American Mathematical Society

Compactifying the space of stable maps

By Dan Abramovich and Angelo Vistoli

Abstract

In this paper we study a notion of twisted stable map, from a curve to a tame Deligne-Mumford stack, which generalizes the well-known notion of stable map to a projective variety.

1. Introduction

We fix a noetherian base scheme double-struck upper S .

1.1. The problem of moduli of families

Consider a Deligne-Mumford stack script upper M (definition in Section 2.1) admitting a projective coarse moduli scheme bold upper M . Given a curve upper C , it is often natural to consider morphisms f colon upper C right-arrow script upper M (or equivalently, objects f element-of script upper M left-parenthesis upper C right-parenthesis ); in case script upper M is the moduli stack of geometric objects, these morphisms correspond to families over upper C . For example, if script upper M equals script upper M overbar Subscript gamma , the stack of stable curves of genus gamma , then morphisms f colon upper C right-arrow script upper M correspond to families of stable curves of genus gamma over upper C ; and if script upper M equals script upper B upper G , the classifying stack of a finite group upper G , we get principal upper G -bundles over upper C . It is interesting to study moduli of such objects; moreover, it is natural to study such moduli as upper C varies and to find a natural compactification for such moduli.

One approach is suggested by Kontsevich’s moduli of stable maps.

1.2. Stable maps

First consider a projective scheme bold upper M right-arrow double-struck upper S with a fixed ample sheaf script upper O Subscript bold upper M Baseline left-parenthesis 1 right-parenthesis . Given integers g comma n comma d , it is known that there exists a proper algebraic stack, which here we denote by script upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis , of stable, n -pointed maps of genus g and degree d into bold upper M . (See ReferenceKo, ReferenceB-M, ReferenceF-P, Reference normal alef -O, where the notation ModifyingAbove script upper M With bar Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis is used. Here we tried to avoid using script upper M comma bold upper M with two meanings.) This stack admits a projective coarse moduli space bold upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis . If one avoids “small” residue characteristics in double-struck upper S , which depend on g comma n comma d and bold upper M , then this stack is in fact a proper Deligne-Mumford stack.

1.3. Stable maps into stacks

Now fix a proper Deligne-Mumford stack script upper M right-arrow double-struck upper S admitting a projective coarse moduli space bold upper M right-arrow double-struck upper S on which we fix an ample sheaf as above. We further assume that script upper M is tame, that is, for any geometric point s colon upper S p e c normal upper Omega right-arrow script upper M , the group upper A u t Subscript upper S p e c normal upper Omega Baseline left-parenthesis s right-parenthesis has order prime to the characteristic of the algebraically closed field normal upper Omega .

It is tempting to mimic Kontsevich’s construction as follows: Let upper C be a nodal projective connected curve; then a morphism upper C right-arrow script upper M is said to be a stable map of degree d if the associated morphism to the coarse moduli scheme upper C right-arrow bold upper M is a stable map of degree d .

It follows from our results below that the category of stable maps into script upper M is a Deligne-Mumford stack. A somewhat surprising point is that it is not complete.

To see this, we fix g equals 2 and consider the specific case of script upper B upper G with upper G equals left-parenthesis double-struck upper Z slash 3 double-struck upper Z right-parenthesis Superscript 4 . Any smooth curve upper C of genus 2 admits a connected principal upper G -bundle, corresponding to a surjection upper H 1 left-parenthesis upper C comma double-struck upper Z right-parenthesis right-arrow upper G , thus giving a map upper C right-arrow script upper B upper G . If we let upper C degenerate to a nodal curve upper C 0 of geometric genus 1 , then upper H 1 left-parenthesis upper C 0 comma double-struck upper Z right-parenthesis asymptotically-equals double-struck upper Z cubed , and since there is no surjection double-struck upper Z cubed right-arrow upper G , there is no connected principal upper G -bundle over upper C 0 . This means that there can be no limiting stable map upper C 0 right-arrow script upper B upper G as a degeneration of upper C right-arrow script upper B upper G .

1.4. Twisted stable maps

Our main goal here is to correct this deficiency. In order to do so, we will enlarge the category of stable maps into script upper M . The source curve script upper C of a new stable map script upper C right-arrow script upper M will acquire an orbispace structure at its nodes. Specifically, we endow it with the structure of a Deligne-Mumford stack.

It is not hard to see how these orbispace structures come about. Let upper S be the spectrum of a discrete valuation ring upper R of pure characteristic 0, with quotient field upper K , and let upper C Subscript upper K Baseline right-arrow eta element-of upper S be a nodal curve over the generic point, together with a map upper C Subscript upper K Baseline right-arrow script upper M of degree d , whose associated map upper C Subscript upper K Baseline right-arrow bold upper M is stable. We can exploit the fact that script upper K Subscript g comma 0 Baseline left-parenthesis bold upper M comma d right-parenthesis is complete; after a ramified base change on upper S the induced map upper C Subscript upper K Baseline right-arrow bold upper M will extend to a stable map upper C right-arrow bold upper M over upper S . Let upper C Subscript s m be the smooth locus of the morphism upper C right-arrow upper S ; Abhyankar’s lemma, plus a fundamental purity lemma (see Lemma 2.4.1 below) shows that after a suitable base change we can extend the map upper C Subscript upper K Baseline right-arrow script upper M to a map upper C Subscript s m Baseline right-arrow script upper M ; in fact the purity lemma fails to apply only at the “new” nodes of the central fiber, namely those which are not in the closure of nodes in the generic fiber. On the other hand, if p element-of upper C is such a node, then on an étale neighborhood upper U of p , the curve upper C looks like

u v equals t Superscript r Baseline comma

where t is the parameter on the base. By taking r -th roots,

u equals u 1 Superscript r Baseline comma v equals v 1 Superscript r Baseline comma

we have a nonsingular cover upper V 0 right-arrow upper U where upper V 0 is defined by u 1 v 1 equals t . The purity lemma applies to upper V 0 , so the composition upper V 0 Subscript upper K Baseline right-arrow upper C Subscript upper K Baseline right-arrow script upper M extends over all of upper V 0 . There is a minimal intermediate cover upper V 0 right-arrow upper V right-arrow upper U such that the family extends already over upper V ; this upper V will be of the form x y equals t Superscript r slash m , and the map upper V right-arrow upper U is given by u equals x Superscript m , v equals y Superscript m . Furthermore, there is an action of the group bold-italic mu Subscript m of roots of 1, under which alpha element-of bold-italic mu Subscript m sends x to alpha x and y to alpha Superscript negative 1 Baseline y , and upper V slash bold-italic mu Subscript m Baseline equals upper U . This gives the orbispace structure script upper C over upper C , and the map upper C Subscript upper K Baseline right-arrow script upper M extends to a map script upper C right-arrow script upper M .

This gives the flavor of our definition.

We define a category script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis , fibered over script upper S c h slash double-struck upper S , of twisted stable n -pointed maps script upper C right-arrow script upper M of genus g and degree d . This category is given in two equivalent realizations: one as a category of stable twisted script upper M -valued objects over nodal pointed curves endowed with atlases of orbispace charts (see Definition 3.7.2); the other as a category of representable maps from pointed nodal Deligne-Mumford stacks into script upper M , such that the map on coarse moduli spaces is stable (see Definition 4.3.1). In our treatment, both realizations are used in proving our main theorem:

Theorem 1.4.1

(1)

The category script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis is a proper algebraic stack.

(2)

The coarse moduli space bold upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis of script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis is projective.

(3)

There is a commutative diagram StartLayout 1st Row 1st Column script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis 2nd Column right-arrow 3rd Column script upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 3rd Row 1st Column bold upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis 2nd Column right-arrow 3rd Column bold upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis EndLayout

where the top arrow is proper, quasifinite, relatively of Deligne-Mumford type and tame, and the bottom arrow is finite. In particular, if script upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis is a Deligne-Mumford stack, then so is script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis .

1.5. Some applications and directions of further work

(1)

In our paper Reference normal alef -V2 we studied the situation where script upper M equals script upper M overbar Subscript gamma , which gives a complete moduli for fibered surfaces. Some further results for elliptic surfaces are obtained in G. La Nave’s thesis ReferenceLa.

