Frobenius-stable lattices in rigid cohomology of curves€¦ · case the curve is a ne, the second cohomology group vanishes. To see this, use, e.g. the comparison theorems with crystalline

FROBENIUS-STABLE LATTICES INRIGID COHOMOLOGY OF CURVES

vorgelegt vonDiplom-Mathematiker und Diplom-Informatiker

MORITZ MINZLAFFKarlsruhe

Von der Fakultät II – Mathematik und Naturwissenschaftender Technischen Universität Berlin

zur Erlangung des akademischen GradesDoktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. John Sullivan, Ph. D.Berichter: Prof. Dr. rer. nat. Florian HeßBerichter: Prof. Dr. rer. nat. Remke Kloosterman

Tag der wissenschaftlichen Aussprache: 8. März 2013

Berlin 2013

D83

MAILTO:[email protected]

Document History

Version Remarks Date

1.1 Accepted version; minor revisions 2013-04-06

1.0 Submitted version; Section 2.2 and Section 2.3 finished,Proposition 3.13 fixed, minor revisions

2012-12-18

0.9 Complete and proofread draft; except for Section 2.2and Section 2.3

2012-05-31

Please send comments and corrections to me at [email protected].

mailto:[email protected]

Ich bedanke mich bei Florian Heß für den spannenden Themenvorschlag und beiRemke Kloosterman für die

”vor Ort“-Betreuung seit Florian Hess nicht mehr in

Berlin war. Desweiteren bedanke ich mich bei der Berlin Mathematical Schoolsowie meinen Eltern Angelika und Volker Minzlaff für die finanzielle Unterstützungwährend der Anfertigung dieser Doktorarbeit. Anja Bewersdorff, Tanja Fagel, Do-minique Schneider, Mariusz Szmerlo und Nadja Wisniewski vom BMS One-StopOffice möchte ich dafür danken, dass jegliche administrativen Gänge so angenehmwaren, wie man es sich nur wünschen kann. Viel Dank gilt auch John Sullivan fürdie kurzfristige Übernahme des Vorsitzes im Promotionsausschuss.

Bei Claus Diem, Felix Fontein, Frank Herrlich und Gabriela Weitze-Schmithüsen,Kiran Kedlaya, Thorsten Lagemann, Alan Lauder und Jan Tuitman, George Walkerund Stefan Wewers bedanke ich mich für Einladungen zu Seminarvorträgen und dieZeit verschiedene Aspekte meiner Arbeit mit mir zu diskutieren. Ganz besondersdanke ich Nils Bruin dafür, daß er mir einen mehrmonatigen Besuch bei ihm an derSimon Fraser University in British Columbia ermöglicht hat.

Für das Korrekturlesen und für Verbesserungsvorschläge bedanke ich mich bei FabianJanuszewski, Stefan Keil, Thorsten Lagemann, Fabian Müller, Christopher Özbek undOsmanbey Uzunkol. Christina Neuhaus schließlich gebührt ein ganz großer Dank fürdie moralische Unterstützung auf der Zielgeraden dieser Arbeit.

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CONTENTS

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. State of the art. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2. This work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Notations and conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Lifting birational equivalence classes of curves. . . . . . . . . . . . . . . . . . . . . . . . . 72.1. Teissier’s criterion for lifting classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2. The equimultiple local Hilbert functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3. Computing equimultiple proper liftings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. An explicit description of Frobenius-stable lattices. . . . . . . . . . . . . . . . . . . . 313.1. De Rham cohomology of smooth proper pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2. Integral Monsky-Washnitzer cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4. Computing the absolute Frobenius action in larger characteristic . . . 514.1. A basis for the cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2. Approximating the absolute Frobenius on differentials. . . . . . . . . . . . . . . . . . . 554.3. Reducing differentials modulo exact differentials. . . . . . . . . . . . . . . . . . . . . . . . . 574.4. Correctness and complexity analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. Implementations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1. Computing equimultiple proper liftings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2. Computing the absolute Frobenius action in larger characteristic. . . . . . . . . 70

A. Curves over complete discrete valuation rings. . . . . . . . . . . . . . . . . . . . . . . . . 73

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

CHAPTER 1

INTRODUCTION

It is probably fair to say that a major part of mathematics is ultimately motivated

by the desire to solve equations or to study their sets of solutions. The present thesis

is no exception. Its guiding problem is that of efficiently counting the number of

solutions to bivariate polynomial equations

f(x, y) = 0

over finite fields. Formulated from the viewpoint of algebraic geometry, the problem

is that of efficiently counting the number of rational points on plane curves. More

generally, let X be any curve over a finite field. If N1, N2, N3, . . . are the numbers of

rational points of X over the extensions of the finite field of degrees 1, 2, 3, . . ., then

the zeta function of X is defined as

ZX(t) = exp

( ∞∑r=1

Nrrtr

).

It is uniquely determined by N1, . . . , Nm whenever m is large enough. For example,

if X is smooth, proper, geometrically integral, then one may take m equal to the

genus [Sti09, Cor. V.1.17]. Conversely, the zeta function is a rational function in t

and if αi are the reciprocal roots of the numerator and βj the reciprocal roots of the

denominator, then one recovers the number of points as Nr =∑j β

rj −∑i α

ri . A more

detailed account of these statements is given by Wan [Wan08]. The aim of this thesis

is to improve the state of the art of computing zeta functions.

The zeta function has a cohomological interpretation: Let X be a smooth, geomet-

rically integral curve over the finite field k. Its rigid cohomlogy H•rig(X) consists of

finite dimensional vector spaces over the fraction field K of the ring of Witt vec-

tors W (k) [Ber97, Thm. 3.1]. The Frobenius morphism F on X induces linear maps

2 CHAPTER 1. INTRODUCTION

on those spaces and

ZX(t) =det(id− Ft | H1rig(X)

)det(

id− Ft | H0rig(X))

det(

id− Ft | H2rig(X))

whenever X is proper. If X is affine, then the relation is

ZX(t) =det(id−#k · F−1t | H1rig(X)

)det(

id−#k · F−1t | H0rig(X))

det(

id−#k · F−1t | H2rig(X)) .

The only interesting map in the above determinants is the one on the first cohomology

group. Indeed, in both cases the zeroeth cohomology group is 1-dimensional and the

Frobenius acts as the identity. If the curve is proper, then the second cohomology

group is also 1-dimensional and the Frobenius action is multiplication by #k. In

case the curve is affine, the second cohomology group vanishes. To see this, use, e.g.

the comparison theorems with crystalline cohomology and with Monsky-Washnitzer

cohomology [Ber97, Prop. 1.9 & 1.10] and the properties of the respective cohomolo-

gies [Ber74, MW68, vdP86].

1.1. State of the art

The task of computing the Frobenius action on H1rig(X) has received a lot of attention

over the past few years. First came Kedlaya’s algorithm for hyperelliptic curves in

odd characteristic [Ked01] which has been generalised several times to superelliptic

curves [GG01], Ca,b-curves [DV06a], and nondegenerate curves [CDV06]. Further-

more there are algorithms for hyperelliptic curves in even characteristic [DV06b] and

smooth plane curves (as special case of smooth hypersurfaces) [AKR11].

More recent are algorithms that add a deformation step: The given curve is first

embedded in a family of curves with a particular “nice” fibre. The Frobenius action

on the cohomology of this fibre is then computed and finally transformed to give the

action on the cohomology of the original curve. By now this has been applied to most

classes of curves mentioned above [Ger07, Hub07, Hub08, CHV08, Tui11].

Related to the approach with a deformation step is the approach that uses a fibration

on X: Computing the Frobenius action for a given variety is reduced to the same prob-

lem for a smooth hyperplane section. This method was introduced by Lauder [Lau06]

and improved by Walker [Wal09]. In the case of curves, Walker focuses on smooth

plane curves and on Ca,b-curves.

Typically the runtimes of these algorithms are polynomial in the genus g of the curve

and the degree n of k, but linear or worse in the characteristic p. In particular, it

1.1. STATE OF THE ART 3

is exponential in log(p). For example, the time complexities of the algorithms for

hyperelliptic, superelliptic, and Ca,b-curves (without deformation or fibration) lie in

Õ(p1n3g5).

To contrast, if X is proper, then O(log(p)ng2) bits suffice to determine X [Hes,

Thm. 56] and to write down the determinant of 1−Ft on H1rig(X) [Sti09, Thm. V.1.15].Using deformation can yield a lower exponent for n as demonstrated in case of hyper-

elliptic curves. Regarding p, Harvey describes an algorithm for hyperelliptic curves

in odd characteristic which improves the dependence on the characteristic to Õ(p0.5)

at the cost of higher exponents for the other parameters [Har07].

Fixing the characteristic, the most general class of curves for which a practical al-

gorithm exists arguably is the class of nondegenerate curves. If the cardinality of k

is large enough, then this class includes all Ca,b-curves and thus all superelliptic and

hyperelliptic curves. Nonetheless, nondegenerancy is quite special: The moduli space

of nondegenerate genus g curves has dimension 2g + 1 whereas the moduli space of

all genus g curves has dimension 3g − 3 [CV09].(1)

Further algorithms. — If the only goal is to compute the zeta function and not

the Frobenius action on rigid cohomology, there are further methods available. Most

notably are those using “Dwork cohomology”, étale (or `-adic) cohomlogy, or the

canonical lift of the Jacobian. Currently the most general class of curves these meth-

ods can practically handle are ordinary hyperelliptic curves in even and superelliptic

curves in any characteristic.

Lauder and Wan proposed an algorithm based on Dwork cohomology for arbitrary

affine (not necessarily smooth) varieties. It runs in polynomial time when the char-

acteristic is kept fixed [LW08], but is deemed not very practical [LW04, p. 332].

Related algorithms for superelliptic curves [Lau03], Artin-Schreier curves [LW04],

and smooth hypersurfaces with a deformation step [Lau04] have time complexity

comparable to those using rigid cohomology. In particular, they share the bad be-

haviour with respect to log(p).

Schoof described an algorithm for elliptic curves using étale cohomology [Sch85].

Apart from several optimisations, there is an adaption to genus 2 [GH00] and even

a generalisation to arbitrary genus [Pil90], but the latter runs doubly-exponential in

the genus and is considered impractical [The04, Sec. F.3.b]

(1)The only exceptions are genus 7 where the dimension is 16 and genera less than 4 where 3g− 3 isless than 2g + 1 and all curves are nondegenerate.


