Resolution: classical, relative and weightedtemkin/lectures/basic_notions.pdf · (2) Transforms: one pullbacks X and subtracts a multiple of the exceptional divisor. (3) Choice of

Resolution: classical, relative and weighted

M. Temkin

May 21,28 2020

Basic notions seminar

M. Temkin (Hebrew University) Resolution: classical, relative and weighted 1 / 42

Classical resolution Introduction

History

Until a few years ago there was known an essentially uniquecanonical resolution of singularities of algebraic varieties (andother schemes) over a field k of characteristic, thoughdescriptions and (sometimes) combinatorial details differed.We refer to it as the classical resolution. A bit of history:(1) Hironaka (64) proved existence of resolution by a very complicated

method containing many constructions and implicit ideas. Latersome become not essential for the algorithm (e.g. normal flatnessand the Hilbert-Samuel function), others were refined by him andGiraud in 70ies (e.g the maximal contact),

(2) 80ies – early 90ies Villamayor and Bierstone-Milman constructedthe first canonical resolution via explicit resolution algorithms,

(3) Wlodarczyk (03) showed it is functorial for all smooth morphisms,other clarifications by Kollar and others.



Basic properties

A resolution of a variety X is a proper morphism f : Xres → X s.t.Xres is smooth and f is an isomorphism over the smooth locus Xsm.Functorial resolution satisfies Xres = Yres ×Y X for any smoothmorphism Y → X . It is automatically

(i) Equivariant for any group action, hence canonical (depends only onthe isomorphism class of X ).

(ii) Compatible with open embeddings, hence a local functorialconstruction automatically globalizes.

The only known construction is embedded: X is locally embeddedinto a manifold (i.e. a smooth variety) M. To the pair (M,X ) oneassociates a modification of manifolds f : Mres → M andXres ↪→ X ×M Mres is a certain transform of X under f .Functorial embedded resolution implies functorial non-embeddedone because an embedding X ↪→ M with minimal dim(M) isunique (étale) locally.



The context

Because of (étale) locality, can assume that M = An andX = V (f1, . . . ,fr ) is given by the vanishing of polynomialsfi(t1, . . . ,tn) or of the ideal IX = (f1, . . . ,fr ) ⊂ k [t1, . . . ,tn]. Thehypersurface case is obtained when r = 1, it already reveals allmain features.In notation we will use pairs (M,X ) or (M, IX ), where IX is theideal of functions on M vanishing on X , but all real work can bedone on local charts, and globalization is automatic.Moreover, essentially the same construction works forholomorphic functions in Cn or even formal series in kJt1, . . . ,tnK.Thus, throughout this talk the varieties M and X can be algebraicor complex analytic, whichever you prefer.



Main choices

The following choices are done in the classical resolution:(1) Class of modifications: the algorithm iteratively blows up

submanifolds of M.(2) Transforms: one pullbacks X and subtracts a multiple of the

exceptional divisor.(3) Choice of centers: the order d = d1 of I = IX at x ∈ M is a very

crude primary invariant. The algorithm is constructed by inductionon n = dim(M), so the actual invariant, whose maximal locus isblown up, is closer to (d1, . . . ,dn) with the lex order.

(4) The history: to avoid loops the algorithm encodes history in theiterated exceptional divisor E . The number s(x) of its componentsat x is another primary invariant.


Classical resolution Basic tools

Submanifold blow ups

The modification Mres → M is constructed as a sequence· · · → M2 → M1 → M0 = M, where Mi+1 = BlVi (Mi) is asubmanifold blow up whose center Vi ↪→ Mi is smooth.The submanifold blow up M ′ = BlV (M) replaces V by the normalbundle. We’ll also write M ′ = BlI(M), where I = IV .The pullback E = V ×M M ′ is the exceptional divisor.A local description: M = An = Spec(k [t1, . . . ,tn]), IV = (t1, . . . ,tr )and M ′ is covered by r charts M ′i with coordinates t

′j = tj for j = i

or j > r , and t ′j =tjti

otherwise. Also, E |M′i = V (ti).Similarly, one can blow up an arbitrary ideal on a scheme. LocallyX = Spec(A), I = (f1, . . . ,fr ) and BlI(X ) = Proj(⊕nIn) is covered bythe charts X ′i = Spec(A[

f1fi, . . . , frfi ]).