(2)

The case where script upper M is the classifying stack of a finite group upper G allows one to improve on the spaces of admissible covers, give moduli compactifications of spaces of curves with abelian and nonabelian level structures, and, with a suitable choice of the group upper G , show that there is a smooth, fine moduli space for admissible upper G -covers, which is a finite covering of script upper M overbar Subscript g . This is the subject of our preprint Reference normal alef -C-V with Alessio Corti, with some ideas contributed by Johan de Jong. Some of these applications were indicated in our announcement Reference normal alef -V1. This approach to admissible covers is closely related to the work of Wewers ReferenceWe.

(3)

A similar reasoning applies to curves with r -spin structures, e.g. theta characteristics. This is studied in Reference normal alef -J.

(4)

The recursive nature of the theorem allows one to construct both minimal models and stable reduction for pluri-fibered varieties. This is related to recent work of Mochizuki ReferenceMo and deserves further study.

(5)

In ReferenceC-R, W. Chen and Y. Ruan introduce Gromov-Witten invariants of an orbifold, using a differential geometric counterpart of our stack of twisted stable maps. In our joint work Reference normal alef -G-V with Tom Graber we give an algebraic treatment of these Gromov-Witten invariants, and in the special case of 3-pointed genus 0 maps of degree 0, construct the Chen-Ruan product with integer coefficients. See also ReferenceF-G for an algebraic treatment of global quotients.

(6)

In this paper we verify that script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis is a proper stack by going through the conditions one by one. It may be worthwhile to develop a theory of Grothendieck Quot-stacks and deduce our results from such a theory. It seems likely that some of our methods could be useful for developing such a theory.

While our paper was circulating, we were told in 1999 by Maxim Kontsevich that he had also discovered the stack of twisted stable maps, but had not written down the theory. His motivation was in the direction of Gromov-Witten invariants of stacks.

1.6. Acknowledgments

We would like to thank Kai Behrend, Larry Breen, Barbara Fantechi, Ofer Gabber, Johan de Jong, Maxim Kontsevich, and Rahul Pandharipande, for helpful discussions. We are grateful to Laurent Moret-Bailly for providing us with a preprint of the book ReferenceL-MB before it appeared. The first author thanks the Max Planck Institute für Mathematik in Bonn for a visiting period which helped in putting this paper together.

2. Generalities on stacks

2.1. Criteria for a Deligne-Mumford stack

We refer the reader to ReferenceAr and ReferenceL-MB for a general discussion of algebraic stacks (generalizing ReferenceD-M), and to the appendix in ReferenceVi for an introduction. We spell out the conditions here, as we follow them closely in the paper. We are given a category script upper X along with a functor script upper X right-arrow script upper S c h slash double-struck upper S . We assume

(1)

script upper X right-arrow script upper S c h slash double-struck upper S is fibered in groupoids (see ReferenceAr, §1, (a) and (b), or ReferenceL-MB, Definition 2.1). This means:

(a)

for any morphism of schemes upper T right-arrow upper T prime and any object xi prime element-of script upper X left-parenthesis upper T prime right-parenthesis there is an object xi element-of script upper X left-parenthesis upper T right-parenthesis and an arrow xi right-arrow xi prime over upper T right-arrow upper T prime ; and

(b)

for any diagram of schemes StartLayout 1st Row 1st Column upper T 1 2nd Column Blank 3rd Column long right-arrow 4th Column Blank 5th Column upper T 2 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column Blank 4th Column down left-arrow 5th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper T 3 4th Column Blank 5th Column Blank EndLayout

and any objects xi Subscript i Baseline element-of script upper X left-parenthesis upper T Subscript i Baseline right-parenthesis sitting in a compatible diagram StartLayout 1st Row 1st Column xi 1 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column xi 2 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column Blank 4th Column down left-arrow 5th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column xi 3 4th Column Blank 5th Column Blank EndLayout

there is a unique arrow xi 1 right-arrow xi 2 over upper T 1 right-arrow upper T 2 making the diagram commutative.

We remark that this condition is automatic for moduli problems, where script upper X is a category of families with morphisms given by fiber diagrams.

(2)

script upper X right-arrow script upper S c h slash double-struck upper S is a stack, namely:

(a)

the upper I s o m functors are sheaves in the étale topology; and

(b)

any étale descent datum for objects of script upper X is effective.

See ReferenceAr, 1.1, or ReferenceL-MB, Definition 3.1.

(3)

The stack script upper X right-arrow script upper S c h slash double-struck upper S is algebraic, namely:

(a)

the upper I s o m functors are representable by separated algebraic spaces of finite type; and

(b)

there is a scheme upper X , locally of finite type, and a smooth and surjective morphism upper X right-arrow script upper X .

See ReferenceL-MB, Definition 4.1. Notice that (3a) implies (2a).

These last two conditions are often the most difficult to verify. For the last one, M. Artin has devised a set of criteria for constructing upper X right-arrow script upper X by algebraization of formal deformation spaces (see ReferenceAr, Corollary 5.2). Thus, in case double-struck upper S is of finite type over a field or an excellent Dedekind domain, condition (3b) holds if

(A)

script upper X is limit preserving (see ReferenceAr, §1);

(B)

script upper X is compatible with formal completions (see ReferenceAr, 5.2 (3));

(C)

Schlessinger’s conditions for pro-representability of the deformation functors hold (see ReferenceAr, (2.2) and (2.5)); and

(D)

there exists an obstruction theory for script upper X (see ReferenceAr, (2.6)) such that

(i)

the deformation and obstruction theory is compatible with étale localization (ReferenceAr, 4.1 (i));

(ii)

the deformation theory is compatible with formal completions (ReferenceAr, 4.1 (ii)); and

(iii)

the deformation and obstruction theory is constructible (ReferenceAr, 4.1 (iii)).

Furthermore, we say that script upper X is a Deligne-Mumford stack if we can choose upper X right-arrow script upper X as in (3b) to be étale. This holds if and only if the diagonal script upper X right-arrow script upper X times script upper X is unramified (ReferenceL-MB, Théorème 8.1). A morphism script upper X right-arrow script upper X 1 is of Deligne-Mumford type if for any scheme upper Y and morphism upper Y right-arrow script upper X 1 the stack upper Y times Subscript script upper X 1 Baseline script upper X is a Deligne-Mumford stack.

For the notion of properness of an algebraic stack see ReferenceL-MB, Chapter 7. Thus a stack script upper X right-arrow double-struck upper S is proper if it is separated, of finite type and universally closed. In ReferenceL-MB, Remarque 7.11.2, it is noted that the weak valuative criterion for properness using traits might be insufficient for properness. However, in case script upper X has finite diagonal, it is shown in ReferenceE-H-K-V, Theorem 2.7, that there exists a finite surjective morphism from a scheme upper Y right-arrow script upper X . In such a case the usual weak valuative criterion suffices (ReferenceL-MB, Proposition 7.12).

2.2. Coarse moduli spaces

Recall the following result:

Theorem 2.2.1 (Keel-Mori ReferenceK-M).

Let script upper X be an algebraic stack with finite diagonal over a scheme upper S . There exists an algebraic space bold upper X and a morphism script upper X right-arrow bold upper X such that

(1)

script upper X right-arrow bold upper X is proper and quasifinite;

(2)

if k is an algebraically closed field, then script upper X left-parenthesis k right-parenthesis slash upper I s o m right-arrow bold upper X left-parenthesis k right-parenthesis is a bijection;

(3)

whenever upper Y right-arrow upper S is an algebraic space and script upper X right-arrow upper Y is a morphism, then the morphism factors uniquely as script upper X right-arrow bold upper X right-arrow upper Y ; more generally

(4)

whenever upper S prime right-arrow upper S is a flat morphism of schemes, and whenever upper Y right-arrow upper S prime is an algebraic space and script upper X times Subscript upper S Baseline upper S Superscript prime Baseline right-arrow upper Y is a morphism, then the morphism factors uniquely as script upper X times Subscript upper S Baseline upper S Superscript prime Baseline right-arrow bold upper X times Subscript upper S Baseline upper S Superscript prime Baseline right-arrow upper Y ; in particular

(5)

pi Subscript asterisk Baseline script upper O Subscript script upper X Baseline equals script upper O Subscript bold upper X .

