for n ≥ n0,

A

4+Aab0nε

+AabD2

nε≤ A

4+A

4+A

4=

3A

4.

Adding the bounds for J(P0, S), and choosing A sufficiently large, we get that (26) holds for P and S.This establishes the induction step and thereby completes the proof. 2

Remark. The dimensionality s of S is used in the proof in two different steps, once in establishing a linearbound when m < n1/s, and once in deriving the bound on J(P0, S), using Theorem 1.7. As in the remarkfollowing Theorem 1.7, the values of s used in these two steps need not be the same, and the latter one istypically smaller (because it is reduced, with respect to intersection curves with a variety). Unfortunately,

this in itself does not lead to an improvement in the bound, because the leading term m2s

3s−1n3s−33s−1

+ε dependson the former value of s. If this value of s could also be improved, say by additional assumptions on thepoints and/or the surfaces, the bound in the theorem would improve too.

8 Discussion

In this paper we have made significant progress on major incidence problems involving points and curvesand points and surfaces in three dimensions. We have also obtained several applications of these results toproblems involving repeated and distinct distances in three dimensions, with significantly improved lowerand upper bounds, in cases where the points, or in the bipartite versions, the points in one of the two givensets, lie on a constant-degree algebraic surface.

The study in this paper raises several interesting open problems.

(i) A long-standing open problem is that of establishing the lower bound of Ω(n2/3) for the number ofdistinct distances determined by a set of n points in R3, without assuming them to lie on a constant-degreesurface. The best known lower bound, close to Ω(n3/5), which follows from the work of Solymosi andVu [63], still falls short of this bound. In the present study we have obtained some partial results (withbetter lower bounds) for cases where (all or some of) the points do lie on such a surface. We hope thatsome of the ideas used in this work could be applied in more general contexts, or in other special situations.

(ii) Another major long-standing open problem is that of improving the upper bound O(n3/2), establishedin [43, 69], on the number of unit distances determined by a set of n points in R3, again without assuming(all or some of) them to lie on a constant-degree surface. It would be interesting to make progress on thisproblem.

(iii) As remarked above, a challenging open problem is to characterize all the surfaces that are infinitelyruled by algebraic curves of degree at most E (or by certain classes thereof), extending the known char-acterizations for lines and circles. A weaker, albeit still hard problem is to reduce the upper bound 100E2

on the degree of such a surface, perhaps all the way down to E, or at least to O(E).

(iv) It would also be interesting to find additional applications of the results of this paper, like the onewith an improved bound on the number of similar triangles in R3, given in Section 3. One direction tolook at is the analysis of other repeated patterns in a point set, such as higher-dimensional congruent orsimilar simplices, which can sometimes be reduced to point-sphere incidence problems; see [1, 3].

(v) Concerning Theorem 1.12, we note that it is stated only for families F of surfaces of a given dimen-sionality s. It would be interesting to obtain a variant in which we assume instead that F has k degrees offreedom, after one comes up with a definition of this notion that is both (a) natural and simple to state,and (b) makes (a suitable variant of) the analysis work. We are currently studying such an extension.

30

(vi) A potentially weak issue in our analysis, manifested in the proof of all our main theorems, is that inorder to bound the number of incidences between points and curves on some variety V of constant degree,we project the points and curves on some generic plane and use a suitable planar bound, from Theorem 1.1or Theorem 1.2, to bound the number of incidences between the projected points and curves. It would bevery interesting if one could obtain an improved bound, exploiting the fact that the points and curves lieon a variety V in R3, under suitable (natural) assumptions on V .

(vii) Finally, it would be challenging to extend the results of this paper to higher dimensions.

References

[1] P. K. Agarwal, R. Apfelbaum, G. Purdy and M. Sharir, Similar simplices in a d-dimensional point set,Proc. 23rd Annu. ACM Sympos. Comput. Geom. (2007), 232–238.

[2] P. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in arrangements ofpseudocircles and their applications, J. ACM 51 (2004), 139–186.

[3] P. Agarwal and M. Sharir, On the number of congruent simplices in a point set, Discrete Comput.Geom. 28 (2002), 123–150.

[4] R. Apfelbaum and M. Sharir, Large bipartite graphs in incidence graphs of points and hyperplanes,SIAM J. Discrete Math. 21 (2007), 707–725.

[5] R. Apfelbaum and M. Sharir, Non-degenerate spheres in three dimensions, Combinat. Probab. Comput.20 (2011), 503–512.

[6] B. Aronov, V. Koltun and M. Sharir, Incidences between points and circles in three and higher dimen-sions, Discrete Comput. Geom. 33.2 (2005), 185–206.

[7] B. Aronov, J. Pach, M. Sharir and G. Tardos, Distinct distances in three and higher dimensions,Combinat. Probab. Comput. 13 (2004), 283–293.

[8] B. Aronov, M. Pellegrini and M. Sharir, On the zone of a surface in a hyperplane arrangement, DiscreteComput. Geom. 9 (1993), 177–186.

[9] B. Aronov and M. Sharir, Cutting circles into pseudo-segments and improved bounds for incidences,Discrete Comput. Geom. 28 (2002), 475–490.

[10] A. Basit and A. Sheffer, Incidences with k-non-degenerate sets and their applications, J. Comput.Geom. 5 (2014), 284–302.

[11] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions, Discrete Comput. Geom. 55.1 (2016), 158–184.

[12] L. Bien, Incidences between points and curves in the plane, M.Sc. Thesis, School of Computer Science,Tel Aviv University, 2007.

[13] P. Brass, Exact point pattern matching and the number of congruent triangles in a threedimensionalpoint set, Proc. European Sympos. Algorithms, 2000, Springer LNCS 1879, pp. 112–119.

[14] P. Brass and Ch. Knauer, On counting point-hyperplane incidences, Comput. Geom. Theory Appls.25 (2003), 13–20.

[15] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer Verlag, New York,2005.

31

[16] P. Burgisser, M. Clausen and A. Shokrollahi, Algebraic Complexity Theory, Vol. 315, Springer Scienceand Business Media, Berlin-Heidelberg, 2013.

[17] J. Cardinal, M. Payne, and N. Solomon, Ramsey-type theorems for lines in 3-space, Discrete Math.Theoret. Comput. Sci. 18 (2016).

[18] T.M. Chan, On levels in arrangements of curves, Discrete Comput. Geom. 29 (2003), 375-393.

[19] T.M. Chan, On levels in arrangements of curves, II: A simple inequality and its consequences, DiscreteComput. Geom. 34 (2005), 11–24.

[20] B. Chazelle, Cuttings, in Handbook of Data Structures and Applications, (D. P. Mehta and S. Sahni,Eds.), Chapman and Hall/CRC, 2004.

[21] B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry,Combinatorica 10 (1990), 229–249.

[22] F. R. K. Chung, Sphere-and-point incidence relations in high dimensions with applications to unitdistances and furthest-neighbor pairs, Discrete Comput. Geom. 4 (1989), 183–190.

[23] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexity boundsfor arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), 99–160.

[24] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to ComputationalAlgebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, 2007.

[25] H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many cells in arrangements of planesand related problems, Discrete Comput. Geom. 5 (1990), 197–216.

[26] Gy. Elekes and Cs. D. Toth, Incidences of not-too-degenerate hyperplanes, Proc. 21st Annu. ACMSympos. Comput. Geom., 2005, 16–21.

[27] J. Ellenberg, J. Solymosi, and J. Zahl, New bounds on curve tangencies and orthogonalities, DiscreteAnalysis, 22 (2016), 1–22.

[28] P. Erdos, On sets of distances on n points in Euclidean space, Magyar Tud. Akad. Mat. Kutato Int.Kozl. 5 (1960), 165–169.

[29] P. Erdos, D. Hickerson, and J. Pach, A problem of Leo Moser about repeated distances on the sphere,Amer. Math. Monthly 96 (1989), 569–575.

[30] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz’s problem,J. European Math. Soc., to appear. Also in arXiv:1407.5705.

[31] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Amer.Math. Soc. Press, Providence, RI, 2007.

[32] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from theCBMS Regional Conference Held at George Mason University, June 27–July 1, 1983, Vol. 54. AMSBookstore, 1984.

[33] M. Green and I. Morrison, The equations defining Chow varieties, Duke Math. J. 53 (1986), 733–747.

[34] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Comput. Geom.53 (2015), 428–444.

[35] L. Guth, Polynomial partitioning for a set of varieties, Math. Proc. Cambridge Phil. Soc. 159 (2015),459–469. Also in arXiv:1410.8871.

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[36] L. Guth, Ruled surface theory and incidence geometry, in arXiv:1606.07682.

[37] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64, Amer. Math. Soc.Press, Providence, RI, 2016.

[38] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem, AdvancesMath. 225 (2010), 2828–2839. Also in arXiv:0812.1043v1.

[39] L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, Annals Math. 181(2015), 155–190. Also in arXiv:1011.4105.

[40] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly-ruled surfaces, Amer. J. Math., inpress. Also in arXiv:1503.02173.

[41] C. G. A. Harnack, Uber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876),189–199.

[42] J. Harris, Algebraic Geometry: A First Course, Vol. 133, Springer-Verlag, New York, 1992.

[43] H. Kaplan, J. Matousek, Z. Safernova and M. Sharir, Unit distances in three dimensions, Combinat.Probab. Comput. 21 (2012), 597–610. Also in arXiv:1107.1077.

[44] J. Kollar, Szemeredi–Trotter-type theorems in dimension 3, Advances Math. 271 (2015), 30–61. Alsoin arXiv:1405.2243.

[45] N. Lubbes, Families of circles on surfaces, Contributions to Algebra and Geometry, accepted. Also inarXiv:1302.6710.

[46] A. Marcus and G. Tardos, Intersection reverse sequences and geometric applications, J. Combinat.Theory Ser. A 113 (2006), 675–691.

[47] J. Matousek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, 2002.

[48] F. Nilov and M. Skopenkov, A surface containing a line and a circle through each point is a quadric,Geom. Dedicata 163:1 (2013), 301–310.

[49] J. Pach and M. Sharir, On the number of incidences between points and curves, Combinat. Probab.Comput. 7 (1998), 121–127.

[50] O. E. Raz, M. Sharir, and F. de Zeeuw, Polynomials vanishing on Cartesian products: The Elekes–Szabo theorem revisited, Duke Math. J. 165.18 (2016), 3517–3566.

[51] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition, Hodges,Figgis and co. Ltd., Dublin, 1915.

[52] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry 42.1 (2001), 71–87.

[53] M. Sharir, A. Sheffer, and N. Solomon, Incidences with Curves in Rd, Electronic J. Combinat. 23(4):(2016). Also in Proc. European Sympos. Algorithms (2015), Springer LNCS 9294, pp. 977–988, and inarXiv:1512.08267.

[54] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and circles, Com-binat. Probab. Comput. 24 (2015), 490–520. Also in arXiv:1208.0053.

[55] M. Sharir and N. Solomon, Incidences between points and lines in R4, Discrete Comput. Geom. 57(3)(2017), 702–756. Also in Proc. 56th IEEE Sympos. Foundations of Computer Science 2015, 1378–1394,and in arXiv:1411.0777.

(A preliminary version appeared in Proc. 30th Symp. on Computational Geometry (2014), 189–197.)

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[56] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, in IntuitiveGeometry (J. Pach, Ed.), to appear. Also in Proc. 31st Annu. Sympos. Computat. Geom. (2015), 553–568, and in arXiv:1501.02544.