Usually, BlI(M) is singular, this is why one has to blow up onlysubmanifolds of M.



Examples

Example (A weighted blow up)

Let M = A2 = Spec(k [x , y ]), I = (x2, y), M ′ = BlI(M). Then the y -chartM ′y = Spec(k [

x2y , x , y ]) = Spec(k [x , y , z]/(x

2 − yz)) has a simpleorbifold singularity at 0.

Example (A submanifold blow up and resolution of an orbifold)

Let M = A3 = Spec(k [x , y , z]), f = x2 − yz, X = V (f ) a cone,V = (x , y , z), M ′ = BlV (M). Then X ×M M ′ = X ! ∪ 2E where E is theexceptional divisor and the strict transform X ! is smooth. E.g.f = (x ′y ′)2 − y ′2z ′ = y ′2(x ′2 − z ′) and E = V (y ′) on the y -chart M ′y .

The strict transform is X ! = BlV×M X (X ), but it is hard to control. Inembedded resolution one uses a controlled/principal transformX ′ = X ×M M ′ − nE , i.e. IX ′ = IX×M M′/InE .



Dream fails

The classical algorithm has a subtle inductive structure andencodes history of the process in the exceptional divisor.A dream algorithm would use an invariant of singularities of X sothat the maximal invariant locus V is smooth and the invariant ofthe transform of X on M ′ = BlV (M) drops. Then one would simplyblow up the new maximal locus, etc.Unfortunately, a dream canonical resolution does not exist:

Example (No progress.)

Let f = x2 − yzt and X = V (f ) in M = A4. By the S3-symmetry V = 0is the only smooth equivariant locus in Xsing, but M ′ = BlV (M) hascharts with X ! having the same singularity, e.g. in M ′y we havef = x ′2y ′2 − y ′(y ′z ′)(y ′t ′) = y ′2(x ′2 − y ′z ′t ′).A similar computation shows that blowing up the pinch point ofWhitney umbrella V (x2 − y2z) yields a pinch point again.



The boundary

After a blow up h : M ′ → M each point x ∈ E = V (t) has a godgiven coordinate t (unique up to a unit) coming from the history ofthe resolution. We will only use coordinate systems which includet .Inductively, for a sequence hi : Mi+1 → Mi we setEi+1 = h−1i (Ei) ∪ Ehi and call it the accumulated boundary of M.We always work with coordinates t1, . . . ,tn s.t. Vi = V (ti1 , . . . ,tir )and Ei = (t1 . . . tr ). So, Ei is an snc (simple normal crossings)divisor and Vi has simple normal crossings with Ei (lies in fewcomponents and is transversal to others).We call the boundary coordinates exceptional and denote thems1, . . . ,sr .



The role of the boundary

Good news:

Using canonical monomial coordinates decreases choices, makesthe construction more canonical, helps to avoid loops.Boundary can accumulate parts of I: we set I = ImonIpure, whereImon = (sl11 . . . s

lrr ) and Ipure is purely non-monomial.

Bad news/another side of the coin:Must treat E and monomial coordinates with a special care.Less possibilities for coordinates, centers must have snc with E .

RemarkMany technical complications of the classical algorithm are due to abad separation of regular and exceptional coordinates (e.g. in thenotion of order).



The order

One uses only a very weak singularity invariant: order ordx (I) atx ∈ M is the minimal d such that I * md+1x , e.g.ord0(x5 + y3z, y10) = 4.Let I = IX . Then ordx (I) ≤ 1 iff x ∈ X and the equality holds iff X iscontained a submanifold N of a smaller dimension – takeN = V (f ), where f ∈ I, ordx (f ) = 1.If ord(X ) ≥ d along V , then X ′ − dE is defined on M ′ = Bl(V (M))because each monomial is of degree n ≥ d and it accumulates tnion the chart M ′i .Moreover, it is easy to see that if ord(X ) = d along V , thenord(X ′ − dE) ≤ d along the preimage E of V .The main work will be when d = ord(I) = maxx∈X ordx (I), but forinductive reasons we cannot restrict to this case.