Recall that an algebraic space bold upper X along with a morphism pi colon script upper X right-arrow bold upper X satisfying properties (2) and (3) is called a coarse moduli space (or just moduli space). In particular, the theorem of Keel and Mori shows that coarse moduli spaces of algebraic stacks with finite diagonal exist. Moreover, from (4) and (5) above we have that the formation of a coarse moduli space behaves well under flat base change:

Lemma 2.2.2

Let script upper X right-arrow upper X be a proper quasifinite morphism, where script upper X is a Deligne-Mumford stack and upper X is a noetherian scheme. Let upper X prime right-arrow upper X be a flat morphism of schemes, and denote script upper X prime equals upper X prime times Subscript upper X Baseline script upper X .

(1)

If upper X is the moduli space of script upper X , then upper X prime is the moduli space of script upper X prime .

(2)

If upper X prime right-arrow upper X is also surjective and upper X prime is the moduli space of script upper X prime , then upper X is the moduli space of script upper X .

Proof.

Given a proper quasifinite morphism pi colon script upper X right-arrow upper X , it then exhibits upper X as a moduli space if and only if pi Subscript asterisk Baseline script upper O Subscript script upper X Baseline equals script upper O Subscript upper X . If upper R right paired arrows upper U is an étale presentation of script upper X , and f colon upper U right-arrow upper X and g colon upper R right-arrow upper X are the induced morphisms, then this condition is equivalent to the exactness of the sequence

0 long right-arrow script upper O Subscript upper X Baseline long right-arrow f Subscript asterisk Baseline script upper O Subscript upper U Baseline right paired arrows g Subscript asterisk Baseline script upper O Subscript upper R Baseline period

From this the statement follows.

The prototypical example of a moduli space is given by a group quotient: Let upper V be a scheme and normal upper Gamma a finite group acting on upper V . Consider the stack left-bracket upper V slash normal upper Gamma right-bracket ; see ReferenceL-MB, 2.4.2. The morphism left-bracket upper V slash normal upper Gamma right-bracket right-arrow upper V slash normal upper Gamma exhibits the quotient space upper V slash normal upper Gamma as the moduli space of the stack left-bracket upper V slash normal upper Gamma right-bracket . The following well-known lemma shows that étale-locally, the moduli space of any Deligne-Mumford stack is of this form.

Lemma 2.2.3

Let script upper X be a separated Deligne-Mumford stack, and bold upper X its coarse moduli space. There is an étale covering StartSet bold upper X Subscript alpha Baseline right-arrow bold upper X EndSet , such that for each alpha there is a scheme upper U Subscript alpha and a finite group normal upper Gamma Subscript alpha acting on upper U Subscript alpha , with the property that the pullback script upper X times Subscript bold upper X Baseline bold upper X Subscript alpha is isomorphic to the stack-theoretic quotient left-bracket upper U Subscript alpha Baseline slash normal upper Gamma Subscript alpha Baseline right-bracket .

Sketch of proof.

Let x 0 be a geometric point of bold upper X . Denote by bold upper X Superscript s h the spectrum of the strict henselization of bold upper X at the point x 0 , and let script upper X Superscript s h Baseline equals script upper X times Subscript bold upper X Baseline bold upper X Superscript s h . If upper V right-arrow script upper X is an étale morphism, with upper V a scheme, having x 0 in its image, there is a component upper U of the pullback upper V times Subscript bold upper X Baseline bold upper X Superscript s h which is finite over bold upper X Superscript s h . Denote upper R equals upper U times Subscript script upper X Sub Superscript s h Baseline upper U . We have that under the first projection upper R right-arrow upper U , the scheme upper R splits as a disjoint union of copies of upper U . Let normal upper Gamma be the set of connected components of upper R , so that upper R is isomorphic to upper U times normal upper Gamma . Then the product upper R times Subscript upper U Baseline upper R right-arrow upper R induces a group structure on normal upper Gamma , and the second projection upper R asymptotically-equals upper U times normal upper Gamma right-arrow upper U defines a group action of normal upper Gamma on upper U , such that script upper X Superscript s h is the quotient upper U slash normal upper Gamma .

We need to descend from script upper X Superscript s h to get the statement on script upper X . This follows from standard limit arguments.

2.3. Tame stacks and their coarse moduli spaces

Definition 2.3.1

(1)

A Deligne-Mumford stack script upper X is said to be tame if for any geometric point s colon upper S p e c normal upper Omega right-arrow script upper X , the group upper A u t Subscript upper S p e c normal upper Omega Baseline left-parenthesis s right-parenthesis has order prime to the characteristic of the algebraically closed field normal upper Omega .

(2)

A morphism script upper X right-arrow script upper X 1 of algebraic stacks is said to be tame if for any scheme upper Y and morphism upper Y right-arrow script upper X 1 the stack upper Y times Subscript script upper X 1 Baseline script upper X is a tame Deligne-Mumford stack.

A closely related notion is the following:

Definition 2.3.2

An action of a finite group normal upper Gamma on a scheme upper V is said to be tame if for any geometric point s colon upper S p e c normal upper Omega right-arrow upper V , the group upper S t a b left-parenthesis s right-parenthesis has order prime to the characteristic of normal upper Omega .

The reader can verify that a separated Deligne-Mumford stack is tame if and only if the actions of the groups normal upper Gamma Subscript alpha on upper V Subscript alpha in Lemma 2.2.3 are tame.

In case script upper X is tame, the formation of coarse moduli spaces commutes with arbitrary morphisms:

Lemma 2.3.3

Let script upper X be a tame Deligne-Mumford stack, script upper X right-arrow bold upper X its moduli space. If bold upper X prime right-arrow bold upper X is any morphism of schemes, then bold upper X prime is the moduli space of the fiber product bold upper X prime times Subscript bold upper X Baseline script upper X . Moreover, if bold upper X prime is reduced, then it is also the moduli space of left-parenthesis bold upper X prime times Subscript bold upper X Baseline script upper X right-parenthesis Subscript r e d .

Proof.

By Lemma 2.2.2, this is a local condition in the étale topology of bold upper X , so we may assume that script upper X is a quotient stack of type left-bracket upper V slash normal upper Gamma right-bracket , where normal upper Gamma is a finite group acting on an affine scheme upper V equals upper S p e c upper R . Moreover, since script upper X is tame, we may assume that the order of normal upper Gamma is prime to all residue characteristics. Then bold upper X equals upper S p e c upper R Superscript normal upper Gamma ; if bold upper X prime equals upper S p e c upper S , then the statement is equivalent to the map upper S right-arrow left-parenthesis upper R circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S right-parenthesis Superscript normal upper Gamma being an isomorphism. This (well-known) fact can be shown as follows: Recall that for any upper R Superscript normal upper Gamma Baseline left-bracket normal upper Gamma right-bracket -module upper M the homomorphism

StartLayout 1st Row 1st Column upper M 2nd Column long right-arrow Overscript upper A v Subscript upper M Endscripts 3rd Column upper M Superscript normal upper Gamma 2nd Row 1st Column y 2nd Column right-arrow from bar 3rd Column StartFraction 1 Over StartAbsoluteValue normal upper Gamma EndAbsoluteValue EndFraction sigma-summation Underscript gamma element-of normal upper Gamma Endscripts gamma dot y EndLayout

is a projector exhibiting upper M Superscript normal upper Gamma as a direct summand in upper M . Thus the induced morphism

upper A v Subscript upper R Baseline circled-times upper I d Subscript upper S Baseline colon upper R circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S right-arrow upper R Superscript normal upper Gamma Baseline circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S equals upper S

shows that upper S right-arrow left-parenthesis upper R circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S right-parenthesis Superscript normal upper Gamma is injective. The morphism upper A v Subscript upper R Baseline circled-times upper I d Subscript upper S is a lifting of

upper A v Subscript upper R circled-times Sub Subscript upper R Sub Sub Superscript normal upper Gamma Sub Subscript Subscript upper S Baseline colon upper R circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S right-arrow left-parenthesis upper R circled-times Subscript upper R Sub Superscript normal upper Gamma Subscript Baseline upper S right-parenthesis Superscript normal upper Gamma Baseline comma

which is surjective.