[57] M. Sharir and N. Solomon, Incidences between points on a variety and planes in three dimensions, inarXiv:1603.04823.

[58] M. Sharir and N. Solomon, Distinct and repeated distances on a surface and incidences between pointsand spheres, in arXiv:1604.01502.

[59] M. Sharir and N. Solomon, Incidences between points and lines on two- and three- dimensional vari-eties, in arXiv:1609.09026.

[60] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applications, J. Combinat.Theory Ser. A 150 (2017), 1–35.

[61] A. Sheffer, Distinct distances: Open problems and current bounds, in arXiv:1406:1949.

[62] M. Skopenkov and R. Krasauskas, Surfaces containing two circles through each point and Pythagorean6-tuples, in arXiv:1503.06481.

[63] J. Solymosi and V. H. Vu, Near optimal bounds for the Erdos distinct distances problem in highdimensions, Combinatorica 28 (2008), 113–125.

[64] J. Spencer, E. Szemeredi, and W. T. Trotter, Unit distances in the Euclidean plane, in: Graph Theoryand Combinatorics (B. Bollobas, ed.), Academic Press, New York, 1984, 293–303.

[65] L. Szekely, Crossing numbers and hard Erdos problems in discrete geometry, Combinat. Probab. Com-put. 6 (1997), 353–358.

[66] E. Szemeredi and W.T. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983),381–392.

[67] T. Tao, https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/.

[68] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer. Math. Soc. 133(1968), 167–178.

[69] J. Zahl, An improved bound for the number of point-surface incidences in three dimensions, Contrib.Discrete Math. 8(1) (2013), 100–121.

[70] http://math.stackexchange.com/questions/95670/

surjective-morphism-of-affine-varieties-and-dimension.

A On surfaces ruled by curves

In this appendix we review, and sketch the proofs, of several tools from algebraic geometry that are requiredin our analysis, the main one of which is Theorem 1.13. These tools are presented in Guth and Zahl [40,Section 6], but we reproduce them here, in a somewhat sketchy form, for the convenience of the reader andin the interest of completeness. We work over the complex field C, but the results here also apply to oursetting over the real numbers (see [40, 55] and a preceding remark for discussions of this issue).

A subset of CN described by some polynomial equalities and one non-equality, of the form

p ∈ CN | f1(p) = 0, . . . , fr(p) = 0, g(p) 6= 0, for f1, . . . , fr, g ∈ C[x1, . . . , xN ],

34

is called locally closed. We recall that the (geometric) degree of an algebraic variety V ⊂ CN is defined asthe number of intersection points of V with the intersection of N −dim(V ) hyperplanes in general position(see, e.g., Harris [42, Definition 18.1]). Locally closed sets have the following property.

Theorem A.1 (Bezout’s inequality; Burgisser et al. [16, Theorem 8.28]). Let V be a nonempty locallyclosed set in CN , and let H1, . . . ,Hr be algebraic hypersurfaces in CN . Then

deg(V ∩H1 ∩ · · · ∩Hr) ≤ deg(V ) · deg(H1) · · · deg(Hr).

A constructible set C is easily seen to be a union of locally closed sets. Moreover, one can decomposeC uniquely as the union of irreducible locally closed sets (namely, sets that cannot be written as the unionof two nonempty and distinct locally closed sets). By Burgisser et al. [16, Definition 8.23]), the degree ofC is the sum of the degrees of its irreducible locally closed components. Theorem A.1 implies that when aconstructible set C has complexity O(1), its degree is also O(1). We also have the following corollaries.

Corollary A.2. Let

X = p ∈ CN | f1(p) = 0, . . . , fr(p) = 0, g(p) 6= 0, for f1, . . . , fr, g ∈ C[x1, . . . , xN ].

If X contains more than deg(f1) · · · deg(fr) points, then X is infinite.

Proof. Assume that X is finite. Put V = p ∈ Cn | g(p) 6= 0. By Bezout’s inequality (Theorem A.1), wehave

deg(X) ≤ deg(V ) · deg(f1) · · · deg(fr) = deg(f1) · · · deg(fr),

where the equality deg(V ) = 1 follows by the definition of the degree of locally closed sets (see, e.g.,Burgisser et al. [16, Definition 8.23]). When X is finite, i.e., zero-dimensional, its degree is equal to thenumber of points in it, counted with multiplicities. This implies that the number of points in X is at mostdeg(f1) · · · deg(fr), contradicting the assumption of the theorem. Therefore, X is infinite. 2

As an immediate consequence, we also have:

Corollary A.3. Let C ⊂ CN be a constructible set and write it as the union of locally closed sets⋃ti=1Xi,

where

Xi = p ∈ CN | f i1(p) = 0, . . . , f iri(p) = 0, gi(p) 6= 0, for f i1, . . . , firi , g

i ∈ C[x1, . . . , xN ].

If C contains more than∑t

i=1 deg(f i1) · · · deg(f iri) points, then C is infinite.

For a constructible set C, let d(C) denote the minimum of∑t

i=1 deg(f i1) · · · deg(f iri), as in Corollary A.3,over all possible decompositions of C as the union of locally closed sets. By Bezout’s inequality (Theo-rem A.1), it follows that deg(C) ≤ d(C). Corollary A.3 implies that if C contains more than d(C) points,then it is infinite.

Following Guth and Zahl [40, Section 4], we call an algebraic curve γ ⊂ C3 a complete intersection ifγ = Z(P,Q) for some pair of polynomials P,Q. We let C[x, y, z]≤E denote the space of complex trivariate

polynomials of degree at most E, and choose an identification of C[x, y, z]≤E with14 C(E+33 ). We use the

variable α to denote an element of (C[x, y, z]≤E)2, and write

α = (Pα, Qα) ∈ (C[x, y, z]≤E)2 =(C(E+3

3 ))2.

14Here(E+33

)is the maximum number of monomials of the polynomials that we consider. For obvious reasons, the actual

representation should be in the complex projective space CP(E+33 ), but we use the many-to-one representation in C(E+3

3 ) forconvenience.

35

Given an irreducible curve γ, we associate with it a choice of α ∈(C(E+3

3 ))2

such that γ is contained in

Z(Pα, Qα), and the latter is a curve (one can show that such an α always exists; see Guth and Zahl [40,Lemma 4.2] and also Basu and Sombra [10]). Let x ∈ γ be a non-singular point15 of γ; we say that α isassociated to γ at x, if α is associated to γ, and ∇Pα(x) and ∇Qα(x) are linearly independent. We referthe reader to [40, Definition 4.1 and Lemma 4.2] for details. This is analogous to the works of Guth andKatz [39] and of Sharir and Solomon [55] for the special cases of parameterizing lines in three and fourdimensions, respectively.

Before applying this machinery to derive the main result of this appendix, we fill in the missing partin the proof of Theorem 1.8.

Lemma A.4. Let V be some irreducible algebraic surface in C3, and let F be a constructible family ofalgebraic surfaces in C3 of constant degree E. Let C be the family of the irreducible components of theintersection curves of pairs of surfaces in F . Then C is a constructible family of curves.

Proof. We use the preceding identification of C[x, y, z]≤E with C(E+33 ), and use the variable α to denote

an element of (C[x, y, z]≤E)2, writing, as before

α = (Pα, Qα) ∈ (C[x, y, z]≤E)2 =(C(E+3

3 ))2.

Since F is constructible, the following set

W = α = (Pα, Qα) ∈ (C[x, y, z]≤E)2 | Pα, Qα ∈ F

is also contructible. By definition, recalling that C3,D denotes the set of all irreducible algebraic curves ofdegree at most D in C2, we have

C = γ ∈ C3,E2 | ∃α ∈W,γ ∈ Pα ∩Qα.

It then follows that C is irreducible by a simple variant of Lemma 4.3 of Guth and Zahl [40]. 2

We now go on to the main result of the appendix. In what follows, we fix a constructible set C0 ⊂ C3,E

of irreducible curves of degree at most E in 3-dimensional space (recall that the entire family C3,E isconstructible). Following [40, Section 9], we call a point p ∈ Z(f), for a given polynomial f ∈ C[x, y, z],a (t, C0, r)-flecnode, if there are at least t curves γ1, . . . , γt ∈ C0, such that, for each i = 1, . . . , t, (i) γi isincident to p, (ii) p is a non-singular point of γi, and (iii) γi osculates to Z(f) to order r at p. This is ageneralization of the notion of a flecnodal point, due to Salmon [51, Chapter XVII, Section III] (see also[39, 55] for details).

With all this machinery, we can now present a (sketchy) proof of Theorem 1.13. The theorem is statedin Section 1, and we recall it here. It is adapted from Guth and Zahl [40, Corollary 10.2], serves as ageneralization of the Cayley–Salmon theorem on surfaces ruled by lines (see, e.g., Guth and Katz [39]),and is closely related to Theorem 1.3 (also due to Guth and Zahl [40]).

Theorem 1.13. For given integer parameters c, E, there are constants c1 = c1(c, E), r = r(c, E), andt = t(c, E), such that the following holds. Let f be a complex irreducible polynomial of degree D E, andlet C0 ⊂ C3,E be a constructible set of complexity at most c. If there exist at least c1D

2 curves of C0, suchthat each of them is contained in Z(f) and contains at least c1D points on Z(f) that are (t, C0, r)-flecnodes,then Z(f) is infinitely ruled by curves from C0. In particular, if Z(f) is not infinitely ruled by curves fromC0 then, except for at most c1D

2 exceptional curves, every curve in C0 that is fully contained in Z(f) is

15Given an irreducible curve in R3, a point x ∈ γ is non-singular if there are polynomials f1, f2 that vanish on γ such that∇f1(x) and ∇f2(x) are linearly independent.

36

incident to at most c1D points that are incident to at least t curves in C0 that are also fully contained inZ(f).

Proof. For the time being, let r be arbitrary. By Guth and Zahl [40, Lemma 8.3 and Equation (8.1)], sincef is irreducible, there exist r polynomials16 hj(α, p) ∈ C[α, x, y, z], for j = 1, . . . , r of degree at most bj inα (where bj is a constant depending on j and on E), and of degree O(D) in p = (x, y, z), with the followingproperty: let γ be an irreducible curve, let p be a non-singular point of γ, and let α be associated to γ atp, then γ osculates to Z(f) to order r at p if and only if hj(α, p) = 0 for j = 1, . . . , r. (These polynomialsare suitable representations of the first r terms of the Taylor expansion of f at p along γ; see [40, Section6.2] for this definition, and also [39, 55] for the special cases of lines in R3 and R4, respectively.)

Regarding p as fixed, the system hj(α, p) = 0, for j = 1, . . . , r, in conjunction with the constructible

condition that α ∈ C0, defines a constructible set Cp. By definition, we have d(Cp) ≤(∏r

j=1 bj

)· d(C0),

which is a constant that depends only on r and E. By Corollary A.3, Cp is either infinite or contains atmost d(Cp) = O(1) points. By Guth and Zahl [40, Corollary 12.1], there exist a Zariski open set O, and asufficiently large constant r0, that depend on C0 and E (see [40, Theorem 8.1] for the way r0 is obtained),such that if p ∈ O is a (t, C0, r)-flecnode, with r ≥ r0, there are at least t curves that are incident to pand are fully contained in Z(f). Since, by assumption, there are at least c1D

2 curves, each containing atleast c1D (t, C0, r)-flecnodes, it follows from [40, Proposition 10.2] that there exists a Zariski open subsetO of Z(f), all of whose points are (t, C0, r)-flecnodes. As noted above, [40, Corollary 12.1] then impliesthat every point of O is incident to at least t curves of degree at most E that are fully contained in Z(f).