Canonical lifts of Jacobians are used in Satoh’s and Mestre’s algorithms for ordinary

elliptic curves in odd respectively even characteristic [Sat00, Mes00]. The most gen-

eral variants of these algorithms are for ordinary hyperelliptic curves [CL07, LL06].

More in-depth accounts of the various approaches to compute zeta functions can be

found in a book by Cohen et al. [CFA05] and a survey by Chambert-Loir [CL08].

1.2. This work

The previous section indicates that when one wants to compute the Frobenius action

on rigid cohomology with current methods, one has to restrict to special classes of

curves such as the class of nondegenerate curves and to “small” characteristic. This

thesis makes three main contributions to weaken these restrictions.

Lifting birational equivalence classes of curves. — A central ingredient to com-

puting zeta functions with rigid cohomology is a comparison theorem with de Rham

cohomology: Let X be a smooth, proper, geometrically integral curve and X be a

proper lifting. If Z is a reduced strict relative normal crossing divisor on X, then

there is a natural isomorphism

H•dR(XK \ ZK)∼−→ H•rig(X \ Z) (1.1)

between the rigid cohomology of the affine curve X \ Z over k and the de Rhamcohomology of the affine curve XK \ ZK over K (see (3.2)). For the types of curvesmentioned in the previous section a proper lifting is easily found. Indeed, each of

those curves is a smooth hypersurface in some ambient surface which is obviously the

reduction of some proper, integral scheme over the ring of Witt vectors: For smooth

plane curves one may take the projective plane as the ambient scheme, for hyper-

elliptic and Ca,b-curves a weighted projective plane, and for nondegenerate curves a

toric surface. One may identify a hypersurface with an element of the homogeneous

coordinate ring. Under this identification, lifting the hypersurface amounts to picking

a lift of the element in the homogeneous coordinate ring of the lifted surface. The

first contribution of this thesis is a treatment of the case when such a nice embedding

is not known. It suffices to lift the birational equivalence class of the curve and we

present an algorithm that approximates a lifting of the class of any curve (Chapter 2).

An explicit description of Frobenius-stable lattices. — Much like lifting a

root modulo p of a polynomial over the integers in general only yields an element

of the p-adic integers, it seems that one cannot expect a proper lifting of X to be

defined over the integers or a finite extension of them. Rather, one has to consider

the ring of Witt vectors W (k). A Frobenius action on lattices over W (k) is therefore

of interest for two reasons: When one is only able to write down an approximation of

1.3. NOTATIONS AND CONVENTIONS 5

a lifting over Witt vectors WN (k) of finite length, one cannot write down the generic

fibre of the exact lifting as needed in (1.1). In this case, a Frobenius-stable lattice

over W (k) naturally yields a WN (k)-module on which to approximate the Frobenius

action. Another reason is that even in the case when one does obtain an exact lifting,

if the chosen basis of H1rig(X \ Z) does not span a Frobenius-stable lattice, then thematrix representing the Frobenius will not have coefficients in W (k). Since we can

only compute with Witt vectors of finite length, this causes a loss of precision. The

second contribution of this thesis is therefore an explicit description of a Frobenius-

stable lattice in the vector space H1rig(X \Z) for any choice of X and Z (Chapter 3).

Computing the Frobenius action in larger characteristic. — The third con-

tribution of this thesis is an extension of Harvey’s algorithm from hyperelliptic curves

in odd characteristic to more general “Kummer coverings” of the rational line. It

computes the Frobenius action in Õ(p0.5)-time (Chapter 4). Like Harvey’s algorithm,

the runtime is polynomial in the degree of k and the genus.

Finally, the effectiveness of all algorithms in this thesis is demonstrated using imple-

mentations in Sage [S+12] and Magma [BCP97] (Chapter 5).

Original contributions of this thesis are usually enclosed in numbered environments

such as propositions, lemmas, and others. Conversely, to our best knowledge such

numbered environment contain original contribution. Exceptions to this are marked

as such. The main theorems are numbered with capital letters.

1.3. Notations and conventions

We denote by N the monoid of finite cardinals 0, 1, 2, . . . and by Z the ring of integers.

Throughout this thesis, k is a finite field of characteristic p. The ring of Witt vectors

over k is denoted by W (k), its fraction field by K, its valuation by v, and the ring

of Witt vectors of length N by WN (k) = W (k)/pNW (k). Liu’s book “Algebraic

Geometry and Arithmetic Curves” [Liu06] will be our default reference for algebraic

geometry.

Objects over k. — Objects over k usually come with an overline · in their notation,e.g. we write X for a scheme over k. We also sometimes use this notation to denote

the base change of an object over W (k) to k, e.g. given a W (k)-scheme X, we might

write X for the special fibre Xk.

Frobenius morphisms. — Let X be a scheme over k of finite type. The morphism

Fp : X → X given by the identity on the topological space of X and p-th powering onOX is called the absolute Frobenius morphism on X. Unless k is the prime field Fp, Fp


is not a k-morphism. In general, let n be the degree of k. Then we define the (k-linear

or relative) Frobenius morphism F as the n-th iterate of the absolute Frobenius, i.e.

F = Fnp . This is the Frobenius that was already mentioned earlier in this introduction.

Curves. — Let S be a locally Noetherian scheme. A curve over S is a flat, separated

scheme of finite type over S whose fibres are geometrically reduced and equidimen-

sional of dimension 1. Curves are proper unless otherwise mentioned. A curve over S

is plane if it is a hypersurface in P2S . In general, we say that a curve X has a certain

property whenever X has that property as a scheme.

The arithmetic genus of a curve X over a field is pa(X)df= 1−χ(OX). If the normali-

sation X̃ of X is smooth, then the genus g(X) of X is the geometric genus of X̃, i.e.

g(X)df= h0(ΩX̃).

Deformations. — Our main references for deformation theory and its terminology

are Hartshorne’s “Deformation Theory” [Har10] and Sernesi’s “Deformations of Al-

gebraic Schemes” [Ser06]. For example, the deformation of a scheme X over k to a

local Noetherian ring A with residue field k is a scheme X that is flat and of finite

type over A together with an isomorphism of its special fibre with X. We use the

notion of a deformation and a lifting interchangeably, in particular when we think of

lifting objects from characteristic p to characteristic 0.

Abbreviations. — Given a sheaf F on a scheme X, we will usually write Hi(F )

for Hi(X,F ). Similarly, if X is understood to be a scheme over another scheme S, we

will write ΩX for ΩX/S and HidR(X) instead of H

idR(X/S). Further abbreviations like

these may be found throughout the thesis and hopefully will not lead to confusion.

A bold letter usually denotes a tuple or sequence, e.g. xmight be a sequence x1, . . . , xrof variables or α might be a tuple (α1, . . . , αr) ∈ Nr of finite cardinals. In the lattercase, we also have |α| = α1 + . . .+ αr.

Complexity. — In addition to the usual Oh notation to measure the asymptotic be-

haviour of functions, e.g. n2 +n ∈ O(n2), we also use the soft Oh notation [vzGG03,Sec. 25.7] that neglects polylogarithmic factors, e.g. n(log n)2 ∈ Õ(n).

CHAPTER 2

LIFTING BIRATIONAL EQUIVALENCE CLASSES OF

CURVES

This chapter addresses the issue of computing a proper lifting to W (k) of a smooth

curve X over k up to birational equivalence. Our motivation to study this problem

arose in the context of describing the p-adic cohomology of open affine subschemes

of X (Chapter 3). Deformation theory tells us that such a proper lifting always

exists [Har10, Thm. 22.1] and our main result will be to turn this existence statement

into an effective algorithm.

Finding a lifting is easy when the curve is embedded in projective space as com-

plete intersection: Pick any homogeneous liftings of its defining equations [Har10,

Thm. 9.4]. But while every smooth curve over k can be embedded already in P3` for

some finite extension ` of k, in general it cannot be embedded as smooth complete

intersection in P3` . It is conjectured that in general an embedding in P3` does not

lift [EH99, Conj. C].

One idea to circumvent this problem is to embed curves as complete intersections

in more general ambient schemes. This is done by most algorithms that compute

zeta functions with p-adic cohomology. However, this approach also seems to have

limitations: For example, the step from hyperelliptic curves (hypersurfaces in weighted

projective planes) to nondegenerate curves (hypersurfaces in toric surfaces) only gives

a modest increase in the dimension of the moduli space. Beyond toric surfaces, there is

(to us) no clear candidate for ambient schemes that are suitable from a computational

point of view.

Therefore, we take a different approach: We restrict to the simplest ambient scheme,

the projective plane, but allow singularities in a controlled fashion. In particular, we

allow nodes so that X is always birational to such a curve [Har97, Thm IV.3.10]

over some finite extension of k. The idea is to lift the plane curve along with its

singularities.

8 CHAPTER 2. LIFTING BIRATIONAL EQUIVALENCE CLASSES OF CURVES

For this chapter, we use a more general definition for k and K: Let Λ

be a complete discrete valuation ring with maximal ideal m, fraction field

K, and perfect residue field k. Set ΛNdf= Λ/mNΛ and denote by S the

spectrum of Λ.

This chapter is organised as follows: In Section 2.1, we formally define liftings of

birational equivalence classes of curves. We will see that a proper lifting X of a

representative may fail to lift the class. The solution is an adaption of Teissier’s

δ-invariant criterion to our situation: A proper lifting X lifts the class if and only

if the δ-invariants of its fibres coincide (Proposition 2.5). For plane curves whose

singularities are all ordinary, e.g. nodes, the δ-invariant is completely determined by

the multiplicities of the singularities and the degrees of their underlying points. An

“equimultiple proper lifting” of a plane ordinary curve therefore solves the problem of

lifting its birational equivalence class (Proposition 2.11).

The definition of an“equimultiple”local Hilbert functor (Definition 2.15) in Section 2.2

frames the problem of computing such liftings in the context of formal deformation

theory. Finally in Section 2.3, we explain how the infinitesimal lifting property of

smooth schemes can be used to compute equimultiple proper liftings of plane curves

over k to ΛN (Algorithm 1). We use the formal deformation theory introduced in the

previous section to show the algorithm is successful whenever the plane curve has only

nodes and not too many ordinary singularities of higher multiplicity (Corollary 2.27).

Most results of this chapter are classical knowledge when one takes Λ as the ring of

formal power series over the complex numbers. We are not aware, however, of any

references for the general case.