This shows that bold upper X prime is the moduli space of the fiber product bold upper X prime times Subscript bold upper X Baseline script upper X . The statement about left-parenthesis bold upper X prime times Subscript bold upper X Baseline script upper X right-parenthesis Subscript r e d is immediate. This proves the result.

Let script upper X be a separated tame stack with coarse moduli scheme bold upper X . Consider the projection pi colon script upper X right-arrow bold upper X . The functor pi Subscript asterisk carries sheaves of script upper O Subscript script upper X -modules to sheaves of script upper O Subscript bold upper X -modules.

Lemma 2.3.4

The functor pi Subscript asterisk carries quasicoherent sheaves to quasicoherent sheaves, coherent sheaves to coherent sheaves, and is exact.

Proof.

The question is local in the étale topology on bold upper X , so we may assume that script upper X is of the form left-bracket upper V slash normal upper Gamma right-bracket , where upper V is a scheme and normal upper Gamma a finite group of order prime to all residue characteristics, in particular bold upper X equals upper V slash normal upper Gamma . Now sheaves on script upper X correspond to equivariant sheaves on upper V . Denote by q colon upper V right-arrow bold upper X the projection. If script upper E is a sheaf on script upper X corresponding to a normal upper Gamma -equivariant sheaf script upper E overTilde on upper V , then pi Subscript asterisk Baseline script upper E equals left-parenthesis q Subscript asterisk Baseline script upper E overTilde right-parenthesis Superscript normal upper Gamma , which, by the tameness assumption, is a direct summand in q Subscript asterisk Baseline script upper E overTilde . From this the statement follows.

2.4. Purity lemma

We recall the following purity lemma from Reference normal alef -V2:

Lemma 2.4.1

Let script upper M be a separated Deligne-Mumford stack, script upper M right-arrow bold upper M the coarse moduli space. Let upper X be a separated scheme of dimension 2 satisfying Serre’s condition upper S 2 . Let upper P subset-of upper X be a finite subset consisting of closed points, upper U equals upper X minus upper P . Assume that the local fundamental groups of upper U around the points of upper P are trivial.

Let f colon upper X right-arrow bold upper M be a morphism. Suppose there is a lifting f overTilde Subscript upper U Baseline colon upper U right-arrow script upper M :

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartLayout EndLayout EndLayout

Then the lifting extends to upper X :

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel StartLayout EndLayout EndLayout

and f overTilde is unique up to a unique isomorphism.

Proof.

By the descent axiom for script upper M (see 2.1 (2)) the problem is local in the étale topology, so we may replace upper X and bold upper M with the spectra of their strict henselizations at a geometric point; then we can also assume that we have a universal deformation space upper V right-arrow script upper M which is finite. Now upper U is the complement of the closed point, upper U maps to script upper M , and the pullback of upper V to upper U is finite and étale, so it has a section, because upper U is simply connected; consider the corresponding map upper U right-arrow upper V . Let upper Y be the scheme-theoretic closure of the graph of this map in upper X times Subscript bold upper M Baseline upper V . Then upper Y right-arrow upper X is finite and is an isomorphism on upper U . Since upper X satisfies upper S 2 , the morphism upper Y right-arrow upper X is an isomorphism.

Remark 2.4.2

The reader can verify that the statement and proof work in higher dimension. See also related lemmas in ReferenceMo.

Corollary 2.4.3

Let upper X be a smooth surface over a field, p element-of upper X a closed point with complement upper U . Let upper X right-arrow bold upper M and upper U right-arrow script upper M be as in the purity lemma. Then there is a lifting upper X right-arrow script upper M .

Corollary 2.4.4

Let upper X be a normal crossings surface over a field k , namely a surface which is étale locally isomorphic to upper S p e c k left-bracket u comma v comma t right-bracket slash left-parenthesis u v right-parenthesis . Let p element-of upper X be a closed point with complement upper U . Let upper X right-arrow bold upper M and upper U right-arrow script upper M be as in the purity lemma. Then there is a lifting upper X right-arrow script upper M .

Proof.

In both cases upper X satisfies condition upper S 2 and the local fundamental group around p is trivial, hence the purity lemma applies.

2.5. Descent of equivariant objects

Lemma 2.5.1

Let upper R be a local ring with residue field k , let upper U equals upper S p e c upper R , u 0 equals upper S p e c k , let script upper M be a Deligne-Mumford stack, and let eta be an object of script upper M left-parenthesis upper U right-parenthesis . Assume we have a pair of compatible actions of a finite group normal upper Gamma on upper R and on eta , in such a way that the induced actions of normal upper Gamma on k and on the pullback eta 0 equals eta vertical-bar Subscript u 0 Baseline are trivial. Then there exists an object eta prime of script upper M on the quotient upper U slash normal upper Gamma equals upper S p e c left-parenthesis upper R Superscript normal upper Gamma Baseline right-parenthesis , and a normal upper Gamma -invariant lifting eta right-arrow eta prime of the projection upper U right-arrow upper U slash normal upper Gamma . Furthermore, if eta double-prime is another such object over upper U slash normal upper Gamma , there is a unique isomorphism eta prime asymptotically-equals eta double-prime over the identity of upper U slash normal upper Gamma , which is compatible with the two arrows eta right-arrow eta prime and eta right-arrow eta double-prime .

As a consequence of the unicity statement, suppose that we have a triple left-parenthesis alpha comma beta comma gamma right-parenthesis , where gamma colon normal upper Gamma asymptotically-equals normal upper Gamma is a group isomorphism, and alpha colon eta asymptotically-equals eta and beta colon upper U asymptotically-equals upper U are compatible gamma -equivariant isomorphisms. Then the given arrow eta right-arrow eta prime and its composition with alpha both satisfy the conditions of the lemma, so there is an induced isomorphism alpha overbar colon eta prime asymptotically-equals eta prime .

Corollary 2.5.2

Let upper R comma upper U comma k comma u 0 comma eta comma eta 0 be as in the previous lemma. Let upper G be a finite group acting compatibly on upper R and on eta . Let normal upper Gamma be the normal subgroup of upper G consisting of elements acting on k and eta 0 as the identity. Then there exist a upper G slash normal upper Gamma -equivariant object eta prime on the quotient upper U slash normal upper Gamma , and a upper G -equivariant arrow eta right-arrow eta prime compatible with the projection upper U right-arrow upper U slash normal upper Gamma .

Proof of the corollary.

The action is defined as follows. If g is an element of upper G , call alpha colon eta asymptotically-equals eta and beta colon upper U asymptotically-equals upper U the induced arrows, and gamma colon normal upper Gamma asymptotically-equals normal upper Gamma the conjugation by g . Then the image of g in upper G slash normal upper Gamma acts on eta prime via the isomorphism alpha overbar colon eta prime asymptotically-equals eta double-prime defined above. One checks easily that this defines an action with the required properties.

Proof of the lemma.

First note that if upper R Superscript s h is the strict henselization of upper R , the condition on the action of normal upper Gamma allows one to lift it to upper R Superscript s h . Also, the statement that we are trying to prove is local in the étale topology, so by standard limit arguments we can assume that upper R is strictly henselian. Replacing bold upper M by the spectrum of the strict henselization of its local ring at the image of the closed point of upper R , we can assume that script upper M is of the form left-bracket upper V slash upper H right-bracket , where upper V is a scheme and upper H is a finite group. Then the object eta corresponds to a principal upper H -bundle upper P right-arrow upper U , on which normal upper Gamma acts compatibly with the action of normal upper Gamma on upper U , and an upper H -equivariant and normal upper Gamma -invariant morphism upper P right-arrow upper V . Since upper U is strictly henselian, the bundle upper P right-arrow upper U is trivial, so upper P is a disjoint union of copies of upper S p e c upper R , and the group normal upper Gamma permutes these copies; furthermore the hypothesis on the action of normal upper Gamma on the closed fiber over the residue field insures that normal upper Gamma sends each component into itself. The thesis follows easily.