As observed above, when t ≥(∏r

j=1 bj

)· d(C0), a constant depending only on C0 and E, Z(f) is infinitely

ruled on this Zariski open set. By a simple argument (a variant of which is given in [59, Lemma 6.1]), wecan conclude that Z(f) is infinitely ruled (everywhere) by curves from C0, thus completing the proof. 2

16To say that hj is a ploynomial in α (and p) means that it is a polynomial in the 2(E+33

)coefficients of the monomials of

the two polynomials in the pair α (and in the coordinates (x, y, z) of p).

37

Part V

Conclusion

249

9 Discussion

9.1 Summary

In this thesis we have studied several Erdos-type questions in combinatorial geometry, involvingincidences between points and lines, curves, or surfaces, in three or higher dimensions, by applyingtools from algebra and algebraic geometry. In Chapter 2 we presented an almost tight bound onthe number of incidences between points and lines in R4. We then presented, in Chapter 3, aconfiguration of points and lines on a quadratic hypersurface in R4, having more incidences thandoes the previous bound, under its assumptions, showing that the relevant assumptions made inChapter 2 are essential.

In Chapter 4 we presented an elementary proof of the Guth-Katz bound on the number ofincidences between points and lines in R3. We then presented, in Chapter 5, applications of theGuth-Katz bound to Ramsey-type theorems on the intersection pattern of lines in R3.

In Chapter 6, we presented bounds on the number of incidences between points and lines ona two- or three-dimensional variety. The results in this chapter extend to the field of complexnumbers. Moreover, when the configuration of points is strongly two- or three-dimensional (in thesense made precise in that chapter), the number of incidences becomes linear.

In Chapter 7, we presented general bounds on the number of incidences between points andalgebraic curves in Rd , with k degrees of freedom. Then, in Chapter 8, we improved these boundsfor the three-dimensional case, extending, and then sharpening several earlier works on similarproblems. In particular, we obtained a refined bound for point-circle incidences in three dimen-sions.

Still in Chapter 8, we presented bounds on the number of incidences between points andconstant-degree algebraic surfaces, also in three dimensions (such as planes, spheres, paraboloids,etc.). As a corollary of the point-surface bounds, we deduced new bounds on the number of distinctand repeated distances determined by points on a constant-degree variety in R3.

In addition to the new bounds, a significant contribution of the thesis is in the algebraic toolsneeded to obtain these results. These tools are adaptations of various techniques in algebraicgeometry, differential geometry, and topology, as well as development of new specially-tailored

251

machinery from these areas. Some of these techniques go back to the 19th or early 20th centuries,while others are fairly recent.

9.2 Future challenges

Incidences between points and curves in higher dimensions and k-rich points. The thesisconstitutes significant progress in the study of incidence geometry in higher dimensions and inseveral related areas, continuing the thrust set forth by Guth and Katz almost a decade ago. Still,there are many further challenges in this area, some inspired by the work in this thesis, and someof broader interest. Here is a representative sample of such challenges.

(i) Our bound for incidences between points and lines in R4 is almost optimal. It remains opento get rid of the factor 2c

√logm in the bound. We have achieved this improvement when m is not

too close to n4/3, so to speak, allowing us to use weak but non-inductive bounds, and complete theanalysis in one step. See the discussion in Chapter 2 for further details.

(ii) Extend (and sharpen) the bound on the number of k-rich points in a set of n lines in R4, forany value of k. In particular, is it true that the number of intersection points of the lines (this is thecase k = 2; the intersection points are also known as 2-rich points) is O(n4/3 + nq1/2 + ns)? Weconjecture that this is indeed the case. Recently, Guth and Zahl proved, for every ε > 0, that thenumber of 2-rich points is Oε(n4/3+3ε), provided that at most O(n2/3+ε) lines lie on a commonhypersurface of constant degree (depending on ε), and at most O(n1/3+ε) lines lie on a commonplane. The goal is to weaken these fairly restrictive assumptions, in the spirit of similar recentattempts in [57, 102].

A deeper question, extending a similar open problem in three dimensions that has been posedby Guth and others (see, e.g., Katz’s expository note [68]), is whether the above conjectured boundcan be improved when q = o(n2/3) and s = o(n1/3), that is, when the second and third terms in theconjectured bound are subsumed by the term n4/3. We also note that if we could establish such abound for the number of k-rich points, for any constant k (when q and s are not too large), thenthe case of large m (that is, m = Ω(n4/3)) would become vacuous, as only O(n4/3) points could beincident to more than k lines.

(iii) Extend our study of point-line incidences in R4 to point-curve incidences in higher dimen-sions. In Chapter 7, we achieved a general bound in arbitrary dimension, using a constant-degreepartitioning polynomial, with the disadvantages discussed in that chapter and in the introduc-tion (slightly weaker bounds, significantly more restrictive assumptions, and inferior “lower-dimensional” terms). The leading terms in the resulting bounds, for points and curves in Rd ,are

O(m2/(d+1)+ε nd/(d+1)+m1+ε),

for any ε > 0. See also Dvir and Gopi [35], Zahl [127], for recent studies on the number ofrich lines determined by points sets in Cd , for arbitrary dimension d, and an improved bound by

252

Hablicsek and Scherr [59] when the points lie in Rd , for arbitrary dimension d.

Obtaining sharper results in such general settings, like the ones obtained in Chapters 4 and2, is quite challenging algebraically, although some of the tools developed in this thesis seempromising for higher dimensions too. Most of the techniques used in the thesis only work in threeor four dimensions. Extending them to higher dimensions is a major challenge in algebraic anddifferential geometry.

Incidences between points and varieties, and distinct and repeated distances in higher di-mensions.

(i) Our bounds on incidences between points and constant-degree algebraic surfaces are not sharp.A major challenge is to improve these bounds, even only for the special cases of planes or spheres.Such improvements have important implications, some of which have been noted in Chapter 8.It is our hope that this thesis, and other related studies in incidence geometry, would motivateresearchers in these fields to tackle these problems, interesting in their own right.

(ii) A long-standing open problem, which has moved to the research front after Guth and Katz’snearly complete solution of the planar case, is that of establishing the lower bound of Ω(n2/3) forthe number of distinct distances determined by a set of n points in R3, without assuming them tolie on a constant-degree surface. The best known lower bound, close to Ω(n3/5), which followsfrom the work of Solymosi and Vu [114], still falls short of this bound. In the present study, inChapter 8, we have obtained some partial results (with better lower bounds) for cases where (allor some of) the points do lie on such a surface. We hope that some of the ideas used in this workcould be applied in more general contexts, or in other special situations.

This is perhaps the most significant open problem in the research front of this area, and appearsto be an extremely hard problem. Some recent work of Bardwell-Evans and Sheffer [13] reducesthis problem to an incidence problem involving points and 2-planes in R5. The reduction is in-teresting and promising (and extends to arbitrary dimensions as well), but the resulting incidenceproblems still seem to be out of reach with the current arsenal of techniques.

(iii) Another major long-standing open problem is that of improving the upper bound O(n3/2),established in [65, 125], on the number of unit distances determined by a set of n points in R3,again without assuming (all or some of) them to lie on a constant-degree surface. It would beinteresting to make progress on this difficult problem. (A small breakthrough progress in thisdirection was achieved, very recently, by Zahl [128].)

(iv) In the context of studying incidences between points and spheres or surfaces in R3, as pre-sented in Chapter 8, a challenging open problem (already noted there) is to characterize all thesurfaces that are infinitely ruled by algebraic curves of degree at most E (or by certain classesthereof), extending the known characterizations for lines and circles, where the only such surfacesare planes (for lines) and planes and spheres (for circles). A weaker, albeit still hard problem is toreduce the upper bound 100E2 on the degree of such a surface, perhaps all the way down to E, or

253

at least to O(E).

Algebraic and other tools. Following Guth and Katz, many, if not most, of the subsequentpapers (including most chapters of this thesis) use partitioning polynomials. In many cases, thedifficult situation, is when all the points lie on the zero set of a partitioning polynomial f of largedegree. Then, one has to construct a “second-level partitioning polynomial” g, when the goal isto partition Z( f ) \ Z(g), in an appropriate analogous manner (see [65, 125]). Then, a difficultcase is when all the points lie on the variety of co-dimension two Z( f ,g) = Z( f )∩Z(g). This isfine in three dimensions, as Z( f ,g) is then a curve. In four dimensions, Basu and Sombra [16]showed how to construct a “third-level partitioning polynomial” h that partitions Z( f ,g)\Z(h), inan appropriate manner, using bounds on the Hilbert function of a variety. In other recent works,Fox et al. [47] and Matousek and Patakova [79] showed how to construct higher-level partitioningpolynomials, which have weaker properties that do not yield in general the desired conjecturedbound, and it remains a major challenge to generalize the result of Basu and Sombra [16] to higherdimensions, with the conjectured properties.

In Chapter 2, we extended to four dimensions the flecnode polynomial, defined in three di-mensions by Salmon [90], and introduced again by Guth and Katz [56]. In a more recent work,Guth and Zahl [57] extended the notion of the flecnode polynomial, from lines in three dimen-sions, to constant-degree algebraic curves in three dimensions. The next natural steps would beto extend the notion of flecnode polynomials to constant-degree algebraic varieties (or at leastjust for curves) in higher dimensions, and using these polynomials for solving other fundamentalproblems in combinatorial geometry.

In Chapter 8, we extended the notion, for a given family of curves, of having k degrees offreedom, from the plane to three dimensions. Using this notion, we were able to generalize andimprove previous bounds on the number of incidences between points and curves. The importantfeature of having k degrees of freedom is that, given k points, there are only O(1) curves fromthe given family that pass through the k points. In some problems, though, for a generic choiceof the k points, there is no curve of the family that passes through all of them. For example, thefamily of unit circles in R3 certainly has three degrees of freedom, but most triples of points donot span a unit circle. On the other hand, any pair of points (at distance at most 2) determineinfinitely many unit circles. We can say (informally and incorrectly) that unit circles in R3 have“2.5 degrees of freedom”. The goal is to exploit this property and derive smaller incidence boundsfor such families (as has been done for unit circles in R3 in [98]).

It is also appealing to extend this definition from curves to higher-dimensional varieties (like2-surfaces in three dimensions), even though the extension is not straightforward.

Algorithmic questions. Finally, the results in this thesis, as well as the overwhelming portionof results obtained since the Guth-Katz breakthroughs, are combinatorial in nature. Nevertheless,the space decomposition yielded by partitioning polynomials appears to be a useful tool for many

254

algorithmic questions, extending the arsenal of space partitioning techniques from the early 1990’s(based on cuttings or simplicial partitionings [27, 76]) in a significant way.

The problem is that the existence of partitioning polynomials is a consequence of polynomialHam-Sandwich cuts, or more generally, of the the Borsuk-Ulam theorem in very high dimensions,and there are no known efficient (polynomial-time) algorithms for these constructs. Knauer etal. [70] showed that the decision version of the Ham-Sandwich problem is NP-hard and W[1]-hard.

One of the very few algorithmic applications of this machinery, by Agarwal et al. [3], proposesa randomized technique that constructs approximate partitioning polynomials for point sets, butno similar construction is known for Guth’s partitioning polynomial [52] (for higher-dimensionalvarieties). This is a major open problem whose solution is likely to have many algorithmic im-plications. Matousek and Patakova [79] provide a polynomial partitioning method with up tod polynomials in dimension d, which allows for a complete decomposition for a given point set.They then apply it to obtain a new algorithm for the semialgebraic range-searching problem, whichhas similar running time bounds as in [3].