Further notation and terminology. — Let Set be the category of sets. The

category of Artinian local Λ-algebras with residue field k is denoted by Ar. We write

mA for the the maximal ideal of a local ring A. The category of complete Noetherian

local Λ-algebras A such that A/miA is in Ar, all i, will be denoted by Âr. A surjection

A′ � A in Ar is a small extension (of A by I) if the kernel I is annihilated by mA′ .The kernel has thus the structure of a k-vector space. We write Ex(A, I) for the set

of small extensions of A by I. One can give Ex(A,−) the structure of a covariantfunctor. In case of a surjection I � J of finitely generated k-vector spaces with kernelC, this functor maps the extension A′ � A to A′/C � A.

2.1. TEISSIER’S CRITERION FOR LIFTING CLASSES 9

2.1. Teissier’s criterion for lifting classes

The (birational equivalence) class of a curve over a field contains a unique-up-to-

isomorphism normal curve.(1) Consequently, we make

Definition 2.1. — A lifting of a class [X] of a curve X over k to Λ is a class of

curves over Λ that contains a proper lifting of the normal curve of [X].

While a proper lifting Y of a normal curve over k is a normal curve over Λ, there may

be several nonisomorphic normal curves in its class. Nonetheless, the generic fibres of

all normal curves in [Y ] are isomorphic and thus define the same class [YK ] over K.

The existence of a unique representative in [Y ] is not relevant to our goals; we refer

the interested reader to Liu’s book [Liu06, Chp. 9 & 10].

Example 2.2. — Let X be a curve over k. By definition, a proper lifting of the nor-

mal curve in [X] also lifts the class. However, given any representative of [X], a proper

lifting does not need to lift the class. Take, for example, the geometrically integral

genus 1 curve X defined by y2 = x3(x−1)(x−2), where the characteristic of k does notdivide 6. Assume that K is perfect. The equation y2 = x(x−m)(x+m)(x−1)(x−2),m ∈ m nonzero, defines a proper lifting X to Λ. Let Y be a proper lifting of the nor-mal curve in [X]. Then Y is smooth since it is a proper lifting of a normal, hence

smooth, curve over k. So the genera of Yk and YK must coincide and be equal to 1.

But the genus of XK is 2, so Y and X cannot be birational.

The general case. — Let X be a curve over Λ. We want to know when [X] lifts

[Xk]. The above example suggests that the genera of the fibres of X should be equal.

As it turns out, that is the exact criterion except that when Xk is not irreducible we

replace the genera by the arithmetic genera of the normalisations of the fibres. In

this subsection, we follow a paper by Diaz and Harris which treats curves over the

complex numbers [DH88, Sec. 2].

Lemma 2.3. — Let X be a curve over Λ. Its normalisation π : Y → X is a finitemorphism of curves over S.

Proof. — Since X is of finite type over Λ, X is excellent [Gro65, Sch. 7.8.3]. So

its normalisation π is finite [Gro65, Prop. 7.8.6]. Moreover, the structure map

Y → X → S is flat: A reduced scheme over Λ is flat if and only if the generic pointof each irreducible component maps to the generic point of S [Gro66, Prop 14.5.6].

Since X is flat over Λ and since the normalisation induces a bijection between generic

points of Y and X, Y is flat over Λ.

(1)In contrast to the definition in Liu’s book [Liu06, Def. 4.1.1], we do not assume that a normal

scheme is irreducible.


Hence, Y is a proper lifting of Yk and it remains to show that Yk is a curve. Since

Y → X is the normalisation, the normalisation of the special fibre Xk must fac-tor through Yk → Xk and Yk must be equidimensional of dimension 1. To showthat Yk is reduced (this implies Yk geometrically reduced as k is perfect), consider a

point P ∈ Yk. Its local ring is reduced if and only if the zero ideal is radical. Let0 = ∩p∈Ass(OYk,P )q(p) be a reduced primary decomposition, i.e. q(p) is p-primary.Since Y is normal and of dimension 2, it is Cohen-Macaulay [Liu06, Cor. 8.2.22] and

so is Yk. Hence Ass(OYk,P ) is the set of minimal primes [Liu06, Prop. 8.2.15]. Let Z

be the irreducible component of Yk passing through P and corresponding to p. Let

ξ be its generic point. The image π(ξ) is a generic point of Xk. Since Xk must be

regular along an open subset of π(Z), Xk must be regular at π(ξ). Hence, X must be

regular at π(ξ). Thus, OY,ξ and OX,π(ξ) are isomorphic since π is the normalisation.

In particular, OYk,ξ is isomorphic to OXk,π(ξ) and thus reduced. So q(p) equals p. This

argument applies to all minimal primes p of OYk,P and so its zero ideal is radical.

Let X be a curve over Λ and s ∈ S. Let π(s) : X̃s → Xs be the normalisation of thefibre over s. Denote by SXs the quotient sheaf of π(s)∗OX̃s by OXs , i.e. there is the

short exact sequence

0→ OXs → π(s)∗OX̃s → SXs → 0. (2.1)

One defines the δ-invariant of Xs as

δ(Xs)df= h0(SXs).

The δ-invariant of a closed point P of Xs is δP (Xs)df= lengthOXs,P (SXs,P )

= [k(P ) : k(s)]−1 dimk(k) SXs,P . Since SXs is a skyscraper sheaf, there is an equality

δ(Xs) =∑P∈X0s

[k(P ) : k(s)]δP (Xs).

The sequence (2.1) gives an identity χ(π(s)∗OX̃s) = χ(OXs) + χ(SXs) [Liu06,

Lem. 7.3.16]. Since χ(π(s)∗OX̃s) and χ(OX̃s) are equal, one arrives at

pa(Xs) = pa(X̃s) + δ(Xs). (2.2)

Lemma 2.4. — Let X be a curve over Λ. Let X̃k → Xk and X̃K → XK be thenormalisations of its fibres. Then pa(X̃k) is at most pa(X̃K).

Proof. — Let Y → X be the normalisation of X. Then X̃K equals YK and the nor-malisation X̃k → Xk factors through Yk → Xk. Since Y is flat over Λ by Lemma 2.3,the arithmetic genera of its fibres coincide. Hence, the genus formula (2.2) gives

pa(X̃K) = pa(YK) = pa(Yk) = pa(X̃k) + δ(Yk). This proves the claim.

Proposition 2.5. — Let X be a curve over Λ. Let π : Y → X, π(k) : X̃k → Xk, andπ(K) : X̃K → XK be the normalisations. The following are equivalent:


(1) The δ-invariants δ(Xk) and δ(XK) coincide,

(2) The arithmetic genera pa(X̃k) and pa(X̃K) coincide,

(3) the fibres of π are π(k) and π(K), and

(4) the class [X] lifts the class [Xk].

Proof. — The genus formula (2.2) implies that (1) and (2) are equivalent. Now

fix a very ample sheaf OX(1) on X and let s ∈ S. As in the lead-up tothe genus formula, tensoring (2.1) with OXs(n) = OX(n)|Xs yields equalitiesχ(π(s)∗OX̃s(n)) = χ(OXs(n)) + χ(SXs(n)) for all n ∈ N. In terms of Hilbertpolynomials this means P (π(s)∗OX̃s) = P (OXs) + P (SXs). Note that SXs is a

skyscraper sheaf, so χ(SXs(n)) equals δ(SXs) for all n ∈ N. Since X is flat over Λ,P (OXs) is independent of s. Therefore (1) holds if and only if the Hilbert polynomial

P (π(s)∗OX̃s) is independent of s. In this formulation, the equivalence of (1) and (3)

was proven by Chiang-Hsieh and Lipman [CHL06, Cor. 3.4.2]. (In the context of

complex analytic spaces the result goes back to Teissier [Tei80].)

To finish, we will show that (3) implies (4) implies (2). Assume (3) holds. By

Lemma 2.3, Y is a proper lifting of Yk = X̃k. Hence, [X] lifts [Xk] and (4) holds.

Assuming (4), one can now argue as in Example 2.2: If [X] contains a proper lifting Z

of X̃k, then Z is normal. In particular, Zs is isomorphic to X̃s for both points s of S.

Since Z is flat over Λ, the arithmetic genera of its fibres agree, i.e. (2) is satisfied.

Definition 2.6. — A curve over Λ is equinormalisable if it satisfies any of the equiv-

alent properties in Proposition 2.5.

Example 2.7. — In the later sections of this chapter, we will use the δ-invariant to

lift classes of curves. Hurwitz’s theorem gives another approach to finding a lifting:

Let Y be a curve over Λ with normal, geometrically integral fibres. Moreover, let

φ : Y → P1S be a finite morphism and s be a point of S. Hurwitz relates the arithmeticgenus of Ys to the degree a(s) of φs and the degree of the ramification divisor (or

different) Rφs of φs:

2pa(Ys) = 2− 2a(s) + degRφs .Now consider the geometrically integral curve X ⊂ P2k defined by ya = h(x) where thecharacteristic of k does not divide a and h(x) is a polynomial over k. For simplicity,

assume that a and deg h are coprime. Let Y be the normalisation of X. The function x

defines a finite morphism φ : Y → P1k of degree a. Let h =∏i h

rii be the factorisation

into irreducibles and si be the greatest common divisor of a and ri, each i. The degree

of Rφ is∑i(a− si) deg hi [Sti09, Sec. 3.7]. So for each hi, choose a degree-preserving

lifting hi ∈ Λ[x]. Each hi is necessarily irreducible. The equation ya =∏i h

rii defines

a proper lifting X of X to Λ. Let Y be its normalisation. The rational function x

defines a finite, flat morphism φ : Y → P1Λ of degree a whose special fibre is φ. Forsimplicity, assume that K is perfect. The analysis of the ramification divisor of the


generic fibre φK is completely analogous to above, i.e. its degree is∑i(a− si) deg hi.

By Hurwitz, the arithmetic genera of Y and YK agree and since YK is a normalisation

of XK , [X] lifts [X].

Remark 2.8. — Except potentially for “special” cases, all geometrically integral

curves X in Prk have an equinormalisable proper lifting in PrΛ: The normalisation

π : Y → X corresponds to a closed immersion φ : Y → Prk and thus to a base-pointfree linear system on Y . Set L = φ∗OPrk(1). The linear system is given by a subspace

V of H0(L ). Choose a proper lifting Y of Y to Λ and an invertible sheaf L on Y that

restricts to L . If L is acyclic, then so is L and the natural map H0(L )→ H0(L ) issurjective. The inverse image V of V under this map defines a base-point free linear

system. Indeed, let P ∈ Y . Then LP ⊗ k equals L P . Since L is generated by Vat P , L is generated by V at P by Nakayama. Now the set Z of points at which

L is not generated by V is closed. As Y is proper over Λ, Z must be empty (oth-

erwise it would intersect Y ). Hence, V and L define a morphism φ : Y → PrΛ. Byconstruction φ lifts φ and induces a finite morphism π : Y → X = φ(Y ) that lifts π.If ξ denotes the generic point of Y , then π(ξ) is the generic point of X and OY,ξ is a

finite extension of OX,π(ξ). In particular, the genus of XK is at most YK . This gives

g(XK) ≤ g(YK) = g(Yk) = g(Xk), so by Lemma 2.4, the genera of the fibres of Xcoincide, i.e. X is equinormalisable.