We note that Lemma 2.5.1 can be proven for an arbitrary algebraic stack script upper M as an application of the notion of relative moduli spaces, which we did not discuss here. Briefly, the assumptions of the lemma give a morphism left-bracket upper U slash normal upper Gamma right-bracket right-arrow script upper M . Using Lemma 2.2.2, it can be shown that there is a relative coarse moduli space, namely a representable morphism script upper V right-arrow script upper M such that script upper V times Subscript script upper M Baseline upper Y gives the coarse moduli space of left-bracket upper U slash normal upper Gamma right-bracket times Subscript script upper M Baseline upper Y for any scheme upper Y and flat morphism upper Y right-arrow script upper M . The fact that normal upper Gamma acts trivially on eta 0 , together with Lemmas 2.2.2 and 4.4.3, imply that script upper V equals upper U slash normal upper Gamma .

3. Twisted objects

Our goal in this section is to introduce the notion of a stable twisted object. This is a representable script upper M -valued object on a suitable atlas of orbispace charts on a nodal curve. Basic charts have the form left-bracket upper S p e c k left-bracket xi comma eta right-bracket slash bold-italic mu Subscript r Baseline right-bracket at a node, or left-bracket upper S p e c k left-bracket xi right-bracket slash bold-italic mu Subscript r Baseline right-bracket along a marking, where bold-italic mu Subscript r acts freely away from the origin; for completeness of the picture we allow more general charts. The category of stable twisted objects is a concrete, yet somewhat technical, incarnation of our stack script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis , which is convenient in many steps of our proof of Theorem 1.4.1.

3.1. Divisorially marked curves

The following definition is a local version of the standard definition of pointed curve; its advantage is that it is stable under localization in the étale topology.

Definition 3.1.1

A divisorially n -marked nodal curve, or simply n -marked curve left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis , consists of a nodal curve pi colon upper U right-arrow upper S , together with a sequence of n pairwise disjoint closed subschemes normal upper Sigma 1 comma period period period comma normal upper Sigma Subscript n Baseline subset-of upper U whose supports do not contain any of the singular points of the fibers of pi , and such that the projections normal upper Sigma Subscript i Baseline right-arrow upper S are étale. (Any of the subschemes normal upper Sigma Subscript i may be empty.)

If more than one curve is considered, we will often use the notation normal upper Sigma Subscript i Superscript upper U to specify the curve upper U . On the other hand, we will often omit the subschemes normal upper Sigma Subscript i Superscript upper U from the notation left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Superscript upper U Baseline right-parenthesis if there is no risk of confusion.

A nodal n -pointed curve upper C right-arrow upper S is considered an n -marked curve by taking as the normal upper Sigma Subscript i Superscript upper C the images of the sections upper S right-arrow upper C .

Definition 3.1.2

If left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis is an n -marked nodal curve, we define the special locus of upper U , denoted by upper U Subscript sp , to be the union of the normal upper Sigma Subscript i with the singular locus of the projection upper U right-arrow upper S , with its natural scheme structure (this makes the projection upper U Subscript sp Baseline right-arrow upper S unramified). The complement of upper U Subscript sp will be called the general locus of upper U , and denoted by upper U Subscript gen .

Definition 3.1.3

If left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Superscript upper U Baseline right-parenthesis is a marked curve, and upper S prime right-arrow upper S is an arbitrary morphism, we define the pullback to be left-parenthesis upper U prime right-arrow upper S Superscript prime Baseline comma normal upper Sigma Subscript i Superscript upper U prime Baseline right-parenthesis , where upper U prime equals upper S prime times Subscript upper S Baseline upper U and normal upper Sigma Subscript i Superscript upper U prime Baseline equals upper S prime times Subscript upper S Baseline normal upper Sigma Subscript i Superscript upper U .

Definition 3.1.4

If upper U right-arrow upper S and upper V right-arrow upper S are n -marked curves, a morphism of n -marked curves f colon upper U right-arrow upper V is a morphism of upper S -schemes which sends each normal upper Sigma Subscript i Superscript upper U into normal upper Sigma Subscript i Superscript upper V .

A morphism of n -marked curves f colon upper U right-arrow upper V is called strict if the support of f Superscript negative 1 Baseline left-parenthesis normal upper Sigma Subscript i Superscript upper V Baseline right-parenthesis coincides with the support of normal upper Sigma Subscript i Superscript upper U for all i equals 1 comma period period period comma n , and similarly for the singular locus.

We notice that if a morphism of marked curves upper U right-arrow upper V is strict, then there is an induced morphism of curves upper U Subscript gen Baseline right-arrow upper V Subscript gen . Furthermore, if f colon upper U right-arrow upper V is strict and étale, then f Superscript negative 1 Baseline left-parenthesis normal upper Sigma Subscript i Superscript upper V Baseline right-parenthesis equals normal upper Sigma Subscript i Superscript upper U scheme-theoretically.

Definition 3.1.5

Let left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis be an n -marked curve and normal upper Gamma a finite group. An action of normal upper Gamma on left-parenthesis upper U comma normal upper Sigma Subscript i Baseline right-parenthesis is an action of normal upper Gamma on upper U as an upper S -scheme, such that each element on normal upper Gamma acts via an automorphism, in the sense of Definition 3.1.4, of upper U as a marked curve on upper S .

If normal upper Gamma is a finite group along with a tame action on a marked curve upper U right-arrow upper S , then the quotient upper U slash normal upper Gamma right-arrow upper S can be given a marked curve structure by defining normal upper Sigma Subscript i Superscript upper U slash normal upper Gamma Baseline colon equals normal upper Sigma Subscript i Superscript upper U slash normal upper Gamma subset-of-or-equal-to upper U slash normal upper Gamma . The latter inclusion holds because the orders of stabilizers in normal upper Gamma are assumed to be prime to the residue characteristics, so normal upper Sigma Subscript i Superscript upper U Baseline slash normal upper Gamma is indeed a subscheme of upper U slash normal upper Gamma .

Given a morphism f colon upper U right-arrow upper V of marked curves, and a tame action of a finite group normal upper Gamma on upper U , leaving f invariant, then there is an induced morphism upper U slash normal upper Gamma right-arrow upper V of marked curves.

Definition 3.1.6

Let left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis be an n -marked curve, with an action of a finite group normal upper Gamma , and let script upper M be a Deligne-Mumford stack. Given eta element-of script upper M left-parenthesis upper U right-parenthesis , an essential action of normal upper Gamma on left-parenthesis eta comma upper U right-parenthesis is a pair of compatible actions of normal upper Gamma on eta and on left-parenthesis upper U right-arrow upper S comma normal upper Sigma Subscript i Baseline right-parenthesis , with the property that if g is an element of normal upper Gamma different from the identity and u 0 is a geometric point of upper U fixed by g , then the automorphism of the pullback of eta to u 0 induced by g is not trivial.

3.2. Generic objects and charts

Definition 3.2.1

Let upper C right-arrow upper S be an n -pointed nodal curve. A generic object on upper C is an object of script upper M left-parenthesis upper C Subscript gen Baseline right-parenthesis .

We will often write left-parenthesis xi comma upper C right-parenthesis for a generic object xi on a curve upper C .

Definition 3.2.2

Let upper C right-arrow upper S be an n -pointed nodal curve and xi a generic object on upper C . A chart left-parenthesis upper U comma eta comma normal upper Gamma right-parenthesis for xi consists of the following collection of data.

(1)

An n -marked curve upper U right-arrow upper S and a strict morphism phi colon upper U right-arrow upper C .

(2)

An object eta of script upper M left-parenthesis upper U right-parenthesis .

(3)

An arrow eta vertical-bar Subscript upper U Sub Subscript gen Subscript Baseline xi in script upper M compatible with the restriction phi vertical-bar Subscript upper U Sub Subscript gen Subscript Baseline colon upper U Subscript gen Baseline right-arrow upper C Subscript gen Baseline .

(4)

A finite group normal upper Gamma .

(5)

A tame, essential action of normal upper Gamma on left-parenthesis eta comma upper U right-parenthesis .

Furthermore, we require that the following conditions be satisfied.

a

The actions of normal upper Gamma leave the morphism upper U right-arrow upper C and the arrow eta vertical-bar Subscript upper U Sub Subscript gen Subscript Baseline xi invariant.

b

The induced morphism upper U slash normal upper Gamma right-arrow upper C is étale.

The following gives a local description of the action of normal upper Gamma .