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References

[1] P. K. Agarwal, R. Apfelbaum, G. Purdy and M. Sharir, Similar simplices in a d-dimensionalpoint set, Proc. 23rd Annu. ACM Sympos. Comput. Geom. (2007), 232–238.

[2] P. K. Agarwal and J. Matousek, On range searching with semialgebraic sets, Discrete Comput.

Geom., 11 (1994), 393–418.

[3] P.K. Agarwal, J. Matousek and M. Sharir, On range searching with semialgebraic sets. II,SIAM J. Comput. 42 (2013), 2039–2062.

[4] P.K. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir and S. Smorodinsky, Lenses in ar-rangements of pseudocircles and their applications, J. ACM 51 (2004), 139–186.

[5] P. Agarwal and M. Sharir, On the number of congruent simplices in a point set, Discrete

Comput. Geom. 28 (2002), 123–150.

[6] R. Apfelbaum and M. Sharir, Large bipartite graphs in incidence graphs of points and hyper-planes, SIAM J. Discrete Math. 21 (2007), 707–725.

[7] R. Apfelbaum and M. Sharir, Non-degenerate spheres in three dimensions, Combinat. Probab.

Comput. 20 (2011), 503–512.

[8] B. Aronov, V. Koltun and M. Sharir, Incidences between points and circles in three and higherdimensions, Discrete Comput. Geom. 33 (2005), 185–206.

[9] B. Aronov, J. Pach, M. Sharir and G. Tardos, Distinct distances in three and higher dimensions,Combinat. Probab. Comput. 13 (2004), 283–293.

[10] B. Aronov, M. Pellegrini and M. Sharir, On the zone of a surface in a hyperplane arrange-ment, Discrete Comput. Geom. 9 (1993), 177–186.

257

[11] B. Aronov and M. Sharir, Cutting circles into pseudo-segments and improved bounds forincidences, Discrete Comput. Geom. 28 (2002), 475–490.

[12] L. Barba, J. Cardinal, J. Iacono, S. Langerman, A. Ooms, and N. Solomon, Subquadraticalgorithms for algebraic generalizations of 3SUM, Proc. 33rd Annu. Sympos. Comput. Geom.,2017, 13:1–13:15. Also in arXiv:1612.02384.

[13] S. Bardwell-Evans and A. Sheffer, A Reduction for the Distinct Distances Problem in Rd , inarXiv:1705.10963 (2017).

[14] S. Barone and S. Basu, Refined bounds on the number of connected components of signconditions on a variety, Discrete Comput. Geom. 47 (2012), 577–597.

[15] A. Basit and A. Sheffer, Incidences with k-non-degenerate sets and their applications, J.

Comput. Geom. 5 (2014), 284–302.

[16] S. Basu and M. Sombra, Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions, Discrete Comput. Geom. 55 (2016), 158–184.

[17] A. Beauville, Complex Algebraic Surfaces, No. 34, Cambridge University Press, Cambridge,1996.

[18] L. Bien, Incidences between points and curves in the plane, M.Sc. Thesis, School of Com-puter Science, Tel Aviv University, 2007.

[19] J. Bourgain, A. Gamburd, and P. Sarnak, Markoff surfaces and strong approximation: 1, inarXiv:1607.01530 (2016).

[20] P. Brass, Exact point pattern matching and the number of congruent triangles in a threedimen-sional point set, Proc. European Sympos. Algorithms, 2000, Springer LNCS 1879, pp. 112–119.

[21] P. Brass and Ch. Knauer, On counting point-hyperplane incidences, Comput. Geom. Theory

Appls. 25 (2003), 13–20.

[22] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer Verlag,New York, 2005.

[23] P. Burgisser, M. Clausen and A. Shokrollahi, Algebraic Complexity Theory, Vol. 315,Springer Science and Business Media, Berlin-Heidelberg, 2013.

[24] J. Cardinal, M. Payne, and N. Solomon, Ramsey-type theorems for lines in 3-space, Discrete

Math. Theoret. Comput. Sci. 18 (2016).

258

[25] T.M. Chan, On levels in arrangements of curves, Discrete Comput. Geom. 29 (2003), 375–393.

[26] T.M. Chan, On levels in arrangements of curves, II: A simple inequality and its consequences,Discrete Comput. Geom. 34 (2005), 11–24.

[27] B. Chazelle, Cuttings, in Handbook of Data Structures and Applications, (D. P. Mehta andS. Sahni, Eds.), Chapman and Hall/CRC, Boca Raton, 2005.

[28] B. Chazelle, H. Edelsbrunner, L. Guibas, R. Pollack, R. Seidel, M. Sharir and J. Snoeyink,Counting and cutting cycles of lines and rods in space, Comput. Geom. Theory Appl. 1 (1992),305–323.

[29] B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geom-etry, Combinatorica 10 (1990), 229–249.

[30] F. R. K. Chung, Sphere-and-point incidence relations in high dimensions with applicationsto unit distances and furthest-neighbor pairs, Discrete Comput. Geom. 4 (1989), 183–190.

[31] K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl, Combinatorial complexitybounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), 99–160.

[32] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Com-

putational Algebraic Geometry and Commutative Algebra, Springer Verlag, Heidelberg, 2007.

[33] D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Springer Verlag, Heidelberg,2005.

[34] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), 1093–1097.

[35] Z. Dvir and S. Gopi, On the number of rich lines in truly high dimensional sets, in Proc. 30th

Annu. ACM Sympos. Comput. Geom., 2015, 584–598.

[36] H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many cells in arrangements ofplanes and related problems, Discrete Comput. Geom. 5 (1990), 197–216.

[37] W. L. Edge, The Theory of Ruled Surfaces, Cambridge University Press, Cambridge, 2011.

[38] Gy. Elekes, Sums versus products in number theory, algebra and Erdos geometry — a survey,Paul Erdos and his Mathematics II, Bolyai Math. Soc. Stud. 11, Budapest, 2002, 241–290.

259

[39] Gy. Elekes, H. Kaplan and M. Sharir, On lines, joints, and incidences in three dimensions, J.

Combinat. Theory, Ser. A 118 (2011), 962–977. Also in arXiv:0905.1583.

[40] Gy. Elekes and M. Sharir, Incidences in three dimensions and distinct distances in the plane,Combinat. Probab. Comput. 20 (2011), 571–608.

[41] Gy. Elekes and Cs. D. Toth, Incidences of not-too-degenerate hyperplanes, Proc. 21st Annu.

ACM Sympos. Comput. Geom., 2005, 16–21.

[42] J. Ellenberg, J. Solymosi, and J. Zahl, New bounds on curve tangencies and orthogonalities,Discrete Analysis, 22 (2016), 1–22.

[43] P. Erdos, On a set of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

[44] P. Erdos, On sets of distances on n points in Euclidean space, Magyar Tud. Akad. Mat. Kutato

Int. Kozl. 5 (1960), 165–169.

[45] P. Erdos, D. Hickerson, and J. Pach, A problem of Leo Moser about repeated distances onthe sphere, Amer. Math. Monthly 96 (1989), 569–575.

[46] S. Feldman and M. Sharir, An improved bound for joints in arrangements of lines in space,Discrete Comput. Geom. 33 (2005), 307–320.

[47] J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl, A semi-algebraic version of Zarankiewicz’sproblem, J. European Math. Soc., to appear. Also in arXiv:1407.5705.

[48] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathe-

matics, Amer. Math. Soc. Press, Providence, RI, 2007.

[49] W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lecturesfrom the CBMS Regional Conference Held at George Mason University, June 27–July 1, 1983,Vol. 54. AMS Bookstore, 1984.

[50] M. Green and I. Morrison, The equations defining Chow varieties, Duke Math. J. 53 (1986),733–747.

[51] L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Com-

put. Geom. 53 (2015), 428–444.

[52] L. Guth, Polynomial partitioning for a set of varieties, Math. Proc. Cambridge Phil. Soc. 159(2015), 459–469. Also in arXiv:1410.8871.

260

[53] L. Guth, Ruled surface theory and incidence geometry, in arXiv:1606.07682.

[54] L. Guth, Polynomial Methods in Combinatorics, University Lecture Series, Vol. 64, Amer.Math. Soc. Press, Providence, RI, 2016.

[55] L. Guth and N. H. Katz, Algebraic methods in discrete analogs of the Kakeya problem,Advances Math. 225 (2010), 2828–2839. Also in arXiv:0812.1043v1.

[56] L. Guth and N. H. Katz, On the Erdos distinct distances problem in the plane, Annals Math.

181 (2015), 155–190. Also in arXiv:1011.4105.

[57] L. Guth and J. Zahl, Algebraic curves, rich points, and doubly-ruled surfaces, Amer. J. Math.,to appear. Also in arXiv:1503.02173.

[58] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Vol. 52, John Wiley & Sons,New York, 2011.

[59] M. Hablicsek and Z. Scherr, On the number of rich lines in high dimensional real vectorspaces, Discrete Comput. Geom. 55 (2014), 1–8. Also in arXiv:1412.7025.

[60] C. G. A. Harnack, Uber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10(1876), 189–199.

[61] J. Harris, Algebraic Geometry: A First Course, Vol. 133. Springer-Verlag, New York, 1992.

[62] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1983.

[63] T. A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving

Frames and Exterior Differential Systems, Graduate Studies in Mathematics, volume 61, Amer.Math. Soc., Providence, RI, 2003.

[64] S. Kalia, M. Sharir, N. Solomon and B. Yang, Generalizations of Szemeredi-Trotter Theo-rem, Discrete Comput. Geom. 55 (2016), 571–593.

[65] H. Kaplan, J. Matousek, Z. Safernova and M. Sharir, Unit distances in three dimensions,Combinat. Probab. Comput., 21(2012), 597–610.

[66] H. Kaplan, J. Matousek and M. Sharir, Simple proofs of classical theorems in discrete geom-etry via the Guth-Katz polynomial partitioning technique, Discrete Comput. Geom., 48 (2012),499–517.

261

[67] H. Kaplan, M. Sharir and E. Shustin, On lines and joints, Discrete Comput. Geom. 44 (2010),838–843. Also in arXiv:0906.0558.

[68] N. H. Katz, The flecnode polynomial: A central object in incidence geometry, inarXiv:1404.3412.

[69] N. H. Katz and G. Tardos, A new entropy inequality for the Erdos distance problem, inTowards a Theory of Geometric Graphs (J. Pach, ed.), vol. 342 of Contemporary Mathematics,AMS, Providence, RI, 2004, 119–126.

[70] C. Knauer, H.R. Tiwary and D. Werner, On the computational complexity of Ham-Sandwichcuts, Helly sets, and related problems, Sympos. Theor. Aspects Computer Science, 9 (2011).

[71] J. Kollar, Szemeredi–Trotter-type theorems in dimension 3, Advances Math. 271 (2015), 30–61. Also in arXiv:1405.2243.

[72] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer-Verlag,Heidelberg, 2012.

[73] J. M. Landsberg, Is a linear space contained in a submanifold? On the number of derivativesneeded to tell, J. Reine Angew. Math. 508 (1999), 53–60.

[74] N. Lubbes, Families of circles on surfaces, Contributions to Algebra and Geometry, accepted.Also in arXiv:1302.6710.

[75] A. Marcus and G. Tardos, Intersection reverse sequences and geometric applications, J. Com-

binat. Theory Ser. A 113 (2006), 675–691.