Example 2.9. — Let X ⊂ P2k be a geometrically integral curve of degree d andgenus g. With the notation of the above remark, the sheaf L is acyclic whenever d is

at least 2g−1. The genus formula (2.2) for X is (d−1)(d−2)/2 = g+δ(X). So d is atleast 2g−1 if and only if δ(X) is at least 2g2−6g+3. In other words, any sufficientlysingular geometrically integral plane curve over k admits an equinormalisable proper

lifting in P2Λ.

The case of plane ordinary curves. — The δ-invariant of a plane curve whose

singularities are all ordinary and whose underlying points are rational is completely de-

termined by the multiplicities of its singularities. Similar to Wahl’s [Wah74b, Sec. 1]

and Markwig’s [Mar07, Paper V] definitions over the complex numbers, we will now

define equimultiple proper liftings of a plane curve X over k. When all singularities

of X are ordinary and have residue field k, such liftings are equinormalisable.

Let F be a field and U ⊆ P2F be open. Let X ⊆ U be a curve and P be an F -rationalpoint of P2F with maximal ideal mP . The multiplicity of X at P is the largest integer

m such that mmP contains the ideal OU (−X)P . The point P lies on X if and onlyif the multiplicity is at least 1. If the multiplicity of X at P is exactly 1, then the

local ring OX,P is regular. If it is greater than 1, then P is called a multiple point or

singularity of X. To generalise these notions to curves over an algebra A of Âr, note

that the F -rational point P corresponds to an F -section of U . So now let U ⊆ P2Λ be


open and σ be an A-section of U . Then σ : SpecA → U is a closed immersion withideal sheaf Iσ. We define the multiplicity mσ(X) of a curve X ⊆ UA along σ as

mσ(X)df= max{n ∈ N | OP2A(−X) ⊆ I

nσ }.

From now on we assume that the underlying point of a singularity is

rational.

Definition 2.10. — Let A be an object of Âr, U ⊆ P2Λ be open, and σ ∈ U(A). Aplane curve X over A is equimultiple along σ (of multiplicity m) if mσk(Xk) = mσ(X)

(= m). The curve X is called equimultiple if there are sections σ1, . . . , σe ∈ U(A) suchthat: (1) X is equimultiple along σi for each i = 1, . . . , e, and (2) σ1, . . . , σe induce

the k-sections that correspond to the multiple points of Xk.

Let x0, x1, x2 be coordinates of P2Λ over Λ. The image of an A-section σ of P

2Λ

is contained in D(xi) for some i = 0, 1, 2, say for i = 0. Then its ideal sheaf Iσis trivial outside D(x0) and on D(x0) it is given by an ideal of A[x1, x2] of the form

(x1−λ1, x2−λ2) with λ1, λ2 ∈ A. Now let X be a plane curve over A and F (x0, x1, x2)be a defining homogeneous polynomial. Then X has multiplicity m along σ if and

only if the homogeneous parts of degrees 0, 1, . . . ,m−1 of F (1, x1 +λ1, x2 +λ2) vanishand the the degree m part does not vanish. Now let A be equal to k. Assume that

σ is the section corresponding to a singularity P of X. We say that P is an ordinary

singularity if the degree m part of F (1, x1 +λ1, x2 +λ2) factors into m distinct linear

forms over k.(2) If in addition m equals 2, then P is a node of X. We call X an

ordinary curve when all its singularities are ordinary and a nodal curve when all its

singularities are nodes. The δ-invariant of a singularity P of multiplicity mP satisfies

δP (X) ≥ mP (mP − 1)/2. (2.3)

The inequality is an equality if and only if P is ordinary [Hir57, Thm. 1].

Proposition 2.11. — Let X be a plane curve over Λ with ordinary special fibre. If

X is equimultiple, then it is equinormalisable.

Proof. — Lemma 2.4 and the genus formula (2.2) imply that δ(XK) is at most δ(Xk).

Let us use the notation from Definition 2.10 and let mi be the multiplicity of X along

σi. Each section σi gives rise to a section σi,K ∈ P2K(K) such that XK has multiplicitymi along σi,K . In other words, each σi corresponds to a singularity of multiplicity

mi on XK . Moreover, pairwise distinct σi give pairwise distinct singularities. From

the formula (2.3) for the δ-invariant of a singularity, we deduce that δ(XK) is at least

(2)The definition of ordinary singularities given here is more restrictive than the general one. In

particular, ordinarity can be defined when the multiple points are not rational. For our purposes,

the definition given here suffices.


δ(Xk). Hence, the δ-invariants of the fibres ofX must agree andX is equinormalisable.

Example 2.12. — The converse to the proposition does not hold: A trivial example

is the union X of the three lines defined by x, y, and x + y + mz in the projective

plane over Λ, m ∈ m nonzero. Its generic fibre has three singularities, each a node, at[0 : 0 : 1], [m : 0 : 1], and [0 : m : 1]. Therefore, δ(XK) is 3. However, its special fibre

has a unique singularity, an ordinary multiple point of multiplicity 3, at [0 : 0 : 1]. So

δ(Xk) also equals 3. Hence, X is equinormalisable but not equimultiple.

2.2. The equimultiple local Hilbert functor

For this section, we fix a plane curve X of degree d over k and a finite

set Σ ⊆ P2k(k) of sections. Let m = {mσ}σ be the multiplicities of Xalong the σ ∈ Σ.

Due to Proposition 2.5 and Proposition 2.11, we are interested in equimultiple proper

liftings of X to Λ. The goal of this section is to introduce and understand the following

natural diagram of functors. The first half of this section introduces the necessary

terminology and the functors on the top square. The bottom functors are the topic

of the second half.

HX� // P

d(d+3)/2Λ

H(X,Σ)� //

99 99

Pd(d+3)/2Λ ×

∏σ∈Σ P

2Λ

99 99

EX?�

OO

E(X,Σ)� //

?�

OO

88 88

E(d,m)?�

OO

(2.4)

The three functors on the right-hand side are (the functors of sections of) projective

schemes over S. The left-hand square is made up of prorepresentable functors of

Artin rings. They are the “local counterparts” at X and Σ to the functors on the

right. The functors on the top square are smooth. If A is an object of Ar and Σ is

the set of sections corresponding to the multiple points of X, then EX(A) is the set

of equimultiple proper liftings of X to A.

2.2. THE EQUIMULTIPLE LOCAL HILBERT FUNCTOR 15

Formal deformation theory. — A functor of Artin rings is a covariant functor

F : Ar→ Set, #F (k) = 1.

For example, if R is an object of Âr, then HomΛ(R,−) is a functor of Artin rings.A functor of Artin rings that is isomorphic to some HomΛ(R,−) is said to be prorep-resentable or prorepresented by R. The functor F of Artin rings is called smooth if

F (A′) → F (A) is surjective whenever A′ → A is. If F is prorepresented by R, thenF is smooth if and only if R is a power series ring over Λ [Sch68, Prop. 2.5]. Any

functor F of Artin rings can be extended to a functor F̂ on Âr by

F̂ (A)df= lim←−F (A/m

NA ).

Let k[ε] be the ring of dual number of k, i.e. ε2 equals 0. The tangent space of F is

defined as

t(F )df= F (k[ε]).

Given surjections A′ � A and A′′ � A in Ar, there is the natural map

F (A′ ×A A′′)→ F (A′)×F (A) F (A′′). (2.5)

Consider the following “Schlessinger’s conditions”:

(H1) (2.5) is surjective whenever A′′ � A is small,(H2) (2.5) is bijective whenever A = k and A′′ = k[ε],

(H3) t(F ) is a k-vector space of finite dimension, and

(H4) (2.5) is bijective whenever A′′ � A is small.

If F satisfies (H2), then the tangent space t(F ) has a natural structure as k-vector

space [Sch68, Lem. 2.10]. Moreover, let A′ � A be a small extension with kernel I.There is a natural action

F (A′)× (t(F )⊗ I)→ F (A′)×F (A) F (A′)

of t(F ) ⊗ I on the nonempty fibres of F (A′) → F (A). If F satisfies (H1) and (H2),then the action is transitive. If F satisfies (H2) and (H4), then the action is transitive

and free [Sch68, (2.15)]. Finally, a functor of Artin rings is prorepresentable if and

only if it satisfies (H2), (H3), and (H4) [Sch68, Thm. 2.11].

Example 2.13. — Let Z be a closed subscheme of Prk. The local Hilbert functor

HZ : Ar→ Set sends an algebra A to the set

HZ(A)df={Z ⊆ PrA | Z is a deformation of Z to A

}.

The functor HZ is prorepresentable [Har10, Thm. 17.1]. Let NZ/Prkbe the normal

sheaf of the closed immersion Z ↪→ Prk. The tangent space of the local Hilbert functoris t(HZ)

∼= H0(NZ/Prk) [Har10, Thm. 2.4]. A particular case is that of a k-rationalpoint P of Prk. Its local Hilbert functor is smooth and can be identified with a

subfunctor of PrΛ.


When σ is a k-section of PrΛ and P is the corresponding k-rational point,

we write Hσ for HP . Similarly, we may identify a plane curve X of degree

d over an object A of Ar with an A-section of Pd(d+3)/2Λ . In particular,

the local Hilbert functor HX of a plane curve over k is a subfunctor of

some Pd(d+3)/2Λ .

Remark 2.14. — Let {Fi}i∈I be a finite family of functors of Artin rings. Thentheir product F =

∏i∈I Fi in the category of functors Ar→ Set is a functor of Artin

rings. Let (H*) be any of Schlessinger’s conditions or smoothness. If each Fi satisfies

(H*), then so does F .

By the above remark and the previous example, the functor

H(X,Σ)df= HX ×

∏σ∈Σ

Hσ

is prorepresentable and smooth.

Equimultiple Hilbert functors. — We are now able to consider equimultiple

proper liftings of plane curves in the framework just introduced. Following Wahl

again [Wah74b, Sec. 1], we make

Definition 2.15. — The equimultiple local Hilbert functor of (X,Σ) sends an alge-

bra A of Ar to

E(X,Σ)(A)df={

(X,Σ) ∈ H(X,Σ)(A) | For all σ ∈ Σ, X is equimultiple along σ}.