Proposition 3.2.3

Let left-parenthesis upper U comma eta comma normal upper Gamma right-parenthesis be a chart for a generic object xi on a pointed nodal curve upper C right-arrow upper S . Then the action of normal upper Gamma on phi Superscript negative 1 Baseline upper C Subscript gen is free.

Furthermore, if s 0 is a geometric point of upper S and u 0 a nodal point of the fiber upper U Subscript s 0 of upper U over s 0 , then

(1)

the stabilizer normal upper Gamma prime of u 0 is a cyclic group which sends each of the branches of upper U Subscript s 0 to itself;

(2)

if k is the order of normal upper Gamma prime , then a generator of normal upper Gamma prime acts on the tangent space of each branch by multiplication with a primitive k -th root of 1 .

In particular, each nodal point of upper U Subscript s 0 is sent to a nodal point of upper C Subscript s 0 .

Proof.

The first statement follows from the definition of an essential action and the invariance of the arrow eta vertical-bar Subscript upper U Sub Subscript gen Subscript Baseline upper E .

As for (1), observe that if the stabilizer normal upper Gamma prime of u 0 did not preserve the branches of upper U Subscript s 0 , then the quotient upper U Subscript s 0 Baseline slash normal upper Gamma prime , which is étale at the point u 0 over the fiber upper C Subscript s 0 , would be smooth over upper S at u 0 , so u 0 would be in the inverse image of upper C Subscript gen . From the first part of the proposition it would follow that normal upper Gamma prime is trivial, a contradiction.

So normal upper Gamma prime acts on each of the two branches individually. The action on each branch must be faithful because it is free on the complement of the set of nodes; this means that the representation of normal upper Gamma prime in each of the tangent spaces to the branches is faithful, and this implies the final statement.

Definition 3.2.4

A chart is called balanced if for any nodal point of any geometric fiber of upper U , the two roots of 1 describing the action of a generator of the stabilizer on the tangent spaces to each branch of upper U are inverse to each other.

3.3. The transition scheme

Let xi be a generic object over a nodal curve upper C right-arrow upper S , and left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis , left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis two charts; call pr Subscript i Baseline colon upper U 1 times Subscript upper C Baseline upper U 2 right-arrow upper U Subscript i Baseline the i -th projection. Consider the scheme

upper I equals upper I s o m Underscript upper U 1 times Subscript upper C Baseline upper U 2 Endscripts left-parenthesis pr Subscript 1 Superscript asterisk Baseline eta 1 comma pr Subscript 2 Superscript asterisk Baseline eta 2 right-parenthesis

over upper U 1 times Subscript upper C Baseline upper U 2 representing the functor of isomorphisms of the two objects pr Subscript 1 Superscript asterisk Baseline eta 1 and pr Subscript 2 Superscript asterisk Baseline eta 2 .

There is a section of upper I over the inverse image upper U overTilde of upper C Subscript gen in upper U 1 times Subscript upper C Baseline upper U 2 which corresponds to the isomorphism pr Subscript 1 Superscript asterisk Baseline eta 1 StartAbsoluteValue asymptotically-equals pr Subscript 2 Superscript asterisk Baseline eta 2 EndAbsoluteValue Subscript upper U overTilde Baseline Subscript upper U overTilde coming from the fact that both pr Subscript 1 Superscript asterisk Baseline eta 1 and pr Subscript 2 Superscript asterisk Baseline eta 2 are pullbacks to upper U overTilde of xi . We will call the scheme-theoretic closure upper R of this section in upper I the transition scheme from left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis to left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis ; it comes equipped with two projections upper R right-arrow upper U 1 and upper R right-arrow upper U 2 .

There is also an action of normal upper Gamma 1 times normal upper Gamma 2 on upper I , defined as follows. Let left-parenthesis gamma 1 comma gamma 2 right-parenthesis element-of normal upper Gamma 1 times normal upper Gamma 2 , and let phi colon pr Subscript 1 Superscript asterisk Baseline eta 1 asymptotically-equals pr Subscript 2 Superscript asterisk Baseline eta 2 be an isomorphism over upper U 1 times Subscript upper C Baseline upper U 2 ; then define left-parenthesis gamma 1 comma gamma 2 right-parenthesis dot phi equals gamma 2 ring phi ring gamma 1 Superscript negative 1 . This action of normal upper Gamma 1 times normal upper Gamma 2 on upper I is compatible with the action of normal upper Gamma 1 times normal upper Gamma 2 on upper U 1 times Subscript upper C Baseline upper U 2 , and leaves upper R invariant. It follows from the definition of an essential action that the action of normal upper Gamma 1 equals normal upper Gamma 1 times StartSet 1 EndSet and normal upper Gamma 2 equals StartSet 1 EndSet times normal upper Gamma 2 on upper I is free.

3.4. Compatibility of charts

Definition 3.4.1

Two charts left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis and left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis are compatible if their transition scheme upper R is étale over upper U 1 and upper U 2 .

Let us analyze this definition. First of all, upper R is obviously étale over left-parenthesis upper U 1 right-parenthesis Subscript gen and left-parenthesis upper U 2 right-parenthesis Subscript gen . Also, since the maps upper U Subscript j Baseline right-arrow upper C are strict, it is clear that the inverse image of normal upper Sigma Subscript i Superscript upper U 1 in upper R is set-theoretically equal to the inverse image of normal upper Sigma Subscript i Superscript upper U 2 . If the two charts are compatible, this also holds scheme-theoretically.

Now, start from two charts left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis and left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis . Fix two geometric points

u 1 colon upper S p e c normal upper Omega right-arrow upper U 1 and u 2 colon upper S p e c normal upper Omega right-arrow upper U 2

mapping to the same geometric point u 0 colon upper S p e c normal upper Omega right-arrow upper C , and call normal upper Gamma prime Subscript j Baseline subset-of normal upper Gamma Subscript j the stabilizer of u Subscript j . Also call upper U 1 Superscript s h , upper U 2 Superscript s h and upper C Superscript s h the spectra of the strict henselizations of upper U 1 , upper U 2 and upper C at the points u 1 comma u 2 and u 0 respectively. The action of normal upper Gamma Subscript j on upper U Subscript j induces an action of normal upper Gamma prime Subscript j on upper U Subscript j Superscript s h . Also call eta Subscript j Superscript s h the pullback of eta Subscript j to upper U Subscript j Superscript s h ; there is an action of normal upper Gamma prime Subscript j on eta Subscript j Superscript s h compatible with the action of normal upper Gamma prime Subscript j on upper U Subscript j . The following essentially says that two charts are compatible if and only if for any choice of u 1 and u 2 the two charts are locally isomorphic in the étale topology.

Proposition 3.4.2

The two charts are compatible if and only if for any pair of geometric points u 1 and u 2 as above there exist an isomorphism of groups theta colon normal upper Gamma prime 1 asymptotically-equals normal upper Gamma prime 2 , a theta -equivariant isomorphism phi colon upper U 1 Superscript s h Baseline asymptotically-equals upper U 2 Superscript s h of schemes over upper C Superscript s h , and a compatible theta -equivariant isomorphism psi colon eta 1 Superscript s h Baseline right-arrow eta 2 Superscript s h .

Proof.

Consider the spectrum left-parenthesis upper U 1 times Subscript upper C Baseline upper U 2 right-parenthesis Superscript s h of the strict henselization of upper U 1 times Subscript upper C Baseline upper U 2 at the point left-parenthesis u 1 comma u 2 right-parenthesis colon upper S p e c normal upper Omega right-arrow upper U 1 times Subscript upper C Baseline upper U 2 , and call upper R Superscript s h the pullback of upper R to left-parenthesis upper U 1 times Subscript upper C Baseline upper U 2 right-parenthesis Superscript s h . Assume that the two charts are compatible. The action of normal upper Gamma 1 times normal upper Gamma 2 on upper I described above induces an action of normal upper Gamma prime 1 times normal upper Gamma prime 2 on upper R Superscript s h , compatible with the action of normal upper Gamma prime 1 times normal upper Gamma prime 2 on left-parenthesis upper U 1 times Subscript upper C Baseline upper U 2 right-parenthesis Superscript s h . The action of normal upper Gamma prime 1 equals normal upper Gamma prime 1 times StartSet 1 EndSet on the inverse image of upper C Subscript gen in upper R Superscript s h is free, and its quotient is the inverse image of upper C Subscript gen in upper U 2 Superscript s h ; but upper R Superscript s h is finite and étale over upper U 2 Superscript s h , so the action of normal upper Gamma prime 1 on all of upper R Superscript s h is free, and upper R Superscript s h Baseline slash normal upper Gamma prime 1 equals upper U 2 . Analogously the action of normal upper Gamma prime 2 on upper R Superscript s h is free, and upper R Superscript s h Baseline slash normal upper Gamma prime 2 equals upper U 1 .

