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.
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 , over -bundles 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, , maps of genus -pointed 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 corresponding to a surjection -bundle, 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 , over -bundle 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 roots, -th
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 , maps -pointed of genus and degree . This category is given in two equivalent realizations: one as a category of stable twisted objects over nodal pointed curves endowed with atlases of orbispace charts (see Definition -valued3.7.2); 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:
1.5. Some applications and directions of further work
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.
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 , which is a finite covering of -covers, 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.
A similar reasoning applies to curves with structures, e.g. theta characteristics. This is studied in -spinReference -J.
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.
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.
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.
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 .
for any morphism of schemes and any object there is an object and an arrow over and ;
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.
is a stack, namely:
the functors are sheaves in the étale topology; and
any étale descent datum for objects of is effective.
The stack is algebraic, namely:
the functors are representable by separated algebraic spaces of finite type; and
there is a scheme locally of finite type, and a smooth and surjective morphism ,.
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
is limit preserving (see Reference Ar, §1);
is compatible with formal completions (see Reference Ar, 5.2 (3));
Schlessinger’s conditions for pro-representability of the deformation functors hold (see Reference Ar, (2.2) and (2.5)); and
there exists an obstruction theory for (see Reference Ar, (2.6)) such that
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:
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:
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. .
2.3. Tame stacks and their coarse moduli spaces
A closely related notion is the following:
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:
Let be a separated tame stack with coarse moduli scheme Consider the projection . The functor . carries sheaves of to sheaves of -modules-modules.
2.4. Purity lemma
We recall the following purity lemma from Reference -V2:
2.5. Descent of equivariant objects
As a consequence of the unicity statement, suppose that we have a triple where , is a group isomorphism, and and are compatible isomorphisms. Then the given arrow -equivariant and its composition with both satisfy the conditions of the lemma, so there is an induced isomorphism .
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 object on a suitable atlas of orbispace charts on a nodal curve. Basic charts have the form -valued 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.
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 curve -pointed is considered an curve by taking as the -marked the images of the sections .
We notice that if a morphism of marked curves is strict, then there is an induced morphism of curves Furthermore, if .