[76] J. Matousek, Efficient partition trees, Discrete Comput. Geom. 8 (1992), 315–334.

[77] J. Matousek, Lectures on Discrete Geometry, Springer Verlag, Heidelberg, 2002.

[78] J. Matousek, and B. Gartner, Understanding and Using Linear Programming (Universitext),Springer-Verlag, New York, 2006.

[79] J. Matousek and Z. Patakova, Multilevel polynomial partitions and simplified range search-ing, Discrete Comput. Geom. 54 (2015), 22–41.

[80] E. Mezzetti and D. Portelli, On threefolds covered by lines, in Abhandlungen aus dem Math-

ematischen Seminar der Universitat Hamburg, Vol. 70, No. 1, Springer Verlag, Heidelberg,2000.

262

[81] G. Monge, Application de l’Analyse a la Geometrie, Bernard, Paris, 1809.

[82] F. Nilov and M. Skopenkov, A surface containing a line and a circle through each point is aquadric, Geom. Dedicata 163:1 (2013), 301–310.

[83] J. Pach and M. Sharir, Geometric incidences, in Towards a Theory of Geometric Graphs (J.Pach, ed.), Contemporary Mathematics, Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp.185–223.

[84] J. Pach, G. Tardos and G. Toth, Disjointness graphs of segments, in arXiv:1704.01892(2017).

[85] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series,Springer Verlag, London, 2001.

[86] R. Quilodran, The joints problem in Rn, SIAM J. Discrete Math. 23 (2010), 2211–2213. Alsoin arXiv:0906.0555.

[87] O. E. Raz, M. Sharir, and F. de Zeeuw, Polynomials vanishing on Cartesian products: TheElekes–Szabo theorem revisited, Duke Math. J. 165 (2016), 3517–3566.

[88] S. Richelson, Classifying Varieties With Many Lines, a senior thesis, Harvard University,2008.

[89] E. Rogora, Varieties with many lines, Manuscripta Mathematica 82 (1994), 207–226.

[90] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition,Hodges, Figgis and co. Ltd., Dublin, 1915.

[91] H. Schenck, Computational Algebraic Geometry, Cambridge University Press, Cambridge,2003.

[92] M. R. Spiegel, S. Lipschutz, and D. Spellman, Vector Analysis, 2nd edition, Schaum’s Out-lines, McGraw-Hill, 2009.

[93] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry 42 (2001),71–87.

[94] B. Segre, Sulle Vn contenenti piu di ∞n−kSk, Nota I e II, Rend. Accad. Naz. Lincei 5 (1948),193–197.

263

[95] F. Severi, Intorno ai punti doppi impropri etc., Rend. Cir. Math. Palermo 15 (10) (1901),33–51.

[96] I. R. Shafarevich, Basic Algebraic Geometry, Vol. 197, Springer-Verlag, New York, 1977.

[97] M. Sharir, A. Sheffer, and N. Solomon, Incidences with curves in Rd , Electronic J. Com-

binat., 23 (2016), P4.16. Also in Proc. European Sympos. Algorithms (2015), Springer LNCS9294, pp. 977–988, and in arXiv:1512.08267.

[98] M. Sharir, A. Sheffer, and J. Zahl, Improved bounds for incidences between points and cir-cles, Combinat. Probab. Comput. 24 (2015), 490–520. Also in arXiv:1208.0053.

[99] M. Sharir and N. Solomon, Incidences between points and lines in R4, Proc. 30th Annu.

Sympos. Computational Geometry, 2014, 189–197.

[100] M. Sharir and N. Solomon, Incidences between points and lines in three dimensions, inIntuitive Geometry (J. Pach, Ed.), to appear. Also in Proc. 31st Annu. Sympos. Computat. Geom.

(2015), 553–568, and in arXiv:1501.02544.

[101] M. Sharir and N. Solomon, Incidences between points and lines on a two-dimensional vari-ety, in arXiv 1502.01670.

[102] M. Sharir and N. Solomon, Incidences between points and lines in R4, Discrete Comput.

Geom. 57 (2017): 702–756. Also in Proc. 56th IEEE Sympos. Foundations of Computer Science

2015, 1378–1394, and in arXiv:1411.0777.

[103] M. Sharir and N. Solomon, Incidences between points and lines on two- and three-dimensional varieties, Discrete Comput. Geom., in press. Also in arXiv:1609.09026.

[104] M. Sharir and N. Solomon, Incidences between points on a variety and planes in threedimensions, in arXiv:1603.04823.

[105] M. Sharir and N. Solomon, Distinct and repeated distances on a surface and incidencesbetween points and spheres, in arXiv:1604.01502.

[106] M. Sharir and N. Solomon, Incidences with curves and surfaces and applications to distinctand repeated distances, Proc. 28th ACM-SIAM Sympos. Discrete Algorithms, 2017, 2456–2475.Also in arXiv:1610.01560.

[107] M. Sharir and J. Zahl, Cutting algebraic curves into pseudo-segments and applications, J.

Combinat. Theory Ser. A 150 (2017), 1–35.

264

[108] A. Sheffer, Distinct distances: Open problems and current bounds, in arXiv:1406:1949.

[109] A. Sheffer, E. Szabo, and J. Zahl, Point-curve incidences in the complex plane, Combina-

torica, to appear.

[110] M. Skopenkov and R. Krasauskas, Surfaces containing two circles through each point andPythagorean 6-tuples, in arXiv:1503.06481.

[111] N. Solomon and R. Zhang, Highly incidental patterns on a quadratic hypersurface in R4,Discrete Math., 340(4) (2017), 585–590. Also in arXiv:1601.01817.

[112] J. Solymosi and T. Tao, An incidence theorem in higher dimensions, Discrete Comput.

Geom., 48 (2012), 255–280.

[113] J. Solymosi and F. de Zeeuw, Incidence bounds for complex algebraic curves on Cartesianproducts, in arXiv:1502.05304.

[114] J. Solymosi and V. H. Vu, Near optimal bounds for the Erdos distinct distances problem inhigh dimensions, Combinatorica 28 (2008), 113–125.

[115] J. Spencer, E. Szemeredi and W. T. Trotter, Unit distances in the Euclidean plane, In: Graph

Theory and Combinatorics (Proc. Cambridge Conf. on Combinatorics, B. Bollobas, ed.), 293–308, Academic Press, 1984.

[116] A. H. Stone and J. W. Tukey, Generalized sandwich theorems, Duke Math. J. 9 (1942),356–359.

[117] L. Szekely, Crossing numbers and hard Erdos problems in discrete geometry, Combinat.

Probab. Comput. 6 (1997), 353–358.

[118] E. Szemeredi and W. T. Trotter, Extremal problems in discrete geometry, Combinatorica 3(1983), 381–392.

[119] T. Tao, From rotating needles to stability of waves: Emerging connections between combi-natorics, analysis, and PDE, Notices AMS 48(3) (2001), 294–303.

[120] T. Tao, The Cayley-Salmon theorem via classical differential geometry,https://terrytao.wordpress.com/2014/03/28/

the-cayley-salmon-theorem-via-classical-differential-geometry/

[121] C.D. Toth, The Szemeredi-Trotter theorem in the complex plane, Combinatorica 35 (2015),95–126. Also in arXiv:0305283 (2003).

265

[122] P. Valtr, Strictly convex norms allowing many unit distances and related touching questions,manuscript, 2006.

[123] B. L. van der Waerden, E. Artin, and E. Noether, Modern Algebra, Vol. 2, Springer Verlag,Heidelberg, 1966.

[124] H. E. Warren, Lower bound for approximation by nonlinear manifolds, Trans. Amer. Math.

Soc. 133 (1968), 167–178.

[125] J. Zahl, An improved bound on the number of point-surface incidences in three dimensions,Contrib. Discrete Math., 8 (2013), 100–121. Also in arXiv:1104.4987.

[126] J. Zahl, A Szemeredi-Trotter type theorem in R4, Discrete Comput. Geom. 54 (2015), 513–572.

[127] J. Zahl, A note on rich lines in truly high dimensional sets, Forum of Mathematics, Sigma

4, 2016, 1-13.

[128] J. Zahl, Breaking the 3/2 barrier for unit distances in three dimensions, in arXiv:1706.05118(2017).

[129] http://math.stackexchange.com/questions/95670/

surjective-morphism-of-affine-varieties-and-dimension.

266

תקציר

גיאומטריה קומבינטורית הוא תחום העוסק בבעיות בעלות תוכן גיאומטרי. התחום נהגה ופותח ע"י פאול ארדש, החל מראשית המאה ה-20. בעוד שבעיות אלו (בעיות "מטיפוס ארדש") בדרך כלל קלות מאוד

לניסוח, חלקן קשות מאוד לפתרון ובבסיסן עומדת תיאוריה עמוקה, וחלקן פתוחות (או היו פתוחות) עשורים רבים.

בעשור החולף, פני הנוף של הגיאומטריה הקומבינטורית השתנו מהותית, בעקבות שתי פריצות דרך של גוט וכץ ([55] ב-2008 ו-[56] ב-2010). הם הציגו שיטות פשוטות יחסית בגיאומטריה אלגברית שאפשרו את פתרונן המוצלח של מספר בעיות מרכזיות בגיאומטריה קומבינטורית. מאמרם הראשון השיג פתרון מלא ל"בעיית המחברים" (Joints), בעיה המערבת חילות בין נקודות וישרים בשלושה

מימדים, שהוצעה על ידי שאזל ושות' [28] ב-1992. מאמרם השני של גוט וכץ הציג פתרון כמעט מלא ל"בעיית המרחקים השונים" של ארדש [43] במישור, שהייתה בעיה פתוחה מרכזית החל מהצגתה בשנת

1946. שתי הבעיות הללו (ובמיוחד השנייה שבהן) נחקרו רבות לאורך השנים, באמצעות שיטות מסורתיות יותר, ולאחר מכן בשיטות מורכבות יותר בגיאומטריה קומבינטורית, שהניבו פתרונות חלקיים

ולא מספקים.

ה"נישואין" הללו, בין גיאומטריה אלגברית וגיאומטריה קומבינטורית וחישובית מציגים אתגרים משמעותיים לשני התחומים, וביניהם הבנה מחודשת וזיקוק כלים קיימים בגיאומטריה אלגברית ופיתוח

כלים חדשים, מוכוונים לקראת צד ה"לקוח" החדש, ושימוש בארגז הכלים החדש לפתרון מגוון רחב ועשיר של בעיות בגיאומטריה קומבינטורית וחישובית.

בתזה זו, אנו בונים גשרים נוספים בין שתי הדיסציפלינות, מאמצים ומפתחים טכניקות וכלים אלגברים חדשים, ומפעילים אותם בפתרונן של מספר בעיות בגיאומטריה קומבינטורית. אנו מניחים ידע בסיסי

בגיאומטריה אלגברית, וכל אימת שנדרש ידע יותר מתקדם, נרחיב וניתן רקע מתאים. רוב הכלים המתקדמים יותר מוצגים בסעיף 1.3, אך גם בפרקים נוספים בתזה.

התזה בנויה כאסופת מאמרים, ומורכבת משבעה מאמרים, המוצגים בארבעה חלקים אותם נסקור כעת בארבעת הסעיפים הבאים.

חילות בין נקודות וישרים בארבעה מימדים 1.