Marked ideals

We will have to remember the “karmic order”, that comes from theprevious (inductive) life and forbids transforms with a larger d .A marked ideal is I = (I,d), where d ∈ N is the weight.The singular locus is Ising = {x ∈ M| ordx (I) ≥ d}.A blow up h : M ′ = BlV (M)→ M is I-admissible if V ⊂ Ising, andas usual V is smooth and has snc with E .Then the transform is I ′ = (I′,d), where I′ = h−1(I)/IdE .A test sequence is a sequence of I i -admissible blow upsMn 99K M0 = M s.t. I i are iterative transforms of I.Marked ideals are equivalent, I ∼ J , if they have the same testsequences, in particular, Ising = J sing.One should view (I,d) as a weighted ideal. It is easy to see that(I,d) ∼ (In,nd). One sets (I,d)(J,e) = (IJ,e + d) and(I,d) + (J,e) = (Ie + Jd ,de) – associative up to equivalence.



Order reduction

An order reduction of I is a test sequence Mn 99K M with(In)sing = 0. So, it is just a maximal test sequence.The main case is d = ord(I) – the so-called maximal order case.Order reduction is the main block of classical resolution. Oneheavily uses equivalences I ∼ J (this is algebra, you do not wantto know the relation between highly non-reduced V (I) and V (J)).One more important sum trick: a test sequence for I + J isnothing else but a test sequence for both I and J . In particular,Ising ∩ J sing = (I + J )sing, and an order reduction of I + Jachieves that the (transforms of) I and J have disjoint singularloci and can be resolved further independently.


Classical resolution Advanced tools

Induction on dimension

The algorithm runs by induction on dimension. Here is the mainexample: if X = V (f ), ordx (X ) = d , I = ((f ),d) then one canchoose coordinates at x so that locally f = td + a2td−2 + · · ·+ ad ,where ai = ai(t2, . . . ,tn), t = t1. Can get a1 = 0 since d 6= 0 in k .Now comes the miracle:The maximal contact hypersurface H = V (t) contains Ising due tothe term dtd−1. We use that char(k) = 0 (or at least prime to d).Moreover, Ising = {x ∈ H| ord(ai) ≥ i for 2 ≤ i ≤ d}.The above is obvious, with a bit more work can prove that thesituation persists after admissible blow ups, i.e. (M, I) ∼ (H,C),where (C,d !) =

∑i(ai , i) = (

∑i a

d !/ii ,d !) is called the coefficients

ideal.We’ve restricted the order reduction problem for f to an equivalentone on a magic hypersurface H via weighted coefficients.


Classical resolution Advanced tools

Derivations

In general, everything except monomials (and combinatorialaspects) is conceptually defined through derivation idealsDn(I) = D ◦ · · · ◦ D(I), where D(I) is generated by the elements ofI and their derivatives:(1) ordx (I) is the minimal d such that Dd (Ix ) = Ox .(2) Maximal contact is any H = V (t), where t ∈ Dd−1(Ix ) is of order 1.(3) The coefficient ideal is just C(I) =

∑d−1i=0 (D

i (I))d!/(d−i).The main claims are:(1) (M, I) ∼ (H,C(I)|H),(2) The order reduction of (H,C(I)|H) is independent of H.

The latter is the most subtle point. One of solutions is calledHironaka’s trick – one introduces a slightly stronger equivalencerelation (more tests), and shows that the order reduction dependsonly on the (narrow) equivalence class.


Classical resolution Complications

The boundary

Problem 1: if E 6= ∅, then it does not have to be transversal to H,and in this case E |H is not a boundary. So, we loose a mechanismto ensure that centers have snc with E .Solution: components coming from blow ups with centers on Hare transversal to the (strict) transform of H, so only the oldboundary needs a care. Let s(x) be the number of components ofEold at x , then the snc condition is automatic if s = s(x) is maximalalong V . By the sum trick, we can simply apply the order reductionto I + IdE(s). This reduces the invariant (d , s) with the lex order.