Remark 2.16. — Let EX be the image of E(X,Σ) in HX . Assume that the sections

in Σ correspond to the multiple points of X. Then for any object A of Âr, ÊX(A) is

the set of equimultiple proper liftings of X to A.

The equimultiple local Hilbert functor is prorepresentable and – at least in some cases

– smooth. To see this, we introduce a scheme that represents a “global equimultiple

Hilbert functor”. Let σ be a lifting of σ ∈ Σ to an object A of Âr. Let (x0, x1, x2)be coordinates of P2Λ over Λ. If the image of σ is contained in D(xi), then so is

the image of σ. Hence, the ideal sheaf of σ is trivial outside D(xi) and on D(xi) it

corresponds to an ideal generated by xj − tj , j ∈ {0, 1, 2} \ {i}, for some tj ∈ A. Asnoted following Definition 2.10, the multiplicity of a plane curve V (F ) over A is given

by the least degree among the nonvanishing homogeneous parts of F (1, x1+t1, x2+t2)

(or F (x0 + t0, 1, x2 + t2) or F (x0 + t0, x1 + t1, 1) depending on i). This motivates the

following notation and definition:

Let x = (x0, x1, x2) be fixed coordinates for P2Λ. For each section σ ∈ Σ,

pick a fixed xσ ∈ x such that the image of σ is contained in D(xσ). Letx̂σ the tuple obtained from x by removing xσ.


Definition 2.17. — Pick coordinates λ = (λα)α∈N3,|α|=d and tσ = (tσ,0, tσ,1, tσ,2),

σ ∈ Σ, for the multiprojective space Pd(d+3)/2Λ ×∏σ∈Σ P

2Λ. Set F =

∑α λαx

α.

For each σ ∈ Σ, let fσ ∈ Λ[x̂σ][λ, tσ] be the polynomial F (1, x1 + tσ,1, x2 + tσ,2) orF (x0 + tσ,0, 1, x2 + tσ,2) or F (x0 + tσ,0, x1 + tσ,1, 1) (choose the polynomial that sets

xσ to 1). For each σ ∈ Σ, β ∈ N2, and |β| ≤ mσ − 1, let fσ,β ∈ Λ[λ, tσ] be thecoefficient of x̂βσ in fσ. We set

E(d,m)df= V ({fσ,β}σ,β) ⊆ Pd(d+3)/2Λ ×

∏σ∈Σ

P2Λ.

Even though the notation suggests that the scheme E(d,m) depends only on d and m,

in general it also depends on how the k-rational points specified by Σ are spread out in

the projective plane: Unless k is infinite, there may not be coordinates x for P2Λ such

that a single D(xi) contains all points specified by Σ. For an example over the field of

2 elements, consider the union of the lines V (x0), V (x1), V (x2) and V (x0 + x1 + x2)

and let Σ be the set of sections corresponding to the multiple points.

Remark 2.18. — Let A be an object of Ar. By the discussion preceding Defini-

tion 2.17,

E(X,Σ)(A) = H(X,Σ)(A) ∩ E(d,m)(A),

where the intersection is taken in(Pd(d+3)/2Λ ×

∏σ∈Σ P

2Λ

)(A).

We now turn to prorepresentability and smoothness of E(X,Σ) and EX .

Proposition 2.19. — The functor E(X,Σ) is prorepresented by the completion Ô of

the local ring O = OE(d,m),(X,Σ).

Proof. — We need to show that the functors HomΛ(Ô,−) and E(X,Σ) are isomorphic.So let A be an object of Ar and let N be its length. Write RN = R/m

NR for any

Λ-algebra R. On the one hand, there is a natural bijection between HomΛ(Ô, A) and

HomΛN (ÔN , A). Moreover, ÔN and ON are naturally isomorphic.

On the other hand, by definition an element of E(X,Σ)(A) may be interpreted as an

A-section of E(d,m) that lifts the unique element of E(X,Σ)(k). In other words, the

elements of E(X,Σ)(A) correspond bijectively to local morphisms O � ON → A of Λ-algebras. These in turn correspond bijectively to the elements of HomΛN (ON , A).

Lemma 2.20. — If Σ contains at most one section, then E(X,Σ) is smooth.

Proof. — If Σ is empty, then EX,Σ equals HX and we already mentioned that this

functor is smooth in Example 2.13. So let Σ = {σ}. Let A′ � A be a surjectionin Ar and let (X,Σ) ∈ E(X,Σ)(A). We need to show that there exists an element


(X ′,Σ′) ∈ E(X,Σ)(A′) that maps to (X,Σ). We may choose coordinates (x, y, z) forP2Λ such that the ideal sheaf of σ is the ideal generated by x and y in A[x, y]. Let∑di=0 Fi(x, y)z

i be a defining polynomial of X and Fi(x, y) be homogeneous part of

degree d − i. Let m be the unique member of m, i.e. the multiplicity of X along σ.Since (X,Σ) is a section of E(d,m), the polynomials F0, F1, . . . , Fm−1 vanish. For

i = m, . . . , d, pick a homogeneous lifting F ′i of Fi to A′. Let X ′ be the plane curve

over A′ defined by∑di=m F

′izi and σ′ be the A′-section of P2Λ that corresponds to

the ideal generated by x and y in A′[x, y]. Then (X ′,Σ′) is the desired element of

E(X,{σ})(A′).

In general, the question of smoothness seems more complicated. Since E(X,Σ) maps

onto EX by definition, the latter functor is smooth whenever the first is. The converse

holds when all points corresponding to the sections in Σ are ordinary singularities

on X. In fact, we have

Lemma 2.21. — Assume that the point corresponding to the section σ ∈ Σ is anordinary singularity on X. Then the natural map E(X,Σ) → E(X,Σ\{σ}) is injective.In particular, when all sections in Σ correspond to ordinary singularities on X, then

the natural surjection E(X,Σ) � EX is an isomorphism.

For Λ the power series ring over the complex numbers and X an algebroid curve, the

lemma was proven by Wahl [Wah74b, Prop. 1.9]. We follow his proof.

Proof. — Since E(X,Σ) and E(X,Σ\{σ}) are both prorepresentable by the previous

proposition, it suffices to show that the map on tangent spaces t(E(X,Σ))→ t(E(X,Σ\{σ}))is injective [Wah74a, Prop. 1.1.4]. (Wahl’s proof is given in a more restrictive setting

than needed here. However, his proof can be used verbatim also in our context.)

Let (X,Σ) and (X,Σ′) be elements of t(E(X,Σ)) that map to the same element of

t(E(X,Σ\{σ})), e.g. Σ = Θ ∪ {σ} and Σ′ = Θ ∪ {σ′} and σ, σ′ are liftings of σ. Let mbe the multiplicity of X along σ. We may choose coordinates (x, y, z) of P2Λ such that

σ and σ′ correspond to the ideals (x, y) and (x− εa, y − εb) of k[ε][x, y] respectively.We may also assume that k is algebraically closed. Then in addition we can choose

the coordinates so that X is defined by a polynomial F (x, y, z) such that the degree

m part of f(x, y) = F (x, y, 1) is equal to yh(x, y).

We may assume that a and b are elements of k and thus must show that both

equal 0. So let F + εG be a defining polynomial of X in k[ε][x, y, z] and write

g(x, y) = G(x, y, 1). Since (X,Σ′) is an element of t(E(X,Σ)), we calculate

f(x+εa, y+εb)+εg(x+εa, y+εb) = f(x, y)+εa∂f

∂x(x, y)+εb

∂f

∂y(x, y)+εg(x, y) ∈ (x, y)m.

(2.6)


But we also know that (X,Σ) is an element of t(E(X,Σ)), so f(x, y) + εg(x, y) also

belongs to (x, y)m. Hence,

a∂f

∂x(x, y) + b

∂f

∂y(x, y) ∈ (x, y)m.

For the degree m part of f(x, y) written as yh(x, y), this means

ay∂h

∂x(x, y) + bh(x, y) + by

∂h

∂y(x, y) = 0.

Therefore, y divides bh(x, y). Since σ corresponds to an ordinary singularity, the

degree m part of f splits into pairwise distinct linear factors. Hence, y does not

divide h(x, y), but b and so b is equal to 0. This in turn implies that a(∂h/∂x)(x, y)

vanishes. So either a equals 0 as desired or h is a p-th power. (Use that y does

not divide h(x, y).) The latter cannot happen since σ corresponds to an ordinary

singularity.

Remark 2.22. — With the notions from the proof, we can identify the tangent space

of EX when Σ contains a unique section: Indeed, let I denote the ideal sheaf in OP2kdefined by the ideal (f, ∂f/∂x, ∂f/∂y)+(x, y)m ⊆ k[x, y]. Recall that there is a shortexact sequence

0→ OP2k → I (X)→ IX(X)→ 0, (2.7)

where I (X) is the tensor product I ⊗ OP2k(X). Equation (2.6) implies

t(EX)∼= H0(IX(X)). (2.8)

In general, define an ideal sheaf for each section in Σ as above and let I be their

sum. Then an isomorphism as in (2.8) still exists. Note also that this identification is

independent of the types of singularities that the elements of Σ correspond to.

Definition 2.23. — The ideal sheaf I defined in the remark above is called the

equimultiple ideal sheaf for X and Σ. If Σ is the set of all sections of X along which

X has multiplicity greater than 1, then we also say that I is the equimultiple ideal

sheaf of X.

Proposition 2.24. — Let I be the equimultiple ideal sheaf. The functor EX is

smooth if H1(IX(X)) vanishes.

For curves over complex surfaces (in fact, characteristic 0 seems to suffice) Greuel

and Lossen show that EX is smooth already when Im(H1(I (X))→ H1(IX(X))

)vanishes [GL01, Prop. 1.3]. In case that no sections are considered, i.e. Σ is the

empty set, their statement is the “Severi-Kodeira-Spencer” theorem. A proof is in

Mumford’s “Lectures on Curves on an Algebraic Surface” [Mum66, Lec. 23]. We

follow this proof.


Proof. — To begin, let us fix some notation: Let {Ui}i∈I be a covering of P2k by openaffines such that each Ui contains the image of at most one section σ ∈ Σ. Let Σibe the subset of Σ that consists of this unique section or is empty when Ui does not

contain the image of any section. Set Uij = Ui ∩ Uj and Uij` = Ui ∩ Uj ∩ U` fori, j, ` ∈ I. Note that P2A has the same topological space as P2k for any object A of Ar.