Now, each of the connected components of upper R Superscript s h maps isomorphically onto both upper U 1 Superscript s h and upper U 2 Superscript s h , because upper U Subscript j Superscript s h is the spectrum of a strictly henselian ring and the projection upper R Superscript s h Baseline right-arrow upper U Subscript j Superscript s h is étale; this implies in particular that the order of normal upper Gamma 1 is the same as the number k of connected components, and likewise for normal upper Gamma 2 . Fix one of these components, call it upper R 0 Superscript s h ; then we get isomorphisms upper R 0 Superscript s h Baseline asymptotically-equals upper U Subscript j Superscript s h , which yield an isomorphism phi colon upper U 1 Superscript s h Baseline asymptotically-equals upper U 2 Superscript s h .

Call normal upper Gamma prime the stabilizer of the component upper R 0 Superscript s h inside normal upper Gamma prime 1 times normal upper Gamma prime 2 ; the order of normal upper Gamma prime is at least StartAbsoluteValue normal upper Gamma prime 1 times normal upper Gamma prime 2 EndAbsoluteValue slash k equals k squared slash k equals k . But the action of normal upper Gamma prime 2 on upper R Superscript s h is free, and so normal upper Gamma prime intersection normal upper Gamma 2 equals StartSet 1 EndSet ; this implies that the order of normal upper Gamma prime is k , and the projection normal upper Gamma prime right-arrow normal upper Gamma prime 1 is an isomorphism. Likewise the projection normal upper Gamma prime right-arrow normal upper Gamma prime 2 is an isomorphism, so from these we get an isomorphism theta colon normal upper Gamma prime 1 right-arrow normal upper Gamma prime 2 , and it is easy to check that the isomorphism of schemes phi colon upper U 1 Superscript s h Baseline asymptotically-equals upper U 2 Superscript s h is theta -equivariant.

There is also an isomorphism of the pullbacks of eta 1 Superscript s h and eta 2 Superscript s h to upper R 0 Superscript s h , coming from the natural morphism upper R 0 Superscript s h Baseline right-arrow upper I , which induces an isomorphism psi colon eta 1 Superscript s h Baseline right-arrow eta 2 Superscript s h . This isomorphism is compatible with phi , and it is also theta -equivariant.

Let us prove the converse. Suppose that there exist theta , phi and psi as above. Then there is a morphism sigma colon upper U 1 Superscript s h Baseline times normal upper Gamma prime 1 right-arrow upper I which sends a point left-parenthesis u 1 comma gamma 1 right-parenthesis of upper U 1 Superscript s h Baseline times normal upper Gamma prime 1 into the point of upper I lying over the point left-parenthesis u 1 comma phi gamma 1 u 1 right-parenthesis equals left-parenthesis u 1 comma theta left-parenthesis gamma 1 right-parenthesis phi u 1 right-parenthesis corresponding to the isomorphism gamma 1 psi of the pullback of eta 1 to u 1 with the pullback of eta 2 to phi gamma 1 u 1 . The morphism sigma is an isomorphism of upper U 1 Superscript s h Baseline times normal upper Gamma prime 1 with upper R Superscript s h in the inverse image of upper C Subscript gen ; also, from the fact that the action of normal upper Gamma prime on left-parenthesis eta 1 comma upper U 1 right-parenthesis is essential, it follows that sigma is injective. Since the inverse image of upper C Subscript gen is scheme-theoretically dense in upper R Superscript s h and upper U 1 Superscript s h Baseline times normal upper Gamma 1 is unramified over upper U 1 we see that sigma is an isomorphism of upper U 1 Superscript s h Baseline times normal upper Gamma prime 1 with upper R Superscript s h . It follows that upper R Superscript s h is étale over upper U 1 Superscript s h ; analogously it is étale over upper U 2 Superscript s h . So upper R is étale over upper U 1 and upper U 2 at the points u 1 and u 2 ; since this holds for all u 1 and u 2 mapping to the same point of upper C the conclusion follows.

Compatibility of charts is stable under base change:

Proposition 3.4.3

(1)

Let left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis , left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis be two compatible charts for a generic object xi on upper C right-arrow upper S . If upper S prime right-arrow upper S is an arbitrary morphism, then left-parenthesis upper S prime times Subscript upper S Baseline upper U 1 comma eta prime 1 comma normal upper Gamma 1 right-parenthesis

and left-parenthesis upper S prime times Subscript upper S Baseline upper U 2 comma eta prime 2 comma normal upper Gamma 2 right-parenthesis comma

where eta prime 1 and eta prime 2 are the pullbacks of eta 1 and eta 2 to upper S prime times Subscript upper S Baseline upper U 1 and upper S prime times Subscript upper S Baseline upper U 2 , are compatible charts for the pullback of xi to left-parenthesis upper S prime times Subscript upper S Baseline upper C right-arrow upper S prime right-parenthesis Subscript gen .

(2)

If upper S prime right-arrow upper S is étale and surjective, then the converse holds.

The proof is immediate from Proposition 3.4.2.

3.5. The product chart

Given two compatible charts left-parenthesis upper U 1 comma eta 1 comma normal upper Gamma 1 right-parenthesis , left-parenthesis upper U 2 comma eta 2 comma normal upper Gamma 2 right-parenthesis , set eta equals pr Subscript 1 Superscript asterisk Baseline eta 1 in script upper M left-parenthesis upper R right-parenthesis . There is an action of normal upper Gamma , coming from pulling back the action of normal upper Gamma 1 on eta 1 ; also the tautological isomorphism alpha colon pr Subscript 1 Superscript asterisk Baseline eta 1 asymptotically-equals eta 2 induces an action of normal upper Gamma 2 on eta . These two actions commute, and therefore define an action of normal upper Gamma 1 times normal upper Gamma 2 on eta , compatible with the action of normal upper Gamma 1 times normal upper Gamma 2 on rho . Also, upper R has a structure of an n -marked curve, by defining normal upper Sigma Subscript i Superscript upper R to be the inverse image of normal upper Sigma Subscript i Superscript upper U 1 , and the map upper R right-arrow upper C is strict. Then

left-parenthesis upper R comma eta comma normal upper Gamma 1 times normal upper Gamma 2 right-parenthesis

is a chart, called the product chart. It is compatible with both of the original charts.

3.6. Atlases and twisted objects

Definition 3.6.1

Fix two nonnegative integers g and n . An n -pointed twisted object left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis of genus g consists of

(1)

a proper, n -pointed curve upper C right-arrow upper S of finite presentation, with geometrically connected fibers of genus g ;

(2)

a generic object xi over upper C right-arrow upper S ; and

(3)

a collection script upper A equals StartSet left-parenthesis upper U Subscript alpha Baseline comma eta Subscript alpha Baseline comma normal upper Gamma Subscript alpha Baseline right-parenthesis EndSet of mutually compatible charts, such that the images of the upper U Subscript alpha cover upper C .

A collection of charts script upper A as in (3) is called an atlas.

A twisted object is called balanced if each chart in its atlas is balanced (Definition 3.2.4).

Lemma 3.6.2

If two charts for a twisted object are compatible with all the charts in an atlas, they are mutually compatible.

Furthermore, if the twisted object is balanced, then any chart which is compatible with every chart of the atlas is balanced.

Proof.

Both statements are immediate from the local characterization of compatibility in Proposition 3.4.2.

Remark 3.6.3

The lemma above allows one to define a twisted object using a maximal atlas, if one prefers.