Classical resolution Complications

Non-maximal order

Problem 2: one might have ord(C(I)|H) > d !, e.g. for I = (x2 + y3)on H = V (x). So, have to deal with general marked ideals andthen no maximal contact exists.Solution: reduce to the maximal order case via I = ImonIpure.(1) If d ′ = ord(Ipure) ≥ d simply reduce the order of G(I) := (Ipure,d ′).(2) If 0 < d ′ < d we want to reduce d ′ but have to restrict to the locus

where ord(I) ≥ d , that is, ord(Imon) ≥ d − d ′. By the sum trick,enough to reduce the order of G(I) := (Ipure,d ′) + (Imon,d − d ′).

(3) If d ′ = 0 resolve (I,d) = (Imon,d) combinatorially.


Classical resolution The algorithm

Induction on dimension

It is time to tie everything together.The non-boundary maximal order case: (I,d) ∼ (C(I)|H ,d !), therhs is resolved by induction on dim(M) and blowing up the samecenters in M reduces d .The general maximal order case: run induction on the number ofold boundary components s, on each step resolve thenon-boundary maximal order (I + IdE(s),d)

The general case: reduce d = ord(Ipure) by resolving the maximalorder marked ideal G(I). If d = 0 apply the combinatorial step toI = Imon.So, up to induction on dimension the work is done in two cycles –the outer one reduces d = ord(Ipure), the inner one reduces s. Thesingle primary invariant is (d , s) with the lex order.



The algorithm and its invariant

The whole algorithm is obtained just by iterating the reductiondimension step e = dim(M)− dim(X ) times.So, it consists of 2e cycles and the invariant is (d0, s0,d1, s1, . . . )where (di , si) is the invariant of the i-th coefficient ideal Ci on thei-th maximal contact Hi , and H0 = M, C0 = I.The sequence ends either with d = 0 and then the monomial stepis applied (it has a combinatorial invariant of its own), or d =∞.The latter means that X = M and IX = 0; in this case we stop.The classical principalization algorithm modifies M so that thepullback of I becomes an snc divisor. It proceeds precisely onemore step – one blows up He when d =∞.The invariant just records our place in the nested cycle, its s-partis not a meaningful invariant of the singularity, but the sequence(d1, . . . ,de) is.



Complexity

The complexity is horrible for the following reasons.The invariant is only lex ordered.Each coefficient ideal at the very least replaces d by d ! (could uselcm(1,2, . . . ,d) instead, but this is not a real release).The number of charts also growthes exponentially.

In addition, the marking d is determined by I = I0, so one usuallyworks with non-maximal order on Hi with i > 0 and makesnon-efficient transforms. This results in combinatorial stepsperformed all the time.



Example

Let us consider the case of Whitney umbrellaX = V (x2 − y2z) ⊂ M = A3. Recall that blowing up (x , y , z) creates apinch point again on the z-chart.

(0) Enough to resolve C = (y2z,2) on H = V (x).(1) G(C) = (C,3) is resolved by blowing up (y , z) (we omit the

computation on H2). So, H ′ = Bl(y ,z)(H) and on the z-chartC′ = (z ′−2(y ′2z ′3)) = (y ′2z ′).

(2) C′pure = (y ′2), so G(C′) = (y ′2,2) and we simply blow up y ′. ThenH ′′ = Bl(y ′)(H ′) = H ′ but C′′ = y ′′−2y ′′2z ′′ = z ′′ is resolved.

(3) The corresponding order reduction of I = (x2 − y2z) blows up(x , y , z) first. On the z-chart the pinch point reappears asI′ = (x ′2 − y ′2z ′), but the history forces us to blow up (x ′, y ′) thistime, and this produces a resolution, e.g. X ′′ = V (x ′′2 − z ′) on they -chart.


Logarithmic algorithms Introduction

Why log resolution?

Log resolution was constructed in 2017,2020, j/w Abramovich andWlodarczyk: it resolves morphisms, provides better functorialityand clarifies the role of log geometry in resolution.The algorithm is faster, simpler and more functorial even forresolution of varieties (absolute setting), but the tools are morecomplicated: log varieties and orbifolds.The main philosophy of the project was that one has to replacesmoothness by log smoothness, both for resolution andfunctoriality. In [ATW17] such resolution of (log) varieties wasconstructed. As was expected, the same algorithm works in therelative setting, [ATW20].For technical simplicity we’ll outline the algorithm for log varieties.The idea is to transfer the classical algorithm to the log setting.