Now let A′ � A be a surjection in Ar. We need to show that any element of EX(A)lifts to an element of EX(A

′). It suffices to do this when A′ � A is a small extensionwhose kernel is generated by a single element η as k-vector space. (Since any surjection

in Ar is the composite of finitely many such extensions.) So let X ∈ EX(A) be given.For each i ∈ I, let E(i)

Xbe the image of E(X,Σi) in HX . By Lemma 2.20 and the

fact that E(X,Σi) → E(i)

Xis surjective, E

(i)

Xis smooth. Moreover, there are natural

injections EX ↪→ E(i)

X. Hence, for each i ∈ I, there exists a lifting X ′i ∈ E

(i)

X(A′) of X.

There exists a lifting X ′ ∈ E(X,Σ)(A′) of X if and only if there exist liftingsX ′i ∈ E

(i)

X(A′) of X that glue along the Uij . Assume that X = V (F ) for some

homogeneous polynomial F (x, y, z) and for each i ∈ I, let X ′i = V (F ′i ), i ∈ I, be anylifting of X in E

(i)

X(A′). We may assume that each F ′i is a lifting of F to A

′. Write

f ′i for the element induced in OP2A′

(Ui) by F′i and fi for the image of f

′i in OP2A(Ui).

Then

f ′i = u′ijf′j + ηh

′ij (2.9)

over Uij for some unit u′ij ∈ OP2

A′(Uij)

∗ and h′ij ∈ OP2A′

(Uij). We must show that

the f ′i and u′ij can be chosen so that the h

′ij vanish. First, we calculate

η(h′ij + u′ijh′j`) = f

′i − u′ijf ′j + u′ij(f ′j − u′j`f ′`)

= f ′i − u′iju′j`f ′`= ηh′i` + (u

′i` − u′iju′j`)f ′`.

Given an element a′ or a of OP2A′

(Uij), let a denote its image in OP2k(Uij). From the

above we get

hij + uijhj` = hi` +

(u′i` − u′iju′j`

η

)f `.

Moreover, f i = uijf j . Hence,

hij

f i+hj`

f j=hj`

f i+

(1− u′iju′j`u′i`

−1

η

).

Since X ′i and X′j are equimultiple along the unique sections (if any) of Ui and Uj , the

sections hij lie in I (Uij). Therefore, {hij/f i}i,j represents an element of H1(IX(X))which, by assumption, is equal to 0.

2.3. COMPUTING EQUIMULTIPLE PROPER LIFTINGS 21

Hence, there exist elements gi ∈ I (Ui) such that hij/f i equals gj/f j − gi/f i assection of IX(X)(Uij). Pick any liftings g

′i ∈ OP2

A′(Ui) of the gi. Then f

′i + ηg

′i is an

equimultiple(!) lifting of fi and

f ′i + ηg′i = (u

′ij + η(hij + g

′i − g′ju′ij)/f ′j)(f ′j + ηg′j).

Since uij is a unit, so is u′ij + η(hij + g

′i − g′ju′ij)/f ′j and the f ′i + ηg′i glue along the

Uij as desired.

2.3. Computing equimultiple proper liftings

We are now ready to describe an algorithm that computes equimultiple proper liftings

of a given plane curve X over k to WN (k). It is an application of formal smoothness

of E(d,m) in neighbourhoods of smooth points.

Since E(X,Σ) is prorepresented by the completion of OE(d,m),(X,Σ), the functor is

smooth if and only if the completion is a power series ring over Λ. This in turn happens

if and only if E(d,m) is smooth at (X,Z) [Bou06, Prop VIII.5.1 & Thm. VIII.5.2].

Now let A be an algebra of Ar. We may identify an element of E(X,Σ)(A) with an

A-section of E(d,m). If E(d,m) is smooth at (X,Z), then it is formally smooth in

a neighbourhood of (X,Z) [Gro67, Thm. 17.5.1]. Hence, for any small extension

A′ � A in Ar, there exists an A′-section of E(d,m) that lifts the A-section [Gro67,Def. 17.1.1]. This is also sometimes called the infinitesimal lifting property.

Proposition 2.25 (Effective infinitesimal lifting property)

Let V be a scheme of finite type over Λ, σ ∈ V (k), and Imσ be a smooth point of V .Given an integer N ≥ 1 and a presentation ΛN [x]/(f) of an open affine neighbourhoodU of Imσ, Algorithm 1 computes a section σ ∈ U(ΛN ) whose special fibre is σ.

Proof. — The case N equal to 1 is trivial. Assume that N is at least 2. Let

x = (x1, . . . , xr), f = (f1, . . . , fs), t be a generator of the maximal ideal of Λ, and

n be an integer between 2 and N . A section in U(Λn) corresponds to an r-tuple of

elements of Λn, i.e. an element of the free module Λrn. We need to show that if a

represents a lifting of σ to U(Λn−1) at the beginning of the loop in Algorithm 1, then

a represents a lifting to U(Λn) at the end of the loop.

To find a lifting of a to Λrn that represents a section in U(Λn) the algorithm proceeds

as follows: First, it picks any lifting a ∈ Λrn, i.e. a section in Ar(Λn). This sectionrestricts to an element of U(ΛN ) if and only if o = f(a) vanishes. Now any other

lifting of a to Λrn is of the form a′ = a+ tN−1∆, ∆ ∈ ΛrN . This element a′ defines a

section in U(Λn) if and only if f(a′) = f(a+ tN−1∆) = 0. This happens if and only


Algorithm 1: Applies the infinitesimal lifting property to lift a k-section σ of

an affine scheme U ⊆ ArΛN to a ΛN -section if Imσ is a smooth point.

J ← Jacobian(U)1a← represent in kr(σ)2for n = 2 to N do3

// Choose lifting to Ar(Λn)

a← lift(Λn,a)4// Compute obstruction for lifting to lie in U(Λn)

o← f(a)5// Modify lifting so that obstruction vanishes

∆← solve(J(a),−o/tn−1)6a← a+ tn−1 · lift(Λn,∆)7

// Now a represents a lifting of σ to U(ΛN )

return a8

if

tN−1J(a) ·∆ = −o,where J denotes the Jacobian matrix of U . Regard this as an equation in ∆. Both

sides are multiples of tN−1, so the solutions to the equation are in one-to-one corre-

spondence to the elements tN−1∆ where ∆ is any lifting to Λrn of a solution ∆ to the

equation

J(a) ·∆ = −o/tN−1. (2.10)Since U is formally smooth in a neighbourhood of Imσ, a lifting of a to Λrn that

defines a section in U(Λn) has to exist, i.e. (2.10) has a solution ∆. The algorithm

can now pick any lifting ∆ of ∆ and replace the initial choice of a by a + tN−1∆.

By the above, this a represents the desired lifting of σ.

Remark 2.26. — Li describes this algorithm in his thesis“Anwendung deformations-

theoretischer Methoden zur Liftung des Frobeniusmorphismus” [Li08] when U is an

affine plane curve. In this case, J(a) is a matrix in k1×2. Li argues that a solution

to (2.10) always exists when U is smooth at Imσ since smoothness is equivalent to

J(a) being of full rank. In the higher dimensional case of course, J(a) does not need

to be of full rank if U is smooth at Imσ. A priori it is not clear that (2.10) has a

solution and this is why we argue with formal smoothness.

Theorem A. — Let X be an ordinary, plane curve over k with irreducible compo-

nents Xi of degrees di, i = 1, . . . , s, Σ ⊆ P2k(k) be the set of sections that correspondto the multiple points of X, and I be the equimultiple ideal sheaf of X and Σ. As-

sume that the curves Xi are geometrically irreducible. The space H1(IX(X)) vanishes


whenever

3di >∑

mσ(X)>2

mσ(Xi), for all i.

Before we spent the remainder of this chapter on the proof of this theorem, let us first

discuss the result. There is

Corollary 2.27. — In the situation of Theorem A, E(d,m) is smooth at (X,Σ) and

Algorithm 1 (applied to a suitable affine patch of E(d,m)) computes a lifting of (X,Σ)

to ΛN .

Proof. — Since H1(IX(X)) vanishes, EX is smooth (Proposition 2.24). As X is or-

dinary, E(X,Σ) is smooth as well (Lemma 2.21). So by the arguments at the beginning

of this section, the scheme Ed,m is smooth at (X,Σ). So Proposition 2.25 shows that

the algorithm computes a lifting of (X,Σ) to ΛN .

Remark 2.28. — Beerenwinkel also considered the problem of effectively lifting

curves. In his Diplomarbeit [Bee99], he derives the same lifting algorithm as ours

but with two restrictions: All multiple points of the given plane curve must be nodes

and only at most b(d + 1)/2c nodes are allowed (where d is as usual the degree ofthe curve). This is due to the fact that Beerenwinkel relies on a Hensel-lifting lemma

instead of formal deformation theory. Therefore, he needs that the Jacobian of Ed,mat (X,Σ) has full rank. However, when there are more than b(d + 1)/2c nodes, therank might no longer be full. (As example consider three nodes on a degree 4 curve

that lie on a line).

The first restriction – that all multiple points are nodes – is a mild one from a the-

oretical point of view. However, in the context of lifting a smooth curve in practice,

it is of course of interest to allow more general (ideally, any) plane images. A more

serious restriction is the one on the number of nodes. If there are at most b(d+ 1)/2csuch points, then the genus formula (2.2) and formula (2.3) force

d2 − 4d+ 3 ≤ 2g. (2.11)

Hence, the Brill-Noether number ρ(g, 2, d) = g − 3(g − d + 2) is negative wheneverthe genus is at least 7. In particular, in general it is not clear whether smooth

geometrically integral curves of large genus can be mapped birationally to a plane

curve so that (2.11) is satisfied. Hence, it is not clear whether Beerenwinkel’s result

is sufficient to lift arbitrary curves.

Let us now turn to the proof of Theorem A. We will use the notion of isomorphism

defect : Let F and G be coherent sheaves on a curve Y over k and assume that both

have the same rank on each irreducible component of Y . In the following, a ·̃ denotesthe reduction of a sheaf modulo torsion. Let Z be a closed subset of Y and denote


by ·Z the restriction of a sheaf to Z. The local isomorphism defect of F in G at a(closed) point P of Z is

isodZ,P (F ,G )df= min

(lengthOZ,P

(Coker(F̃Z,P −→ G̃Z,P )

)),

where the minimum is over all injective local homomorphisms induced by local homo-

morphisms FP → GP . The local isomorphism defect is nonzero at only finitely manypoints of Z and one defines

isodZ(F ,G )df=∑P∈Z

[k(P ) : k]isodZ,P (F ,G )

as the isomorphism defect of F in G along Z. When Z is the whole curve, then we

drop the Z in the indices. A key lemma for the proof of Theorem A is

Lemma 2.29. — Let Y be a curve on a smooth surface(3) over k, Y 1, . . . , Y s its

irreducible components, ωY its canonical sheaf, and F be a coherent OY -module of

rank 1 on each irreducible component of Y . Then H1(F ) vanishes if for all i = 1, . . . , s

χ(F̃Y i) > χ(ωY ,Y i)− isodY i(F , ωY ).

