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 .

1.1. The problem of moduli of families

Consider a Deligne-Mumford stack (definition in Section 2.1) admitting a projective coarse moduli scheme . Given a curve , it is often natural to consider morphisms (or equivalently, objects ); in case is the moduli stack of geometric objects, these morphisms correspond to families over . For example, if , the stack of stable curves of genus , then morphisms correspond to families of stable curves of genus over ; and if , the classifying stack of a finite group , we get principal -bundles over . It is interesting to study moduli of such objects; moreover, it is natural to study such moduli as 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 with a fixed ample sheaf . Given integers , it is known that there exists a proper algebraic stack, which here we denote by , of stable, -pointed maps of genus and degree into . (See Reference Ko, Reference B-M, Reference F-P, Reference -O, where the notation is used. Here we tried to avoid using with two meanings.) This stack admits a projective coarse moduli space . If one avoids “small” residue characteristics in , which depend on and , then this stack is in fact a proper Deligne-Mumford stack.

1.3. Stable maps into stacks

Now fix a proper Deligne-Mumford stack admitting a projective coarse moduli space on which we fix an ample sheaf as above. We further assume that is tame, that is, for any geometric point , the group has order prime to the characteristic of the algebraically closed field .

It is tempting to mimic Kontsevich’s construction as follows: Let be a nodal projective connected curve; then a morphism is said to be a stable map of degree if the associated morphism to the coarse moduli scheme is a stable map of degree .

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

To see this, we fix and consider the specific case of with . Any smooth curve of genus admits a connected principal -bundle, corresponding to a surjection , thus giving a map . If we let degenerate to a nodal curve of geometric genus , then , and since there is no surjection , there is no connected principal -bundle over . This means that there can be no limiting stable map as a degeneration of .

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 . The source curve of a new stable map 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 be the spectrum of a discrete valuation ring of pure characteristic 0, with quotient field , and let be a nodal curve over the generic point, together with a map of degree , whose associated map is stable. We can exploit the fact that is complete; after a ramified base change on the induced map will extend to a stable map over . Let be the smooth locus of the morphism ; 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 to a map ; 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 is such a node, then on an étale neighborhood of , the curve looks like

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

we have a nonsingular cover where is defined by . The purity lemma applies to , so the composition extends over all of . There is a minimal intermediate cover such that the family extends already over ; this will be of the form , and the map is given by , . Furthermore, there is an action of the group of roots of 1, under which sends to and to , and . This gives the orbispace structure over , and the map extends to a map .

This gives the flavor of our definition.

We define a category , fibered over , of twisted stable -pointed maps of genus and degree . This category is given in two equivalent realizations: one as a category of stable twisted -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 , 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 is a proper algebraic stack.

(2)

The coarse moduli space of is projective.

(3)

There is a commutative diagram

where the top arrow is proper, quasifinite, relatively of Deligne-Mumford type and tame, and the bottom arrow is finite. In particular, if is a Deligne-Mumford stack, then so is .

1.5. Some applications and directions of further work

(1)

In our paper Reference -V2 we studied the situation where , which gives a complete moduli for fibered surfaces. Some further results for elliptic surfaces are obtained in G. La Nave’s thesis Reference La.

(2)

The case where is the classifying stack of a finite group 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 , show that there is a smooth, fine moduli space for admissible -covers, which is a finite covering of . This is the subject of our preprint Reference -C-V with Alessio Corti, with some ideas contributed by Johan de Jong. Some of these applications were indicated in our announcement Reference -V1. This approach to admissible covers is closely related to the work of Wewers Reference We.

(3)

A similar reasoning applies to curves with -spin structures, e.g. theta characteristics. This is studied in Reference -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 Reference Mo and deserves further study.

(5)

In Reference C-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 -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 Reference F-G for an algebraic treatment of global quotients.

(6)

In this paper we verify that 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 Reference L-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 Reference Ar and Reference L-MB for a general discussion of algebraic stacks (generalizing Reference D-M), and to the appendix in Reference Vi for an introduction. We spell out the conditions here, as we follow them closely in the paper. We are given a category along with a functor . We assume

(1)

is fibered in groupoids (see Reference Ar, §1, (a) and (b), or Reference L-MB, Definition 2.1). This means:

(a)

for any morphism of schemes and any object there is an object and an arrow over ; and

(b)

for any diagram of schemes

and any objects sitting in a compatible diagram

there is a unique arrow over making the diagram commutative.

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

(2)

is a stack, namely:

(a)

the functors are sheaves in the étale topology; and

(b)

any étale descent datum for objects of is effective.

See Reference Ar, 1.1, or Reference L-MB, Definition 3.1.

(3)

The stack is algebraic, namely:

(a)

the functors are representable by separated algebraic spaces of finite type; and

(b)

there is a scheme , locally of finite type, and a smooth and surjective morphism .