בפרק 2, אנחנו מרחיבים את המחקר של גוט וכץ [56], מחילות בין נקודות וישרים בשלושה מימדים, לחילות בין נקודות וישרים בארבעה מימדים, ומוכיחים חסמים הדוקים או כמעט הדוקים על מספר

החילות הללו במקרה הגרוע ביותר. בעיה זו, שהינה יותר קשה משמעותית מהגרסה התלת-ממדית, דורשת פיתוח של טכניקות וכלים נוספים בגיאומטריה אלגברית, שאת רובם נסקור בסעיף 1.3

בהקדמה, ושימוש במגוון שיטות קומבינטוריות שפיתחנו בכדי לפתור את הבעיות הרלוונטיות. בצורה חופשית, ניתן לתאר את המחקר בסעיף זה, כחקר התבניות שבהן הישרים יכולים לגעת האחד ברעהו

1

כאשר הם "נזרקים" לחלל הארבע-מימדי, כאשר תת-בעיה משמעותית היא להבין את אותן תבניות כאשר הישרים הללו מוכלים ביריעה אלגברית תלת-ממדית מדרגה קבועה.

המשפט המרכזי אותו הוכחנו, שהופיע ב- [102] (לאחר עבודה מקדימה [99]), הינו המשפט הבא.

משפט 1.

> פרמטרים, כך ש-(i) כל , ויהיו < > ישרים ב > קבוצה של > נקודות ו- > קבוצה של תהא , ו-(ii) כל 2-מישור מכיל לכל < > ישרים מתוך על-מישור או על-משטח ריבועי מכיל לכל היותר

. אז מתקיים < > ישרים של היותר

<

, אנו מקבלים את החסם החזק < > או > הם קבועים מתאימים. כאשר > ו- כאשר יותר

<

> בחסם הראשון, החסם הדוק במקרה הגרוע ביותר, לכל טווח הערכים באופן כללי, פרט לגורם . < > ועבור ערכים מתאימים של של

בפתרון שלנו, השתמשנו בכלים כבדים מגיאומטריה אלגברית, חלקם בני מעל 150 שנים. למשל, משפט מ-1846 של קיילי וסלמון (שהתקבל באופן בלתי-תלוי גם ע"י מונז') [81,90], מבטיח שמשטח

אלגברי בשלושה מימדים יכול להכיל לכל היותר מספר קבוע של ישרים, אלא אם כן הוא "נשלט" על ידי ישרים. המחקר של משטחים נשלטים (ruled surfaces), כולל האופן בו ניתן לשכנם במימדים ארבע

ומעלה, הינו נושא מרכזי במחקרנו. משפט קשור אחר שאנו משתמשים בו, משנת 1901, של סגרה וסברי [94,95], מאפיין על-משטחים במרחב המרוכב ה-4-מימדי שהינם "נשלטים אינסוף-פעמים" ע"י

ישרים. הכלים הללו מוצגים גם בסעיף 1.3, וגם בפרק 2.

מספר רב של ישרים על משטח ריבועי עשוי להגדיל את מספר החילות.

בעבודת המשך עם רויז'יאנג ז'אנג [111], המוצגת בפרק 3, אנו מראים שההגבלות שנעשו במשפט 1 הינן הכרחיות, כלומר שאם נוותר עליהן, נקבל מספר חילות גדול יותר. קונקרטית, אנו מראים

שההנחה שעשינו בתנאי (i) במשפט 1, על כך שאף על-משטח ריבועי לא יכיל יותר מדי ישרים, הינה > ישרים המוכלים > נקודות ו- > קבוצה של הכרחית, על ידי כך שאנו מציגים, לכל זוג טבעיים

> המוגדר על ידי > ב- במשטח הריבועי

, <

. < כך שמספר החילות בין הנקודות והישרים הינו

, מתקבל מספר חילות גדול יותר מהחסם שמופיע במשפט 1. < כאשר

PmLnℝ4q, s ≤ nqL

sL

I(P, L) ≤ 2c logm(m2/5n4/5 + m) + A(m1/2n1/2q1/4 + m2/3n1/3s1/3 + n),

Acm ≥ n5/3m ≤ n6/7

I(P, L) ≤ A(m2/5n4/5 + m + m1/2n1/2q1/4 + m2/3n1/3s1/3 + n) .

2c logm

, m , nq, s

, m , nmnSℝ4

S = (x1, x2, x3, x4) ∈ ℝ4 |x1 = x22 + x32 − x42Ω(m2/3n1/2 + m + n)

n9/8 < m < n3/2

2

חילות בין נקודות וישרים בשלושה מימדים, עם שימושים 2.

בפרק 4, בפרק 4, שהתפרסם ב-[100], אנחנו מציגים הוכחה אלמנטרית ופשוטה יחסית לחסם של גוט , כך שאף < > ישרים ב- > נקודות ל- וכץ בשלושה מימדים, כלומר, אנחנו מראים שמספר החילות בין

> ישרים, הוא מישור לא מכיל יותר מאשר

(בניסוח גרסת המשפט המדויקת, קבועי הפרופורציה במחוברים הראשון והשלישי, תלויים, בצורה .( < > ו- חלשה, בקשר בין

. 7 משפטי רמזי עבור ישרים ב-

בפרק 5, אנחנו מוכיחים, במשותף עם מייקל פיין וז'אן קרדינל [24], משפטים מטיפוס רמזי עבור קבוצת ישרים במרחב התלת-מימדי. המשפטים הללו מבטיחים קיום של קליקה או קבוצה בלתי-תלויה עבור ה(היפר)גרפים המושרים על ידי חילות בין ישרים, נקודות, ומשטחים ריבועיים מטיפוס רגולוס

(משטח ריבועי הנשלט באופן כפול ע"י ישרים) במרחב התלת-מימדי. בין יתר הדברים, אנחנו מוכיחים את התוצאות הבאות:

גרף החיתוכים של ישרים ב- מכיל קליקה או קבוצה בלתי-תלויה בגודל . א.

כל קבוצה של ישרים ב- מכילה תת-קבוצה של ישרים שכולם נדקרים ע"י ב.

ישר אחד, או תת קבוצה של ישרים כך שאין בה 6 ישרים שנדקרים

כולם ע"י ישר אחד.

כל קבוצה של ישרים ב- במצב כללי מכילה תת-קבוצה של ישרים שמוכלים ג.

ברגולוס, או תת-קבוצה של ישרים, כך שאין בה 4 ישרים המוכלים כולם

ברגולוס.

חילות בין נקודות וישרים על יריעות 3.

> הנקודות נמצאות על יריעה דו-ממדית בתלת מימד, או יריעה תלת-ממדית במרחב כאשר הארבע-מימדי, אשר דרגתן אינה גדולה מדי, אנו מראים, בפרק 6 של התזה, שהתפרסם ב-[103],

> ישרים משמעותית קטן יותר מהחסמים של גוט וכץ והחסם במשפט שמספר החילות בין נקודות אלו ו-1, בהתאמה. יתירה מכך, אין צורך להניח שהיריעות הללו משוכנות במרחבים ה-3 או ה-4 מימדיים, בהתאמה, והחסמים שלנו תקפים גם כאשר היריעות הללו משוכנות במרחבים ממימדים גבוהים יותר.

המשפט המרכזי הראשון בפרק זה של התזה הוא המשפט הבא.

mnℝ3

s

O(m1/2n3/4 + m2/3n1/3s1/3 + m + n),

mn

ℝ3

nℝ3Ω(n1/3)nℝ3Ω( n)

Ω((n / logn)1/5)

nℝ3Ω(n2/3)Ω(n1/3)

m

n

3

משפט 2.

> א. , עבור < > ישרים ב- > קבוצה של > נקודות ו- > קבוצה של המקרה הממשי: תהי > שני פרמטרים טבעיים, כך שכל הנקודות והישרים נמצאים על כלשהוא, ויהיו

> אשר לא מכילה אף 2-מישור, וכך שאף 2-מישור לא מכיל יותר > מדרגה יריעה דו-מימדית . אז מתקיים < > ישרים של מאשר

המקרה המרוכב: תחת אותם תנאים בדיוק, כאשר המרחב הוא המרחב המרוכב , עבור ב.

כלשהוא, מתקיים

. < > קבועים, מתקבל החסם הלינארי > ו- בשני המקרים, כאשר

המשפט המרכזי השני בפרק זה של התזה הוא המשפט הבא.

משפט 3.

המקרה הממשי: תהי קבוצה של נקודות ותהי קבוצה של ישרים ב- , עבור א.

כלשהוא, ויהיו פרמטרים כך ש-(i) כל הנקודות של נמצאות על יריעה

(ii)-תלת-ממדית מדרגה , שאינה מכילה 3-מישורים או משטחים ריבועיים תלת-ממדיים, ו

אף 2-מישור לא מכיל יותר מאשר ישרים של . אז מתקיים

<

המקרה המרוכב: תחת אותם תנאים בדיוק, כאשר המרחב הוא המרחב המרוכב , עבור ב.

כלשהוא, מתקיים

<

. < > קבועים, מתקבל החסם > ו- בשני המקרים, כאשר

מאפיין מרכזי של הניתוח שלנו הוא שהחסמים המתקבלים תקפים גם מעל המרוכבים (כפי שמצוין בסעיף > קטנה מספיק). מנגד, החסמים ב' של משפטים 2 ו-3, עם תוספת קטנה שנהיית זניחה אם הדרגה

הכלליים יותר עבור חילות בין נקודות וישרים בשלושה וארבעה מימדים (כפי שנסקרו לעיל) תקפים אך ורק מעל הממשיים. ההדדיות בין המקרה הממשי והמקרה המרוכב תידון בהרחבה בסעיף 1.3.

PmLnℝdd ≥ 32 ≤ s ≤ D

VDsL

I(P, L) = O(m1/2n1/2D1/2 + m2/3D2/3s1/3 + m + n) .

ℂd

d ≥ 3

I(P, L) = O(m1/2n1/2D1/2 + m2/3D2/3s1/3 + D3 + m + n) .

sDO(m + n)

PmLnℝd

d ≥ 4s, DPD

sL

I(P, L) = O(m1/2n1/2D + m2/3n1/3s1/3 + nD + m) .

ℂd

d ≥ 4

I(P, L) = O(m1/2n1/2D + m2/3n1/3s1/3 + D6 + nD + m) .

sDO(m + n)

D

4

חילות בין נקודות ועקומים וחילות בין נקודות ומשטחים 4.

חילות בין נקודות ועקומים בשלושה מימדים ומעלה.

התוצאות שהוצגו עד כה מערבות חילות בין נקודות וישרים. אוסף התוצאות הבאות שנציג, המוצגות בחלק 4, מערבות חילות בין נקודות ועקומים אלגבריים מדרגה קבועה בשלושה מימדים ומעלה.