Proof. — Greuel and Karras’ proof over the complex numbers [GK89, Prop. 5.2]

works over any field (the fact that k is perfect is not needed): Assume that H1(F )

does not vanish. Since Y is a curve on a smooth surface, it is a local complete

intersection and the canonical sheaf of Y is (isomorphic to) its dualising sheaf [Liu06,

Thm. 6.4.32], so there exists a nonzero morphism ϕ : F → ωY [Liu06, Rem. 6.4.20].Since ωY is torsion-free, the image of ϕ must have rank 1 on at least one irreducible

component Y i of Y . Let ϕ̃i : F̃Y i → ω̃Y ,Y i be the map induced by ϕ. Note thatω̃Y ,Y i equals ωY ,Y i as the latter sheaf is already torsion-free. By the above, ϕ̃i is

injective and thus

χ(F̃Y i) = χ(ωY ,Y i)− χ(Coker ϕ̃i) ≤ χ(ωY ,Y i)− isodY i(F , ωY ).

We introduce one final notion that we will use in the following proof: Given a plane

curve Y over k with equimultiple ideal sheaf I and a closed point P on Y , set

τ emP

(Y )df= lengthOY ,P (OY ,P

/IY ,P )

and

τ em(Y )df= h0(OY ,P

/IY ,P ) =

∑P

[k(P ) : k]τ emP

(Y ).

(3)A surface over k is a geometrically reduced, separated k-scheme of finite type all of whose irre-

ducible components are of dimension 2.


Proof of Theorem A. — We follow the proof of Greuel and Lossen who consider the

case that Λ is the ring of formal power series over the complex numbers [GL96,

Cor. 5.1]. To begin, note that the sheaf IX(X) vanishes outside X, so we may regard

it as OX -module when computing its cohomology [Har97, Lem. III.2.10]. Our goal

is to apply Lemma 2.29. To this end, note that there is an exact sequence

0→ IX(X)→ OX(X)→ Q → 0

which we may take to define Q. By construction, the stalk of Q at a closed point P

of X is

QP∼= OX,P

/IX,P ,

Hence, Q is torsion and vanishes outside the singular locus of X. Let us indicate with

an index i the restriction of a sheaf on X to Xi. Then Qi is also torsion and (with

the ·̃-notation introduced before Lemma 2.29) there is an exact sequence

0→ ĨX(X)i → OX(X)i → Qi → 0 (2.12)

for all i = 1, . . . , s. Since OX(X) is an invertible sheaf, so is OX(X)i and we deduce

from the exact sequences that IX(X) is a coherent OX -module of rank 1 on each

irreducible component of X. Let ωX be the canonical sheaf of X. We apply

Lemma 2.29 and find that H1(IX(X)) vanishes if

χ(ĨX(X)i) > χ(ωX,i)− isodXi(IX(X), ωX) (2.13)

for all i = 1, . . . , s. Let us first consider χ(ωX,i). Let ωP2k be the canonical sheaf

of P2k. The adjunction formula tells us that ωX,i equals (OP2k(X)⊗ ωP2k)|Xi [Liu06,Thm. 9.1.37]. This in turn equals (OP2k(Xi)⊗ωP2k)|Xi⊗OP2k(X

′)|Xi withX = X′+Xi.

By adjunction again, (OP2k(Xi) ⊗ ωP2k)|Xi is isomorphic to the canonical sheaf ωXiof Xi. For the Euler-Poincaré characteristics this means

χ(ωX,i) = χ(ωXi) + χ(OP2k(X′)|Xi)− χ(OXi).

Now, χ(OP2k(X′)|Xi) − χ(OXi) is just the intersection number X

′ · Xi of X′

and

Xi [Liu06, Thm. 9.1.12]. Furthermore, χ(ωXi) equals −χ(OXi) by duality [Liu06,Rem. 6.4.21] and therefore is equal to pa(Xi)− 1. Putting everything together, (2.13)is equivalent to

χ(ĨX(X)i) > pa(Xi)− 1 +X′ ·Xi − isodXi(IX(X), ωX). (2.14)

Let us now turn to χ(ĨX(X)i). By(2.12) and since the support of Qi is 0-dimensional,

χ(ĨX(X)i) = χ(OX(X)i)− χ(Qi)

= χ(OX(X)i)− h0(Qi).

Now, χ(OX(X)i) equals 1−pa(Xi) +X ·Xi [Liu06, Thm. 9.1.12]. Plugging this into(2.14) gives

X ·Xi − 2pa(Xi) + 2 > X′ ·Xi + h0(Qi)− isodXi(IX(X), ωX)


and by adjunction

X ·Xi − (W +Xi) ·Xi > X′ ·Xi + h0(Qi)− isodXi(IX(X),OX),

where W is a canonical divisor of P2k. This simplifies to

−W ·Xi > h0(Qi)− isodXi(IX(X), ωX). (2.15)

Let H be a line on P2k. Then one may take W as −3H. So Bézout tells us that theabove inequality is equivalent to

3di > h0(Qi)− isodXi(IX(X), ωX).

We are left to show that∑mP (X)>2

mP (Xi) is at least h0(Qi)− isodXi(IX(X), ωX).

By Lemma 2.30, the latter equals∑mP (X)>0

τ emP

(Xi) + iP (Xi, X′)− isodXi,P (IX(X), ωX).

By Lemma 2.33 and Corollary 2.31, this simplifies to∑mP (X)>0

τ emP

(Xi)− δP (Xi) + length(

(IX,P/

cond(IX,P ))⊗ OXi,P),

where cond(·) denotes the conductor (see below). Clearly, the individual summandsare each 0 whenever P is a regular point of X, i.e. whenever mP (X) equals 1. So,

together with Lemma 2.34, the above simplifies once more to∑mP (X)>1

min{0,mP (Xi)− 2}+ length(

(IX,P/

cond(IX,P ))⊗ OXi,P).

The theorem is proven with Lemma 2.35.

Lemmas for the Proof. — This subsection collects several lemmas that were used

in the proof of Theorem A. To this end, we assume that k is algebraically closed (this

is not strictly needed for every lemma). Let Y be a plane, ordinary curve over k, P

be a closed point of Y , m be the multiplicity of Y at P , I the equimultiple ideal

sheaf of Y , and ωY be the canonical sheaf of Y . In the application to the proof, Y

will play the role of X or Xi. Let x and y be local coordinates P2k at P . We may

and do assume that Y is defined at P by f ∈ k[x, y] and that x does not divide thedegree m part of f .

Lemma 2.30. — In the situation of the proof Theorem A, let P be a closed point

of X. Then

length(Qi,P ) = τemP

(Xi) + iP (Xi, X′).

Proof. — If P is not a multiple point of X, then the three cardinals in each of the

two equations are each equal to 0. So let P be a multiple point of X. By our

earlier assumption in Section 2.1, the degree of P is 1, so we find defining elements


f i, f′ ∈ k[x, y] of Xi and X

′at P respectively. Let mi,m

′ be the multiplicities of Xiand X

′at P respectively. Then(

OX,P/IX,P

)⊗ OXi,P ∼= OXi,P

/f′ ·(∂f i∂x

,∂f i∂y

)+ (x, y)mi+m

′

∼= ÔXi,P/f′ ·(∂f i∂x

,∂f i∂y

)+ (x, y)mi+m

′,

where ÔXi,P is the completion of OXi,P . Hence, the lemma is a direct consequence

of the snake lemma applied to the following commutative diagram of ÔXi,P -modules

with exact rows:(∂fi∂x ,

∂fi∂y

)+ (x, y)mi

·f ′//

��

f′ ·(∂fi∂x ,

∂fi∂y

)+ (x, y)mi+m

′//

��

0 //

��

0

0 // ÔXi,P·f ′

// ÔXi,P// ÔXi,P

/f′ÔXi,P

// 0

The only question that remains is why the first row is exact. To answer it, we need to

show that (x, y)mi+m′

is in the image of the first map (the rest is immediate). To this

end, it suffices to show that the ideal in k[[x, y]] generated by the degree mi part of

f i and the degree m part of f′

contains (x, y)mi+m′. Now recall that P is an ordinary

multiple point of X. Hence, the degree mi part of f i and the degree m′ part of f

′are

coprime. Some linear algebra in the coefficients of those polynomials shows that the

first row is exact.

Let R be reduced Noetherian ring and R̃ be its normalisation. The conductor

of an ideal I of R is the ideal cond(I)df= {h ∈ I | hR̃ ⊆ I}. Conductors im-

plicitely appeared already earlier in this chapter: The δ-invariant δP (Y ) equals

length(OY ,P/

cond(OY ,P )) [Ful02, Cor. 2.2]. If P is an ordinary singularity, then

cond(OY ,P ) equals (x, y)m−1 [Ful02, Prop. 2.1]. So the same reasoning as in the

proof of the above lemma gives

Corollary 2.31. — In the situation of the proof Theorem A, let P be a closed point

of X. Then

length(

(OX,P/

cond(OX,P ))⊗ OXi,P)

= δP (Xi) + iP (Xi, X′).

The map that sends an element h of cond(I) to the multipliation-by-h map gives a

natural isomorphism between cond(I) and HomR(R̃, I). In particular, any morphism

I → J of ideals of R restricts to a morphism cond(I)→ cond(J).

Lemma 2.32. — Assume that P is an ordinary singularity of Y and let ÕY ,P be the

normalisation of OY ,P . If P is a node, then the conductor of IY ,P is IY ,P = (x, y).


Otherwise m is at least 3 and the conductor is xmÕY ,P = (x, y)m. Moreover, the iso-

morphism ϕ : xmÕY ,P∼−→ xm−1ÕY ,P = (x, y)m−1 extends to an injective morphism

IY ,P ↪→ OY ,P .