Definition 3.6.4

A morphism of twisted objects left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis to left-parenthesis xi prime comma upper C prime right-arrow upper S Superscript prime Baseline comma script upper A prime right-parenthesis consists of a cartesian diagram

StartLayout 1st Row 1st Column upper C 2nd Column long right-arrow Overscript f Endscripts 3rd Column upper C prime 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 3rd Row 1st Column upper S 2nd Column long right-arrow 3rd Column upper S prime EndLayout

and an arrow xi right-arrow xi prime lying over the restriction f vertical-bar Subscript upper C Sub Subscript gen Subscript Baseline colon upper C Subscript gen Baseline right-arrow upper C prime Subscript gen , with the property that the pullback of the charts in script upper A prime are all compatible with the charts in script upper A .

The composition of morphisms of twisted objects is defined to be the one induced by composition of morphisms of curves.

Let left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis be a twisted object, and upper T right-arrow upper S a morphism. Then, using Proposition 3.4.3 one can define the pullback of left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis to upper T in the obvious way.

3.7. Stability

Lemma 3.7.1

Let left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis be a twisted object. Then the morphism upper C Subscript gen Baseline right-arrow bold upper M induced by xi extends uniquely to a morphism upper C right-arrow bold upper M .

Proof.

The unicity is clear from the fact that bold upper M is separated and upper C Subscript gen is scheme-theoretically dense in upper C . To prove the existence of an extension is a local question in the étale topology; but if script upper A equals StartSet left-parenthesis upper U Subscript alpha Baseline comma eta Subscript alpha Baseline comma normal upper Gamma Subscript alpha Baseline right-parenthesis EndSet , then the objects eta Subscript alpha induce morphisms upper U Subscript alpha Baseline right-arrow bold upper M , which are normal upper Gamma Subscript alpha -equivariant, yielding morphisms upper U Subscript alpha Baseline slash normal upper Gamma Subscript alpha Baseline right-arrow bold upper M . These morphisms are extensions of the pullback to left-parenthesis upper U Subscript alpha Baseline right-parenthesis Subscript gen Baseline slash normal upper Gamma Subscript alpha of the morphism upper C Subscript gen Baseline right-arrow bold upper M . Therefore they descend to upper C .

We can now define the main object of this section:

Definition 3.7.2

A twisted object is stable if the associated map upper C right-arrow bold upper M is Kontsevich stable.

3.8. The stack of stable twisted objects

Fix an ample line bundle script upper O Subscript bold upper M Baseline left-parenthesis 1 right-parenthesis over bold upper M . We define a category script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis equals script upper K Subscript g comma n Baseline left-parenthesis script upper M slash double-struck upper S comma d right-parenthesis as follows. The objects are stable twisted objects left-parenthesis xi comma upper C right-arrow upper S comma script upper A right-parenthesis , where upper C right-arrow upper S is a nodal n -pointed curve of genus g , such that for the associated morphism f colon upper C right-arrow bold upper M the degree of the line bundle f Superscript asterisk Baseline script upper O Subscript bold upper M Baseline left-parenthesis 1 right-parenthesis is d . The arrows are morphisms of twisted objects.

As stated in Theorem 1.4.1, this category is a proper algebraic stack which is relatively of Deligne-Mumford type over script upper K Subscript g comma n Baseline left-parenthesis bold upper M comma d right-parenthesis , admitting a projective coarse moduli space bold upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis . The proof of the theorem will begin in Section 5.

We shall also consider the full subcategory script upper K Subscript g comma n Superscript b a l Baseline left-parenthesis script upper M comma d right-parenthesis of balanced twisted objects. It will be shown in Proposition 8.1.1 that this is an open and closed substack in script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis , whose moduli space is open and closed in bold upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis .

4. Twisted curves and twisted stable maps

In this section we give a stack-theoretic description of the category script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis in terms of twisted stable maps. The language of stacks allows one to circumvent many of the technical details involved in twisted objects, and gives a convenient way of thinking about the category script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis . It is also convenient for studying deformation and obstruction theory for script upper K Subscript g comma n Baseline left-parenthesis script upper M comma d right-parenthesis .

4.1. Nodal stacks

Let upper S be a scheme over double-struck upper S . Consider a proper, flat, tame Deligne-Mumford stack script upper C right-arrow upper S of finite presentation, such that its fibers are purely one-dimensional and geometrically connected, with at most nodal singularities. Call upper C the moduli space of script upper C ; by ReferenceK-M this exists as an algebraic space.

Proposition 4.1.1

The morphism upper C right-arrow upper S is a proper flat nodal curve of finite presentation, with geometrically connected fibers.

Proof.

First of all let us show that upper C is flat over upper S . We may assume that upper S is affine; let upper R be its coordinate ring. Fix a geometric point c 0 right-arrow upper C , and call upper C Superscript s h the strict henselization of upper C at c 0 . Let upper U be an étale cover of script upper C , and u 0 a geometric point of upper U lying over c 0 ; denote by upper U Superscript s h the strict henselization of upper U at u 0 . If normal upper Gamma is the automorphism group of the object of script upper C corresponding to u 0 , then normal upper Gamma acts on upper U Superscript s h , and upper C Superscript s h is the quotient upper U Superscript s h Baseline slash normal upper Gamma . Since script upper C is tame, the order of normal upper Gamma is prime to the residue characteristic of u 0 , therefore the coordinate ring of upper C Superscript s h is a direct summand, as an upper R -module, of the coordinate ring of upper U Superscript s h , so it is flat over upper R .

The fact that the fibers are nodal follows from the fact that, over an algebraically closed field, the quotient of a nodal curve by a group action is again a nodal curve. Properness is clear; the fact that the morphism script upper C right-arrow upper C is surjective implies that the fibers are geometrically connected. The fact that upper C is of finite presentation is an easy consequence of the fact that script upper C is of finite presentation.

Following tradition, when we speak of a “family over upper S or “curve over upper S ”, it is always assumed to be of finite presentation.

Definition 4.1.2

A twisted nodal n -pointed curve over upper S is a diagram

StartLayout 1st Row 1st Column normal upper Sigma Subscript i Superscript script upper C 2nd Column subset-of 3rd Column script upper C 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column down-arrow 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper C 4th Row 1st Column Blank 2nd Column Blank 3rd Column down-arrow 5th Row 1st Column Blank 2nd Column Blank 3rd Column upper S EndLayout

where

(1)

script upper C is a tame Deligne-Mumford stack, proper and of finite presentation over upper S , and étale locally is a nodal curve over upper S ;

(2)

normal upper Sigma Subscript i Superscript script upper C Baseline subset-of script upper C are disjoint closed substacks in the smooth locus of script upper C right-arrow upper S ;

(3)

normal upper Sigma Subscript i Superscript script upper C Baseline right-arrow upper S are étale gerbes;

(4)

the morphism script upper C right-arrow upper C exhibits upper C as the coarse moduli scheme of script upper C ; and

(5)

script upper C right-arrow upper C is an isomorphism over upper C Subscript gen .

Note that if we let normal upper Sigma Subscript i Superscript upper C be the coarse moduli spaces of normal upper Sigma Subscript i Superscript script upper C , then, since script upper C is tame, the schemes normal upper Sigma Subscript i Superscript upper C embed in upper C (they are the images of normal upper Sigma Subscript i Superscript script upper C ), and upper C becomes a usual nodal pointed curve. We say that script upper C right-arrow upper S is a twisted pointed curve of genus g if upper C right-arrow upper S is a pointed curve of genus g .

4.2. Morphisms of twisted n -pointed nodal curves

Definition 4.2.1

Let script upper C right-arrow upper S and script upper C prime right-arrow upper S prime be twisted n -pointed nodal curves. A 1 -morphism (or just a morphism) upper F colon script upper C right-arrow script upper C prime is a cartesian diagram

StartLayout 1st Row 1st Column script upper C 2nd Column right-arrow Overscript upper F Endscripts 3rd Column script upper C prime 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 3rd Row 1st Column upper S 2nd Column right-arrow Overscript f Endscripts 3rd Column upper S prime EndLayout

such that upper F Superscript negative 1 Baseline normal upper Sigma Subscript i Superscript script upper C prime Baseline equals normal upper Sigma Subscript i Superscript script upper C .

If upper F comma upper F 1 colon script upper C right-arrow script upper C prime