שני מאמרים שונים מתייחסים לבעיה זו. ראשית, אנחנו חוקרים את הבעיה הכללית, במימד כלשהוא. מאמר זה, במשותף עם אדם שפר, מוצג בפרק 7 והופיע ב-[97]. המאמר השני, המוצג בפרק 8, הופיע ב-[106]. במאמר זה אנחנו מתייחסים למקרה התלת-מימדי בלבד, ובו אנו מקבלים תוצאות חזקות יותר

משמעותית (לעומת התוצאות בפרק 7, במקרה הכללי יותר). פרק 8 גם מכיל מחקר על חילות בין , אותם נסקור בהמשך הסעיף. < נקודות ומשטחים ב-

. 7 חילות בין נקודות ועקומים אלגבריים ב-

בפרק 7, אנו מוכיחים, יחד עם אדם שפר, חסם על מספר החילות בין נקודות ועקומים אלגבריים ב-> עקומים אלגבריים מדרגה > נקודות ובין . ובאופן מפורש, אנו מוכיחים כי מספר החילות בין <

> הוא > דרגות חופש ב- קבועה עם

<

-מימדית לא מכילה < > בהינתן שאף יריעה , כאשר קבוע הפרופורציונליות תלוי ב- < לכל > דרגות חופש > מקיימים אי-אלו קשרים חלשים ביניהם. המושג של > עקומי קלט, ושה- יותר מאשר

במימד כלשהוא מכליל את המושג המקביל במישור, שהוגדר ב-[83] ע"י פאך ושריר. הדרישה היא > נקודות כלשהן, ושכל זוג > עקומים מהמשפחה הנתונה שעוברים דרך שיהיו לכל היותר

> נקודות לכל היותר. נגדיר מושג זה במדויק בפרק 8 של התזה. עקומים ייחתכו ב-

. והוא גם מכליל תוצאות קודמות < החסם הזה מכליל את החסם של פאך ושריר [83] מהמישור ל-רבות (עם הגבלות מסוימות), כולל תוצאות מסוימות שמוצגות בתזה זו.

הכלליות של תוצאה זו טומנת בחובה גם מגבלות, ובמקרים רבים ידוע כי החסם הכללי שקיבלנו במקרה זה אינו הדוק, והתנאים הנדרשים הם מגבילים מאוד במקרים מסוימים. דיון מלא בנושאים אלה מוצג

בפרקים 7 ו-8.

חילות בין נקודות ועקומים אלגבריים בשלושה מימדים.

כעת נעבור לסקור את התוצאות בפרק 8, הפרק האחרון בתזה. בפרק זה נדרשת היכרות עם מספר מושגים מתקדמים יותר בגיאומטריה אלגברית, בהם אנו דנים בהרחבה בהקדמה באנגלית, ובפרק 8

> לכל > מדרגה > נשלט אינסופית ע"י משפחת עקומים עצמו (אך לא כאן). נעיר כאן שמשטח אלגברי . כפי שכבר < > שמוכלים בשלמותם ב- > הינה סמוכה לאינסוף עקומים של היותר, אם כל נקודה הוסבר, הקשר בין משטחים נשלטים לתחום החילות בגיאומטריה קומבינטורית התגלה בעבודה החלוצית

ℝ3

ℝd

ℝdmnkℝd

O mk

dk − d + 1 +εndk − d

dk − d + 1 +d−1

∑j=2

mk

jk − j + 1 +εnd( j − 1)(k − 1)

(d − 1)( jk − j + 1) qj

(d − j)(k − 1)(d − 1)( jk − j + 1) + m + n

ε > 0, k , ε, djqjqjk

μ = O(1)kμ

ℝd

VΓEp ∈ VΓV

5

של גוט וכץ [56], ושב והופיע בעבודות רבות מאז. סקירה מפורטת של נושא זה מופיעה בסעיף 1.3, כמו כן גם במאמר הסקירה של גוט [53] ובספרו החדש [54], ובעבודתו של קולאר [71]. עוד נעיר שמשפחה נוצרת (constructible) מסיבוכיות קבועה היא משפחה שמקיימת תנאי אלגברי שקל

לתארו באמצעות מספר קבוע של שוויונות ואי-שוויונות אלגבריים.

.( < משפט 4 (עקומים ב-

, < > עקומים אלגבריים אי-פריקים מדרגה מקסימלית > קבוצה של > נקודות ותהי > קבוצה של תהי < > דרגות חופש (ו- > מסיבוכיות קבועה של עקומים אלגבריים, בעלת הלקוחים מתוך משפחה נוצרת

< > לא מכיל יותר מאשר , כך שאף משטח הנשלט ע"י אינסוף עקומים של < קבוע כלשהוא) ב-. אז מתקיים < , עבור פרמטר < עקומים של

כאשר קבוע הפרופורציונליות תלוי ב- , ובסיבוכיות של .

הערות.

במקרים מסוימים, כמו למשל במקרים של ישרים ומעגלים, משטחים הנשלטים ע"י אינסוף 1.עקומים של הם בעלי תיאור פשוט (משטחים הנשלטים ע"י אינסוף ישרים הינם מישורים,

ומשטחים הנשלטים ע"י אינסוף מעגלים הינם מישורים או ספרות). במקרים אלו, משפט 4 מתחזק, היות וההנחה היא בעצם שלסוג מאוד קונקרטי של משטחים אסור להכיל יותר מאשר

מספר מקסימלי של עקומים.

משפט 4 מכליל את התוצאה של גוט וכץ [56], שהתקבלה במקרה של ישרים. 2.

שיפור החסם.

. לא < ניתן לשפר את החסם במשפט 4, אם אנחנו גם מוסיפים הנחה על המימד של משפחת העקומים > שאיבריה זו אף זו, כפי שיעלה מן ההוכחה, המימד שנשתמש בו הוא רק זה של תת משפחה של > היא בעלת . אנו נאמר שהמשפחה < מוכלים בשלמותם במשטח נשלט אינסופית ע"י עקומים של

> של , תת-המשפחה של < > הנשלט אינסופית ע"י עקומים של > אם, לכל משטח מימד מצומצם > כנקודות > המוכלים ב- -מימדית (כלומר ניתן לייצג את עקומי < >הינה עקומים המוכלים בשלמותם ב-

-מימדי). במקרה זה אנו מקבלים את המשפט הבא. < במרחב פרמטרי

.( < משפט 5 (עקומים ב-

, < > עקומים אלגבריים אי-פריקים מדרגה מקסימלית > קבוצה של > נקודות ותהי > קבוצה של תהי < > דרגות חופש (ו- > מסיבוכיות קבועה, של עקומים אלגבריים, בעלת הלקוחים מתוך משפחה נוצרת

< > לא מכיל יותר מאשר , כך שאף משטח הנשלט ע"י אינסוף עקומים של < קבוע כלשהוא) ב-. אז מתקיים < > הינה בעלת מימד מצומצם , ונניח בנוסף ש- < , עבור פרמטר < עקומים של

ℝ3

Pm∁nE∁0kμ

ℝ3∁0q∁q < n

I(P, ∁) = O(mk

3k − 2 n3k − 33k − 2 + m

k2k − 1 n

k − 12k − 1 q

k − 12k − 1 + m + n)

k, μ, E∁0

∁0

∁0

∁0

∁0∁0

sV∁0∁0

Vs∁0Vs

ℝ3

Pm∁nE∁0kμ

ℝ3∁0q∁q < n∁0s

I(P, ∁) = O(mk

3k − 2 n3k − 33k − 2 ) + Oε(m2/3n1/3q1/3 + m

2s5s − 4 n

3s − 45s − 4 q

2s − 25s − 4 +ε + m + n)

6

, ובסיבוכיות המקסימלית של תת-משפחה כלשהיא של < כאשר קבועי הפרופורציונליות תלוי ב- , < > שמורכבת מעקומים המוכלים בשלמותם במשטח כלשהוא הנשלט אינסופית ע"י עקומים של

. < והקבוע השני תלוי גם ב-

הערות.

משפט 5 הינו שיפור של משפט 4 כאשר וגם , במקרים בהם מספיק 1.גדול כך שהביטוי השני בחסם של משפט 4 שולט על הביטוי הראשון באותו חסם; לערכים

קטנים יותר של הביטוי כולו לינארי (זה נכון פרט לביטוי ). פרטים נוספים על ההדדיות בין משפטים 4 ו-5 ניתן למצוא בפרק 8.

החסמים במשפטים 4 ו-5 משפרים, בשלושה מימדים, את התוצאה מפרק 7, בשלושה אופנים 2.משמעותיים: (i) הגורמים המובילים בכל הביטויים הינם פחות או יותר זהים, אך החסמים

במשפטים 4 ו-5 לא כוללים את הגורם שמופיע בפרק 7. (ii) ההנחה כאן, לגבי

מספר העקומים המוכלים במשטח מדרגה נמוכה, חלשות משמעותית לעומת ההנחה בפרק 7, , לא 2 אשר דורשת שאף משטח מדרגה (קבועה אך פוטנציאלית מאוד גבוהה) , התלויה ב-

יכיל יותר מאשר עקומים של . (iii) הגורמים הלא מובילים במשפטים 4 ו-5 קטנים יותר משמעותית מאשר אלו בפרק 7, ובמובן מסוים (שיובהר בהרחבה בפרק 8), הם הטובים ביותר

האפשריים.

חילות בין נקודות ומשטחים אלגבריים בשלושה מימדים.

כעת נסקור את אוסף התוצאות האחרון בתזה זו, המערבות חילות בין נקודות ומשטחים אלגבריים מדרגה קבועה בשלושה מימדים. הסקירה תהיה חלקית, והתוצאות מוצגות בפרק 8 בתזה. כאמור, גם התוצאות

האלה התפרסמו ב-[106].

נשים לב, שבמקרה של חילות בין נקודות ומשטחים, גרף החילות בין הנקודות והמשטחים יכול להכיל גרפים דו-צדדיים שלמים, כך שכל אחד מהם מערב נקודות רבות שמוכלות בעקום אלגברי, ומשטחים

רבים המכילים את אותו עקום.במקרים כאלה מספר החילות עשוי להיות רב, ואף להגיע למקסימום . שלא כמו מחקרים קודמים בתחום, אנחנו לא שוללים אפשרות כזו, מה שהופך את < האפשרי

הגישה שלנו לכללית יותר.

החסמים שלנו מעריכים את הגודל הכולל של קבוצת הקודקודים בפירוק של גרף החילות לתתי-גרפים דו-צדדיים שלמים. במקרים מסוימים, החסמים שלנו מיתרגמים לחסמים על מספר החילות ממש. בסך

הכול, גם במקרה הזה התוצאות שלנו מספקות הכללה גורפת של תוצאות קודמות בנושא.

פירוק גרף החילות, עבור נקודות על יריעה ומשטחים.

> מדרגה קבועה. התוצאה המרכזית הראשונה שלנו, היא כאשר הנקודות נמצאות על יריעה אלגברית מלבד העובדה שלתוצאה זו יש עניין עצמאי, היא גם משמשת לניתוח המקרה הכללי בו הנקודות לא

חייבות להיות על יריעה אלגברית.

k , μ, s, E∁0∁0

ε

k ≤ sm > n1/kq

mqε

O(nε)

cεε

q∁

mn

V

7

> של משטחים אלגבריים, > נלקחת מתוך משפחה אינסופית במשפטים הבאים נניח שקבוצת המשטחים , < > ביחס ל- > כלשהוא), או שהיא ממימד מצומצם (עם < > דרגות חופש ביחס ל- שהיא או בעלת

- < > היא משפחה , כשהכוונה היא שהמשפחה < עבור פרמטר קבוע כלשהוא

מימדית של עקומים.

משפט 6.

< > קבוצה של , ותהי < > ב- > מדרגה קבועה > נקודות על יריעה אלגברית > קבוצה של תהי > של משטחים אלגבריים, שהיא , הלקוחים מתוך משפחה < משטחים אלגבריים מדרגה מקסימלית , עבור < > ביחס ל- > קבוע כלשהוא), או שהיא ממימד מצומצם > (ו- > דרגות חופש ביחס ל- בעלת

> לא מכילים רכיבים . אנחנו מניחים בנוסף שהמשטחים של < ), או < , (ו- < פרמטרים קבועים כלשהם < אי-פריקים משותפים (וזה ודאי קורה אם כולם אי-פריקים). אז ניתן לפרק את גרף החילות

כ-

, עבור < > שהם רכיבים אי-פריקים של עקומים מהצורה כאשר האיחוד הוא על-פני כל העקומים < > מייצג את כל המשטחים ב- , ו- < > המוכלות ב- > מייצג את כל הנקודות ב- , וכאשר <

. < המכילים את

> אזי > דרגות חופש ביחס ל- > היא בעלת אם

, < , אז, לכל < > ביחס ל- > בעלת מימד מצומצם ואילו אם

< > במקרה הראשון, או ב- , וב- < , ובסיבוכיות של < כאשר קבועי הפרופורציונליות תלויים ב-> במקרה השני. וב-

< > וב- , כאשר קבוע הפרופורציונליות תלוי ב- < יתירה מכך, בשני המקרים,

הערה.