Proof. — First, let m equal 2, i.e. P is a node. Then IY ,P equals (x, y) and Fulton’s

proof that (x, y) is the conductor of OY ,P also shows that (x, y) is the conductor of

IY ,P . Now let m be at least 3. A single blow-up resolves the singularity. Thus, ÕY ,Pis generated as OY ,P -module by t

j , j = 0, . . . ,m− 1, with xt = y [Ful02, Prop. 2.1].Consequently, xmÕY ,P = (x, y)

m is a subset of the conductor of IY ,P . Moreover

xm−1ÕY ,P equals the conductor (x, y)m−1 of OY ,P . It is then not difficult to see that

ϕ extends as claimed and is injective. So by the remarks above the lemma, the inverse

image of xm−1ÕY ,P under ϕ contains the conductor of IY ,P . Since the inverse image

is exactly xmÕY ,P , the claim is proven.

Lemma 2.33. — In the situation of the proof Theorem A, let P be a closed point

of X. There is the equality

isodXi,P (IX(X), ωX) = length(

(OX,P/


− length(

(IX,P/

cond(IX,P ))⊗ OXi,P),

Proof. — If P is not a multiple point of X, then all three cardinals in the equation are

equal to 0. So let P be a multiple point of X. Since ωX and OX(X) are both invertible

sheaves, isodXi,P (IX(X), ωX) equals isodXi,P (IX(X),OX(X)). The latter, in turn,

equals isodXi,P (IX ,OX). So let ϕ : IX,P → OX,P be a morphism such that theinduced morphism ϕ̃i : ˜IX,P ⊗ OXi,P → ˜OX,P ⊗ OXi,P = OXi,P is injective.

Note that IX,P ⊗ OXi,P is the OXi,P -ideal generated by ∂f/∂x, ∂f/∂y and themonomials of degree at least m. It is therefore torsion-free, so the restriction ϕi of ϕ

to Xi and ϕ̃i agree. Hence, there is the following exact commutative diagram.

0

��

cond(IX,P )⊗ OXi,P //

��

cond(OX,P )⊗ OXi,P //

��

Cokerϕi|cond(...) //

��

0

0 // IX,P ⊗ OXi,Pϕi

// OX,P ⊗ OXi,P // Cokerϕi // 0


By the snake lemma,

length (Cokerϕi) = length(

(OX,P/


− length(

(IX,P/

cond(IX,P ))⊗ OXi,P)

+ length(Cokerϕi|cond(...)

)If we let ϕ be the map from Lemma 2.32 (take ϕ as the natural inclusion if P is a

node), then ϕi is injective and the claim is proven.

Lemma 2.34. — Let P be a multiple point of Y . Then τ emP

(Y ) = m(m+ 1)/2− 2.

Proof. — Recall that τ emP

(Y ) is length(OY ,P /IY ,P ) and that the length equals the

dimension over k since P is rational. Moreover, IY ,P equals (∂f/∂x, ∂f/∂y)+(x, y)m.

Since P is an ordinary singularity, the degree m − 1 parts of ∂f/∂x and ∂f/∂y arelinearly independent over k (even in positive characteristic) and the claim follows.

Lemma 2.35. — In the situation of the proof Theorem A, let P be a multiple point

of X. Then

length(

(IX,P /cond(IX,P ))⊗ OXi,P)≤ min{mP (Xi), 2}.

Moreover, if mP (X) is 2, then the length is 0.

Proof. — If mP (X) equals 2, then the claim immediately follows from Lemma 2.32.

So let mP (X) be larger than 2. If mP (Xi) is 0, then OXi,P vanishes and so does the

length of (IX,P /cond(IX,P ))⊗ OXi,P .

So letmP (Xi) be at least 1. Since P is rational, the length of (IX,P /cond(IX,P ))⊗OXi,Pequals its dimension over k. By Lemma 2.32, the k-space is generated by the images

of ∂f/∂x and ∂f/∂y.

As remarked earlier, IX,P ⊗ OXi,P is the OXi,P -ideal generated by ∂f/∂x and∂f/∂y) + (x, y)mP (X). So if mP (Xi) equals 1, then ∂f/∂x and ∂f/∂y are con-

stant multiples of f′

(f′

as before a defining element of X′

at P ). In particular,

IX,P ⊗ OXi,P /cond(IX,P )⊗ OXi,P is generated by the image of f′.

CHAPTER 3

AN EXPLICIT DESCRIPTION OF FROBENIUS-STABLE

LATTICES

The zeta function of a smooth geometrically integral curve X over k is uniquely

determined by the Frobenius action on the first rigid cohomology group. It is therefore

of interest to have an explicit description of this group. Let X be a proper lifting of

X to W (k) and Z a nontrivial reduced strict relative normal crossing divisor on X.

Then there is the following description for large enough `:

H0(ΩXK ((`+ 1)ZK))/dH0(OXK (`ZK))

∼−→ H1dR(XK \ ZK)∼−→ H1rig(X \ Z). (3.1)

Indeed, this is a consequence of (1.1) and the Riemann-Roch theorem. The remarks

from Section 1.2 lead to the following natural and relevant question:(1)

Does there exist and can one describe a sequence of isomorphic,

Frobenius-stable lattices such that tensoring with K naturally yields(3.1)?

Two approaches to answer this question come to mind. In this chapter we look at

both of them and show that one succeeds while the other fails (and to what extent the

other fails). The successful approach, considered in Section 3.1, is via crystalline and

de Rham cohomology of the pair (X,Z). Crystalline cohomology is a contravariant

functor from pairs (X,Z) to W (k)-modules Hicr(X,Z) [Shi02, Chp. 2 & 3]. Thanks to

natural isomorphisms by Abott et al. [AKR11, Cor. 2.2.6], Kato [Kat89, Thm. 6.4],

and Shiho [Shi02, Cor. 2.4.13 & Thm. 3.1.1], there is the following commutative

diagram

H1dR(X,Z)⊗K∼ // H1dR(XK \ ZK)

∼ // H1rig(X \ Z)∼ // H1cr(X,Z)⊗K

H1dR(X,Z)∼ //

OO

H1cr(X,Z)

OO(3.2)

(1)A lattice in a vector space V of finite dimension n over K is a free sub-W (k)-module of V of

rank n [Bou89a, Sec. VII.4].

32 CHAPTER 3. AN EXPLICIT DESCRIPTION OF FROBENIUS-STABLE LATTICES

in which (by naturality) the top, right-hand map is equivariant with respect to the

Frobenius maps on crystalline and rigid cohomology. The main result of this section

is as follows: Let ϕ be the map sending a differential ω ∈ H0(ΩX((` + 1)Z)) to theformal sum of its Laurent expansions at the closed points in the support of Z, cut off

above t−2dt (where t is a uniformising parameter) and where the coefficient of tidt is

taken modulo i+ 1. Then there is a natural isomorphism

Kerϕ/dH0(OX(`Z))

∼−→ H1dR(X,Z)/

(tor).(2) (3.3)

The statement is made precise below (Theorem B).

The second approach to give an answer is via Monsky-Washnitzer and de Rham coho-

mology of X \Z. The question whether the image of H1dR(X \Z) in H1dR(XK \ZK) isa Frobenius-stable lattice was already asked by van den Bogaart in his thesis [vdB08,

p. 82]. In Section 3.2, we provide this module with a Frobenius action using an integral

definition of Monsky-Washnitzer cohomology (Definition 3.10 and Proposition 3.12).

We will then express the failure of the image of H1dR(X \ Z) to be a lattice in termsof the genus and the Hasse-Witt invariant of X (Proposition 3.13).

We end this overview of the chapter with the following remark: Let Ω(rf)XK

and Ω(rf)

Xbe

the spaces of residue-free differentials on XK respectively X. Let g be the genus of

X. The quotient of Ω(rf)XK

by exact differentials has dimension 2g, while the quotient

of Ω(rf)

Xby exact differentials is infinite-dimensional. There are two ways to get a

finite-dimensional quotient space in characteristic p: One can restrict to differentials

of the second kind [Ros53], i.e. differentials whose Laurent tails are integrable (in

characteristic 0 this notion coincides with being residue-free), or one can take the

quotient by the larger space of pseudo-exact differentials [Lam58]; a definition is

given below. The first approach loosely corresponds to de Rham cohomology of (X,Z)

and the second approach to de Rham cohomology of X \ Z.

Further notation and terminology. — The diagram (3.2) holds for any smooth

proper pair (X,Z) overW (k) and in any degree. Here, a smooth proper pair over W (k)

is a pair (X,Z) consisting of a smooth proper W (k)-scheme X of relative dimension

n and a reduced strict relative normal crossing divisor Z on X. The strict relative

normal crossing condition means that X can be covered by open affine subschemes U

such that each U is étale over AnW (k) and Z|U is the fibre product of U and the unionof some (or all, or none) coordinate hyperplanes. In other words, there exists an étale

map

W (k)[x1, . . . , xn]étale−→ OX(U)

xj 7→ tj

(2)One can show that H1dR(X,Z) is torsion-free (Lemma 3.7).

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 33

such that

OX(−Z)(U) = t1 · · · trOX(U),

for some 0 ≤ r ≤ n depending on U . We always assume that r in the above formula isminimal in the sense that no tj , j = 1, . . . , r, can be left out. For simplicity, we call U

together with the t1, . . . , tn a pleasant open affine of X and a pleasant neighbourhood

of any point P of U .

For the remainder of this chapter, let (X,Z) be a smooth proper pair over

W (k) where X is of relative dimension n and Z nontrivial.

3.1. De Rham cohomology of smooth proper pairs

This section is divided into three subsections. In the first we define differentials with

logarithmic poles and the de Rham cohomology of (X,Z). The middle subsection

contains Theorem B and its proof. The third part specialises to the case of curves.

Differentials with logarithmic poles. — To define the de Rham cohomology of

(X,Z) one introduces the sheaf Ω(X,Z) of differentials on X with logarithmic poles

along Z: Let U , t1, . . . , tn, be a pleasant open affine. Since U is étale over AnW (k), the

differentials dt1, . . . , dtn form an OX(U)-basis of ΩX(U). Set

d̃tjdf=

{dtj/tj if j ≤ rdtj otherwise.

The sections of Ω(X,Z) over U are defined as

Ω(X,Z)(U)df=

n⊕j=1

OX(U)d̃tj (3.4)

⊆ ΩX(Z)(U).

This definition glues and gives rise to a subsheaf Ω(X,Z) of ΩX(Z). We let

∂̃j : OX(U)→ OX(U) (3.5)

be the map sending an element f to the coefficient of d̃tj in df ∈ Ω(X,Z)(U).

As usual, we set Ωi(X,Z)df=∧i

�

Documents

Frobenius-stable lattices in rigid cohomology of curves€¦ · case the curve is a ne, the second cohomology group vanishes. To see this, use, e.g. the comparison theorems with crystalline