See Reference L-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 by algebraization of formal deformation spaces (see Reference Ar, Corollary 5.2). Thus, in case is of finite type over a field or an excellent Dedekind domain, condition (3b) holds if

(A)

is limit preserving (see Reference Ar, §1);

(B)

is compatible with formal completions (see Reference Ar, 5.2 (3));

(C)

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

(D)

there exists an obstruction theory for (see Reference Ar, (2.6)) such that

(i)

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

(ii)

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

(iii)

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

Furthermore, we say that is a Deligne-Mumford stack if we can choose as in (3b) to be étale. This holds if and only if the diagonal is unramified (Reference L-MB, Théorème 8.1). A morphism is of Deligne-Mumford type if for any scheme and morphism the stack is a Deligne-Mumford stack.

For the notion of properness of an algebraic stack see Reference L-MB, Chapter 7. Thus a stack is proper if it is separated, of finite type and universally closed. In Reference L-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 has finite diagonal, it is shown in Reference E-H-K-V, Theorem 2.7, that there exists a finite surjective morphism from a scheme . In such a case the usual weak valuative criterion suffices (Reference L-MB, Proposition 7.12).

2.2. Coarse moduli spaces

Recall the following result:

Theorem 2.2.1 (Keel-Mori Reference K-M).

Let be an algebraic stack with finite diagonal over a scheme . There exists an algebraic space and a morphism such that

(1)

is proper and quasifinite;

(2)

if is an algebraically closed field, then is a bijection;

(3)

whenever is an algebraic space and is a morphism, then the morphism factors uniquely as ; more generally

(4)

whenever is a flat morphism of schemes, and whenever is an algebraic space and is a morphism, then the morphism factors uniquely as ; in particular

(5)

.

Recall that an algebraic space along with a morphism 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 be a proper quasifinite morphism, where is a Deligne-Mumford stack and is a noetherian scheme. Let be a flat morphism of schemes, and denote .

(1)

If is the moduli space of , then is the moduli space of .

(2)

If is also surjective and is the moduli space of , then is the moduli space of .

Proof.

Given a proper quasifinite morphism , it then exhibits as a moduli space if and only if . If is an étale presentation of , and and are the induced morphisms, then this condition is equivalent to the exactness of the sequence

From this the statement follows.

The prototypical example of a moduli space is given by a group quotient: Let be a scheme and a finite group acting on . Consider the stack ; see Reference L-MB, 2.4.2. The morphism exhibits the quotient space as the moduli space of the stack . 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 be a separated Deligne-Mumford stack, and its coarse moduli space. There is an étale covering , such that for each there is a scheme and a finite group acting on , with the property that the pullback is isomorphic to the stack-theoretic quotient .

Sketch of proof.

Let be a geometric point of . Denote by the spectrum of the strict henselization of at the point , and let . If is an étale morphism, with a scheme, having in its image, there is a component of the pullback which is finite over . Denote . We have that under the first projection , the scheme splits as a disjoint union of copies of . Let be the set of connected components of , so that is isomorphic to . Then the product induces a group structure on , and the second projection defines a group action of on , such that is the quotient .

We need to descend from to get the statement on . This follows from standard limit arguments.

2.3. Tame stacks and their coarse moduli spaces

Definition 2.3.1.
(1)

A Deligne-Mumford stack is said to be tame if for any geometric point , the group has order prime to the characteristic of the algebraically closed field .

(2)

A morphism of algebraic stacks is said to be tame if for any scheme and morphism the stack is a tame Deligne-Mumford stack.

A closely related notion is the following:

Definition 2.3.2.

An action of a finite group on a scheme is said to be tame if for any geometric point , the group has order prime to the characteristic of .

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

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

Lemma 2.3.3.

Let be a tame Deligne-Mumford stack, its moduli space. If is any morphism of schemes, then is the moduli space of the fiber product . Moreover, if is reduced, then it is also the moduli space of .

Proof.

By Lemma 2.2.2, this is a local condition in the étale topology of , so we may assume that is a quotient stack of type , where is a finite group acting on an affine scheme . Moreover, since is tame, we may assume that the order of is prime to all residue characteristics. Then ; if , then the statement is equivalent to the map being an isomorphism. This (well-known) fact can be shown as follows: Recall that for any -module the homomorphism

is a projector exhibiting as a direct summand in . Thus the induced morphism

shows that is injective. The morphism is a lifting of

which is surjective.

This shows that is the moduli space of the fiber product . The statement about is immediate. This proves the result.

Let be a separated tame stack with coarse moduli scheme . Consider the projection . The functor carries sheaves of -modules to sheaves of -modules.

Lemma 2.3.4.

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

Proof.