תכונה חשובה של תוצאה זו היא שאיננו מניחים שום הנחות על גרף החילות, כמו למשל הדרישה שלא יכיל עותק של הגרף הדו-צדדי השלם , עבור קבוע , כפי שהניחו במחקרים קודמים

[17,66,125]. התוצאה שלנו כללית יותר, ופירוט נרחב על כך ניתן למצוא בפרק 8 בתזה.

SℱkVμsV

sΓ ≔ σ ∩ V |σ ∈ ℱs

PmVDℝ3SnEℱ

kVμsVkμsS

G (P, S )

G(P, S ) = ⋃γ

Pγ × Sγ

γσ ∩ Vσ ∈ SPγPγSγS

γ

ℱkV

∑γ

Pγ = O(mk

2k − 1 n2k − 22k − 1 + m + n)

ℱsVε > 0

∑γ

Pγ = O(m2s

5s − 4 n5s − 65s − 4 +ℇ + m2/3n2/3 + m + n)

D, Eℱ, k , μsε

∑γ

Sγ = O(n)D. E

Kr,rr

8

חסם חילות משולב (עבור נקודות על יריעה "כללית" ומשטחים כלליים).

> מדרגה קבועה, התוצאה הבאה שלנו היא שיפור של משפט 6, כאשר הנקודות נמצאות על יריעה . במקרה זה, אנו מקבלים חסם < שאינה נשלטת ע"י אינסוף עקומי חיתוך של משטחים מתוך

> ו- > לשני תתי-גרפים "משולב" משופר, שבו ניתן לפצל את גרף החילות , והחסמים על תת-הגרף הראשון הם חסמים על מספר החילות ממש. הוכחנו את המשפט <

הבא.

משפט 7.

< > קבוצה של , ותהי < > ב- > מדרגה קבועה > נקודות על יריעה אלגברית > קבוצה של תהי > של משטחים אלגבריים, שהיא , הלקוחים מתוך משפחה < משטחים אלגבריים מדרגה מקסימלית , עבור < > ביחס ל- > קבוע כלשהוא), או שהיא ממימד מצומצם > (ו- > דרגות חופש ביחס ל- בעלת

> לא מכילים רכיבים . אנחנו מניחים בנוסף שהמשטחים של < ), או < , (ו- < פרמטרים קבועים כלשהם > אינה נשלטת ע"י (הרכיבים אי-פריקים משותפים (וזה ודאי קורה אם כולם אי-פריקים), וש-

> כ- . אז ניתן לפרק את גרף החילות < האי-פריקים של) עקומי החיתוך של משטחים מתוך

, < > שהם רכיבים אי-פריקים של עקומים מהצורה כאשר האיחוד הוא על-פני כל העקומים > מייצג את כל , ו- < > המוכלות ב- > מייצג חלק מהנקודות ב- , וכאשר < עבור

. < > המכילים את המשטחים ב-

>אזי > דרגות חופש ביחס ל- > היא בעלת אם

,< , אז, לכל < > ביחס ל- > בעלת מימד מצומצם ואם

,

> במקרה הראשון, או ב- >, וב- >, ובסיבוכיות של כאשר קבועי הפרופורציונליות תלויים ב-> במקרה השני. >וב-

, כאשר קבוע < > ו- יתירה מכך, בשני המקרים מתקיים

> במקרה >וב- > במקרה הראשון, או ב- >, וב- > ובסיבוכיות של הפרופורציונליות תלוי ב-השני.

. < > את משפחת הספרות ב- ניתן להשתמש במשפט 7 כאשר לוקחים בתור המשפחה

V𝓕

G (P, S )G0(P, S )G1(P, S )

PmVDℝ3SnEℱ

kVμsVkμsS

VℱG (P, S )

G(P, S ) = G0(P, S ) ∪ ⋃γ

Pγ × Sγ

γσ ∩ σ ′ ∩ Vσ ≠ σ ′∈ SPγPγSγ

Sγ

ℱk V

G0(P, S ) = O(mk

2k − 1 n2k − 22k − 1 + m + n)

ℱsVε > 0

G0(P, S ) = O(m2s

5s − 4 n5s − 65s − 4 +ℇ + m2/3n2/3 + m + n)

D, Eℱ , k , μ sε

∑γ

Pγ = O(m)∑γ

Sγ = O(n)

, E, Dℱ , k , μ sε

ℱℝ3

9

> היא משפט זה מופיע בתזה (בפרק 8). נכלול כאן שתי מסקנות מעניינות שנובעות מהמקרה בו . < משפחת הספרות ב-

משפט 8 (מרחקים שונים).

תהי קבוצה של נקודות על יריעה אלגברית מדרגה קבועה ב- , אז, לכל א.

, מספר המרחקים השונים הנקבעים ע"י הנקודות ב- הינו , כאשר <

. < קבוע הפרופורציונליות תלוי ב- וב-

> ב. > קבוצה של > (כמו בסעיף א'), ותהי > נקודות על יריעה אלגברית > קבוצה של תהי , מספר המרחקים השונים הנקבעים ע"י זוגות נקודות ב- < . אז, לכל < נקודות כלשהן ב-

, כאשר קבוע הפרופורציונליות < > הינו

. < > וב- תלוי ב-

משפט 9 (מרחקים נשנים).

תהי קבוצה של נקודות על יריעה אלגברית מדרגה קבועה ב- , אז, מספר א.

מרחקי היחידה הנקבעים ע"י הנקודות ב- הינו , כאשר קבוע הפרופורציונליות

תלוי ב- .

> ב. > קבוצה של > (כמו בסעיף א'), ותהי > נקודות על יריעה אלגברית > קבוצה של תהי , מספר מרחקי היחידה הנקבעים ע"י זוגות נקודות ב- < . אז, לכל < נקודות כלשהן ב-, כאשר קבוע הפרופורציונליות < > הינו

. < > וב- תלוי ב-

> וקבוצת נקודות התוצאה האחרונה שאנו מציגים בתזה מתייחסת למקרה הכללי המערב קבוצת משטחים . הטיפול במקרה כללי זה הינו באמצעות טיעון אינדוקטיבי המתבסס על שיטת החלוקה < כלשהיא ב-

הפולינומיאלית של גוט וכץ [56], שבו משפט 6 משחק תפקיד מרכזי. תוצאה זו מרחיבה תוצאה בעבודה מקדימה שלנו [105, משפט 1.4], מספירות למשטחים כלליים, ומרחיבה תוצאה חדשה של

זאל, עבור משטחים כלליים, כאשר בניסוח המשפט שלנו, אין כלל אילוצים מוקדמים על גרף החילות . הוכחנו את המשפט הבא. <

משפט 10.

, < > משטחים אלגבריים מדרגה מקסימלית > קבוצה של , ותהי < > נקודות ב- > קבוצה של תהי < > של משטחים אלגבריים. אז ניתן לפרק את גרף החילות -מימדית < הלקוחים מתוך משפחה

כ-

ℱℝ3

PmVDℝ3

ε > 0PΩ(n7/9−ε)Dε

P1mVP2nℝ3ε > 0

P1 × P2Ω(minm4/7−εn1/7−ε, m1/2n1/2, m)Dε

PmVDℝ3

PO(n4/3)D

P1mVP2nℝ3ε > 0

P1 × P2O(m6/11n9/11+ε + m2/3n2/3 + m + n)Dε

Sℝ3

G (P, S )

Pmℝ3SnEsℱG (P, S )

G(P, S ) = G0(P, S ) ∪ ⋃γ

Pγ × Sγ

10

< , כאשר < > המוכלים בחיתוכים של לפחות שני משטחים ב- כאשר האיחוד הוא על-פני כל העקומים

< > מייצג את כל (אך לפחות שני) המשטחים ב- , ו- < > המוכלות ב- מייצג את כל הנקודות ב-

. < המכילים את

>, מתקיים יתירה מכך, לכל

<

< ו-

> ובסיבוכיות של . כאשר קבועי הפרופורציונליות תלויים ב-

γSPγ

PγSγSγ

ε > 0

J(P, S ) ≔ ∑γ

( Pγ + Sγ ) = O(m2s

3s − 1 n3s − 33s − 1 +ε + m + n)

G0(P, S ) = O(m + n)

, ε, s, D, Eℱ

11

מציתת

חילות בין נקודות ינטורית, העוסקות במספר בעיות בגיאומטריה קומב חוקריםבתזה זו אנחנו

רים בשלושה וארבעה מימדים, עבור לעקומים ביש ייקטים גיאומטריים אחרים, החלואוב

למשטחים אלגבריים בשלושה מימדים. אנחנו מפתחים ומשתמשים במגוון אלגבריים, ועד

גיאומטריה אלגברית כדי להתמודד עם בעיות אלו, כלים בכלים "תשתיתיים" באלגברה ו

יות אחרות. בנוסף, אנו גם מיישמיםעיות קומבינטורשעשויים למצוא שימושים גם במגוון ב

ה קומבינטורית.חסמים אלו בפתרון בעיות נוספות בגיאומטרי

בחלק הראשון של התזה, אנו מתייחסים לבעיה של קבלת חסמים הדוקים אסימפטוטית על

מספר החילות בין נקודות וישרים במימדים גבוהים, ובכך מכלילים את החסם של סמרדי

אך מרעישה לא עבור המקרה המישורי, ואת התוצאה החדשה יותר 1983וטרוטר משנת

מימדי. העבודה האחרונה הציגה כלים -( עבור המקרה התלת2010שנת פחות של גוט וכץ )מ

מתחום האלגברה המתקדמת ובמיוחד מתחום הגיאומטריה האלגברית, אשר לא נעשה בהם

מלאבאופן )כמעט( תאפשר לגוט וכץ לפתור שימוש בתחום הקומבינטוריקה בעבר. בכך ה

ן על מספר המרחקים השונים את בעיית המרחקים השונים של ארדש, ולהשיג גבול תחתו

נקודות במישור, בעיה פתוחה שעמדה עיקשת nהנקבעים על ידי קבוצה כלשהיא בת

ה. העבודה של גוט וכץ הוסיפה יסיונות אמיצים רבים לפותרשנה, למרות נ 60בסירובה מעל

בעיות רבות, וביניהן תחום גיאומטריית החילות, ואפשרה פתרון של מומנטום משמעותי ל

תירות לפני פריצת הדרך שלהם, באמצעות הבעיות שחקרנו בתזה, אשר נדמו כבלתי פ

החדשות שפותחו.האלגבריות הכלים והטכניקות

לנקודות אנחנו מרחיבים את מחקר החילות בין נקודות וישרים לארבעה מימדים, ולאחר מכן

למנטרית לחסם של מימדיות. בנוסף גם מצאנו הוכחה א-ותלת -ביריעות דווישרים שמוכלים

אנו גם מציגים חסמים תחתונים כץ עבור חילות בין נקודות וישרים בשלושה מימדים.-גוט

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