Why log smoothness?

Example (No smooth resolution of morphisms)

The element π = xy induces a morphism f : N = A2 → B = A1 withf−1(0) = V (xy). By Zariski’s connectedness theorem, for anymodification g : N ′ → N = A2 the set g−1(0) is connected. So,f ′ : N ′ → B has connected reducible f ′−1(0), hence f ′ is not smooth.

In fact, the best situation one can hope for is when N is smooth,and f is semistable: locally given by π = t1 . . . tr for somecoordinates t1, . . . ,td . In particular, f−1(0) is an snc divisor.A more general and robust notion is of a log smooth f . If N issmooth this means that locally π = tm11 . . . t

mrr .

Informally, the non-smoothness of f is of monomial (orcombinatorial) nature. Loosely speaking, constructing a solidcategorical envelope for such examples is one of the main goalsof toroidal and log geometries.



Previous results

The principalization has the following extremely important corollary:

Theorem (Resolution of closed subsets)If N is a manifold and Z ⊂ N is closed, then there exists a functorialmodification f : N ′ → N s.t. N ′ is a manifold and f−1(Z ) is an sncd.

CorollaryIf B is a smooth curve of char 0, X variety, f : X → B dominant, thenthere exists a modification X ′ → X such that f ′ : X ′ → B is log smooth.

Proof.Replace X by Xres and apply the theorem to the singular fibers of f .

This was the only known relative desingularization result.[KKMS] deduces semistable reduction theorem by a heavycombinatorics: after a finite B′ → B, f ′ can be made semistable.


Logarithmic algorithms Log varieties

What is the boundary?

One often confuses embedded subscheme X and the boundaryE , though they play very different roles as hinted already byfunctoriality: for (N ′,E ′)→ (N,E) one typically has no mapE ′ → E . Conversely, there is an embedding f−1(E) ↪→ E ′.Note that E is determined by the sheaf of monoidsMN =MN(log E) = O×N\E ∩ ON ⊂ (ON , ·) of functions on Ninvertible outside of E . We get the right functorialityf−1MN →MN′ . Note also that one uses the monoidMN ratherthan the subvariety E in the principalization.Locally and non-canonicallyMN = O×N × N

r , and the splitting isgiven by fixing exceptional coordinates s1, . . . ,sr .



Logarithmic varieties

DefinitionA log variety (X ,MX ) consists of a variety X and a log structureαX : MX → (OX , ·), whereM×X = α

−1X (O

×X ). A morphism is a

compatible pair f : X ′ → X and f ∗MX →MX ′ .

We’ll only takeMX =MX (logD) ⊆ OX for a divisor D, i.e. MXconsists of functions invertible on X \ D. Morphisms in this caseare just f : X ′ → X s.t. f−1(D) ↪→ D′.Many constructions extend to log geometry, e.g. the sheaf of logdifferentials Ω(X ,MX ) is generated by ΩX and symbolsδm = d log(m) for m ∈ MX subject to relations dm = m · δm.One also naturally defines log smooth morphisms, they havelocally free sheaves of relative differentials of expected rank.



Log manifolds

Log smooth varieties or log manifolds (or toroidal varieties) étale(analytically or formally) locally are modeled by the charts

X = AdM = Spec(k [M][t1, . . . ,td ]),

where M is the monoid of integral points in a rational polyhedralcone (i.e a torsion free fs monoid).The log structureMX is induced by M and Ω(X ,M) is freelygenerated by dti and δmi , where {mi} is any basis of Mgp.Note thatMX =MX (log D), where D = ∪m∈MV (m) is the toricdivisor.We view t1, . . . ,td as regular coordinates (subject to a choice) andall elements of M as monomial coordinates (unique up to units).In particular, M does not have to be free anymore and there is nocanonical base of Mgp. “Monomial democracy”: no element ofM \M× is more equal than another one.