The question is local in the étale topology on , so we may assume that is of the form , where is a scheme and a finite group of order prime to all residue characteristics, in particular . Now sheaves on correspond to equivariant sheaves on . Denote by the projection. If is a sheaf on corresponding to a -equivariant sheaf on , then , which, by the tameness assumption, is a direct summand in . From this the statement follows.

2.4. Purity lemma

We recall the following purity lemma from Reference -V2:

Lemma 2.4.1.

Let be a separated Deligne-Mumford stack, the coarse moduli space. Let be a separated scheme of dimension satisfying Serre’s condition . Let be a finite subset consisting of closed points, . Assume that the local fundamental groups of around the points of are trivial.

Let be a morphism. Suppose there is a lifting :

Then the lifting extends to :

and is unique up to a unique isomorphism.

Proof.

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

Corollary 2.4.3.

Let be a smooth surface over a field, a closed point with complement . Let and be as in the purity lemma. Then there is a lifting .

Corollary 2.4.4.

Let be a normal crossings surface over a field , namely a surface which is étale locally isomorphic to . Let be a closed point with complement . Let and be as in the purity lemma. Then there is a lifting .

Proof.

In both cases satisfies condition and the local fundamental group around is trivial, hence the purity lemma applies.

2.5. Descent of equivariant objects

Lemma 2.5.1.

Let be a local ring with residue field , let , , let be a Deligne-Mumford stack, and let be an object of . Assume we have a pair of compatible actions of a finite group on and on , in such a way that the induced actions of on and on the pullback are trivial. Then there exists an object of on the quotient , and a -invariant lifting of the projection . Furthermore, if is another such object over , there is a unique isomorphism over the identity of , which is compatible with the two arrows and .

As a consequence of the unicity statement, suppose that we have a triple , where is a group isomorphism, and and are compatible -equivariant isomorphisms. Then the given arrow and its composition with both satisfy the conditions of the lemma, so there is an induced isomorphism .

Corollary 2.5.2.

Let be as in the previous lemma. Let be a finite group acting compatibly on and on . Let be the normal subgroup of consisting of elements acting on and as the identity. Then there exist a -equivariant object on the quotient , and a -equivariant arrow compatible with the projection .

Proof of the corollary.

The action is defined as follows. If is an element of , call and the induced arrows, and the conjugation by . Then the image of in acts on via the isomorphism defined above. One checks easily that this defines an action with the required properties.

Proof of the lemma.

First note that if is the strict henselization of , the condition on the action of allows one to lift it to . Also, the statement that we are trying to prove is local in the étale topology, so by standard limit arguments we can assume that is strictly henselian. Replacing by the spectrum of the strict henselization of its local ring at the image of the closed point of , we can assume that is of the form , where is a scheme and is a finite group. Then the object corresponds to a principal -bundle , on which acts compatibly with the action of on , and an -equivariant and -invariant morphism . Since is strictly henselian, the bundle is trivial, so is a disjoint union of copies of , and the group permutes these copies; furthermore the hypothesis on the action of on the closed fiber over the residue field insures that sends each component into itself. The thesis follows easily.

We note that Lemma 2.5.1 can be proven for an arbitrary algebraic stack 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 . Using Lemma 2.2.2, it can be shown that there is a relative coarse moduli space, namely a representable morphism such that gives the coarse moduli space of for any scheme and flat morphism . The fact that acts trivially on , together with Lemmas 2.2.2 and 4.4.3, imply that .

3. Twisted objects

Our goal in this section is to introduce the notion of a stable twisted object. This is a representable -valued object on a suitable atlas of orbispace charts on a nodal curve. Basic charts have the form at a node, or along a marking, where 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 , 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 -marked nodal curve, or simply -marked curve , consists of a nodal curve , together with a sequence of pairwise disjoint closed subschemes whose supports do not contain any of the singular points of the fibers of , and such that the projections are étale. (Any of the subschemes may be empty.)

If more than one curve is considered, we will often use the notation to specify the curve . On the other hand, we will often omit the subschemes from the notation if there is no risk of confusion.

A nodal -pointed curve is considered an -marked curve by taking as the the images of the sections .

Definition 3.1.2.

If is an -marked nodal curve, we define the special locus of , denoted by , to be the union of the with the singular locus of the projection , with its natural scheme structure (this makes the projection unramified). The complement of will be called the general locus of , and denoted by .

Definition 3.1.3.

If is a marked curve, and is an arbitrary morphism, we define the pullback to be , where and .

Definition 3.1.4.

If and are -marked curves, a morphism of -marked curves is a morphism of -schemes which sends each into .

A morphism of -marked curves is called strict if the support of coincides with the support of for all , and similarly for the singular locus.

We notice that if a morphism of marked curves is strict, then there is an induced morphism of curves . Furthermore, if