Log smooth morphisms

Log smooth morphisms of log manifolds (or toroidal morphisms) inchar=0 are (étale, formally, analytically) locally modelled on toric maps

AnN = Spec(k [N][t1, . . . ,tn])→ AmM = Spec(k [M][t1, . . . ,tm])

induced by M ↪→ N.

Example

(i) Semistable maps, e.g. X = A2 → A1 = B given by π = xayb for thelog structures xN × yN and πN. Here ΩlogX/B is free with basisδx = −baδy .(ii) Kummer log-étale covers: M ⊆ N ⊆ 1d M and r = n, Ω

logX/B = 0. The

map is finite, usually non-flat, e.g. Spec k [x , y ]→ Spec k [x2, xy , y2] withthe log structures of monomials in x , y .



Some remarks

Remark(i) In a sense, toroidal geometry studies log smooth log varieties andmorphisms.(ii) Kummer log étale covers are obtained by extracting roots of themonomial coordinates, e.g. in the classical setting they include covers(X ′,D′)→ (X ,D) where X is smooth, D is an sncd, f : X ′ → X isramified along D and D′ = f−1(D).(iii) The most interesting feature of the new algorithm is functorialityw.r.t. log smooth morphisms, including Kummer log étale covers. Thisis out of reach and unnatural for the classical approach.


Logarithmic algorithms Main results and the method

Main results

Ignoring an orbifold aspect, our main result is:

Theorem (Log principalization)

Given a log manifold X with an ideal I ⊂ OX there exists a sequence ofadmissible blow ups of log manifoldss Xn → · · · → X such that theideal IOXn is monomial. This sequence is compatible with log smoothmorphisms X ′ → X.

As in the classical situation this implies non-embedded resolution

Theorem (Log resolution)

For any integral log variety Z there exists a modification Zres → Z suchthat Zres is log smooth. This is functorial w.r.t. log smooth morphismsZ ′ → Z.



The main ingredients

In brief, we want to log-adjust all parts of the classical algorithm. Themain idea: instead of DX use the sheaf D

logX of log derivations, which is

generated by ∂ti and δm = m∂m.

(1) logordx (I) is the minimal d such that (DlogX )

d (Ix ) = Ox .(2) Maximal contact is any H = V (t), where t ∈ (DlogX )

d−1(Ix ) is aregular coordinate. In particular, in the classical case of a smoothX , a non-empty E is allowed, but E |H is a boundary.

(3) The coefficient ideal C(I) is just∑d−1

i=0 ((DlogX )

i(I))d!/(d−i).(4) In addition, there is a combinatorial novelty: we blow up

submonoimal centers J = (t1, . . . ,tl ,m1, . . . ,mr ) for any set ofmonomials. This is forced (see below) and fine, as X ′ = BlJ(X ) islog smooth! An instance of monomial democracy.

(5) J is (I,d)-admissible if I ⊆ Jd . Note that this agrees with theclassical case. The transform of (I,d) is (I′,d) whereI′ = (IOX ′)(JOX ′)−d = h−1I/IdE .



Infinite log order

The main novelty: logord(ti) = 1, but logord(m) =∞, since m is aneigenvector of DlogX . Looks horrible at first, but enables functorialityw.r.t. extracting roots of monomials (e.g. Kummer covers).The price to pay: a special monomial initial cleaning step whenlogord(I) =∞. In fact, it is a single (!) blow up. (It exists in theclassical case, but only when X = H, so IX |H = 0.)Fact (Kollar): M(I) = D∞X (I) is the minimal monomial idealcontaining I. Blowing upM(I) and dividing by its pullback makeslogord finite.Example: if I = (

∑i∈Nr mi t

i) thenM(I) = (mi).Monomial step forces us to blow up any monomial center oniterated max contact, hence by inductive karma we have to blowup general submonomial centers.



The main algorithm: log order reduction

Log order reduction of a marked ideal (I,d) on a log manifold X :(1) Initial cleaning: blow upM(I).(2) The main loop: the splitting I = ImonIpure is defined, where Imon is

invertible monomial and e = logord(Ipure) d .)The invariant is just (d0, . . . ,dn) with d = d0,d1, . . . ,dn−1 ∈ d + Nand dn ∈ {0,∞}. On Hn one performs a cleaning step (initial orfinal).


Logarithmic algorithms Orbifolds

The main complication

Is all this so elementary? Where is the cheating?Well. All monomials are “born equal”. The algorithm does notknow whether m = nd because of log smooth functoriality, butachieves admissibility by using Kummer monomials m1/d . In fact,the initial cleaning blows upM(I)1/d because of the karmic d .At first glance, by functoriality we can work Kummer étale locally –on a finite G-Galois cover where Xd = X [M

1/dX ] (in fact, G ⊂ µ

rd is

abelian). We, indeed, use this to formalize the notion of Kummerideals J, e.g. (t ,m1/d ), their derivatives, etc. – J is an actual idealon Xd . However, to describe the algorithm via modifications of Xwe have to descend from X ′d = BlJ(Xd ) to a Kummer blow upX ′ = X ′d/G of X and ... X

′ does not have to be log smooth.Solution: X ′ = [X ′d/G] is an orbifold! The functorial algorithmproduces a log smooth orbifold. Mild singularities of the underlyingvariety can be easily resolved, but only in a smooth-functorial way.



An example

We had to extend the smooth context once again, this time, by allowingorbifoldic Kummer blow ups. Here is the simplest example:

Example

(i) X = Spec(k [t ,m]) and I = (t2 −m2). Then logord(I) = 2, H = V (t),C(I) = (m2,2), the order reduction of C(I)|H blows up (m2)1/2 = (m),and the order reduction of I is X ′ = Bl(t ,m)(X ). As in the classical case.(ii) Y = Spec(k [t ,n]) and I = (t2 − n). Then logord(I) = 2 (!!!),H = V (t), C(I) = (n,2), the order reduction of C(I)|H blows up (n1/2),and the order reduction of I is Y ′ = [Bl(t ,n1/2)(Y )]. Note that Y

′ isnon-representable and the underlying variety Bl(t2,n)(X ) is not logsmooth.(i)⇐⇒(ii) n = m2 yields a Kummer cover X → Y , then X ′ = X ×Y Y ′ bythe functoriality and Y ′ = [X ′/µ2].



Relative resolution

Morphisms Z → B of log varieties are resolved by embedding Zinto X which is log smooth over B and principalizing IZ on X .Principalization is done in the same way using DlogX/B, e.g. relativeregular coordinates are defined by logordX/B(t) = 1.

Main novelty: functions on B are killed by DlogX/B, so one has tomodify B to make enough of them monomials (depending on IZ ).

Theorem (Resolution of morphisms)There exists a procedure associating to dominant f : Z → B an orbifoldmodification g : Zres → Z or a fail value “modify B” s.t. fres : Zres → B islog smooth, g is compatible with log smooth maps Z ′ → Z andarbitrary base changes B′ → B, and for any f there exists amodification B1 → B s.t. the resolution of f ×B B1 → B1 does not fail.


Weighted algorithms The dream algorithm

Weighted blow ups

The log algorithm forces us to extend the toolkit: Kummer blow upof (t1, . . . ,tn,m

1/d1 , . . . ,m

1/dr ) can be viewed as a blow up of (t ,m)

with weights (1, . . . ,1,d , . . . ,d). The weights are special becausewe work with marked ideals and the same karmic d = d0.It is now natural to rethink everything even in the classical case ofa manifold M without log structure. In fact, blow ups of (t1, . . . ,tn)with weights (w1, . . . ,wn) were occasionally used for decades,especially in the quasi-homogeneous case. But one getsBl

(tw/w11 ,...,t

w/wnn )

(M), where w = w1 . . .wn, which is often singular.

Example: the minimal resolution of x2 + y3 + z6 = 0 is done byblowing up (x , y , z) with weights (3,2,1).



Refined weighted blow ups

As in the log case, a refined weighted blow up M ′ of M along(t1, . . . ,tn) with weights (w1, . . . ,wn) can be defined as a smoothorbifold whose underlying space is the usual weighted blow upBl

(tw/w11 ,...,t

w/wnn )

(M).

One should think of M ′ as the blow up along a generalized idealJ = (t1/w11 , . . . ,t

1/wnn ) (can be formalized, e.g. in h-topology). A

formal definition: extract the roots, blow up, then divide by theGalois group. The pullback of J is an invertible ideal, similarly tothe usual blow ups.Another possibility is to define it as a stacky Proj of a graded ringAJ associated with J. It is defined as the usual Proj, but onedivides by Gm stack-theoretically. This is a manifold if AJ isgenerated in degree 1, but is a smooth orbifold with stabilizers µnin general.



The dream algorithm

Let M be a manifold with an ideal I. An I-admissible center isJ = (xd11 , . . . ,x

dnn )

nor ⊇ I with d1 ≤ d2 ≤ . . . . The associated blowup h : M ′ → M is along (x1/w11 , . . . ,x

1/wnn ), where wi =

∏j 6=i dj .

The transform is I′ = h−1(I) · h−1(J)−1.inv(I) is the maximal such d = (d1, . . . ,dn) with the lex order.

Theorem ([ATW19])There is a unique I-admissible J with the maximal value of theinvariant. If M ′ = BlJ(M) and I′ is the transform, then inv(I′) < inv(I).

CorollaryIn the realm of orbifolds and weighted blow ups there existsmooth-functorial dream algorithms for principalization and resolution.

Iterate the unique blow up above, resolution follows as usual.M. Temkin (Hebrew University) Resolution: classical, relative and weighted 39 / 42


Not only the algorithm has no history, it has no log structures. Thelast exceptional divisor is smooth, but the iterated one does nothave to be snc (not even have smooth components).The formulation is simple, but the proof is not and has to use thatchar=0. We used the usual theory of max contact to get a shortproof.In fact the invariant always was there: if H0 = M, Hi+1 = Hi(ti = 0)are iterated max contacts, Ii+1 = C(Ii)|Hi+1 anddi+1 = ord(Ii+1)/(di − 1)!, then J is generated by powers oft1, . . . ,tn and the weights are inverse to di .Thus, we use the same “best adapted” coordinate system. Themain difference is that we blow up with correct weights, and notwith the same weight 1, . . . ,1 associated to the degrees d , . . . ,d .This makes all monomial steps redundant (as well as the notion ofmonomial).The complexity improves drastically, but is still horrible – the lexorder and factorials are still there.



Example

Example

For the Whitney umbrella f = x2 − y2z = 0 in M = A3 the invariant is(2,3,3), J = (x2, y3, z3), the weighted center is (x1/3, y1/2, z1/2). Thez chart has coordinates z ′ = z1/2, x ′ = x/z ′3, y ′ = y/z ′2, and sincef = z ′6(x ′2 − y ′2) the transform is f ′ = x ′2 − y ′2 and its invariant (2,2)is smaller.

RemarkAny quasi-homogeneous polynomial is resolved similarly. Moregenerally, the geometric meaning is that we choose coordinates andweights which provide best possible quasi-homogeneousapproximation and reduce its order. This can be also described aschoosing a minimal face of an appropriate Newton polyhedron andreducing it by a weighted blow up (to be worked out in detail.)


Weighted algorithms Logarithmic setting

Logarithmic dream algorithm

And what about log structures? Can one combine the dreamalgorithm and the logarithmic algorithms?Surely yes: one should work with log smooth manifolds X (or logsmooth morphisms X → B), and principalize ideals on X byblowing up arbitrary weighted centers of the form(t1/w11 , . . . ,t

1/wnn ,m

1/v11 , . . . ,m

1/vrr ) using log order, log derivations,

log maximal contact and log coefficients ideals.For log varieties this algorithm was constructed by Ming Hao Quek(PhD under supervision of Abramovich). The relative case isplanned to be worked out next.


Classical resolutionIntroductionBasic toolsAdvanced toolsComplicationsThe algorithm

Logarithmic algorithmsIntroductionLog varietiesMain results and the methodOrbifolds

Weighted algorithmsThe dream algorithmLogarithmic setting

Documents

Resolution: classical, relative and weightedtemkin/lectures/basic_notions.pdf · (2) Transforms: one pullbacks X and subtracts a multiple of the exceptional divisor. (3) Choice of