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 is strict and étale, then scheme-theoretically.

Definition 3.1.5.

Let be an -marked curve and a finite group. An action of on is an action of on as an -scheme, such that each element on acts via an automorphism, in the sense of Definition 3.1.4, of as a marked curve on .

If is a finite group along with a tame action on a marked curve , then the quotient can be given a marked curve structure by defining . The latter inclusion holds because the orders of stabilizers in are assumed to be prime to the residue characteristics, so is indeed a subscheme of .

Given a morphism of marked curves, and a tame action of a finite group on , leaving invariant, then there is an induced morphism of marked curves.

Definition 3.1.6.

Let be an -marked curve, with an action of a finite group , and let be a Deligne-Mumford stack. Given , an essential action of on is a pair of compatible actions of on and on , with the property that if is an element of different from the identity and is a geometric point of fixed by , then the automorphism of the pullback of to induced by is not trivial.

3.2. Generic objects and charts

Definition 3.2.1.

Let be an -pointed nodal curve. A generic object on is an object of .

We will often write for a generic object on a curve .

Definition 3.2.2.

Let be an -pointed nodal curve and a generic object on . A chart for consists of the following collection of data.

(1)

An -marked curve and a strict morphism .

(2)

An object of .

(3)

An arrow in compatible with the restriction .

(4)

A finite group .

(5)

A tame, essential action of on .

Furthermore, we require that the following conditions be satisfied.

a

The actions of leave the morphism and the arrow invariant.

b

The induced morphism is étale.

The following gives a local description of the action of .

Proposition 3.2.3.

Let be a chart for a generic object on a pointed nodal curve . Then the action of on is free.

Furthermore, if is a geometric point of and a nodal point of the fiber of over , then

(1)

the stabilizer of is a cyclic group which sends each of the branches of to itself;

(2)

if is the order of , then a generator of acts on the tangent space of each branch by multiplication with a primitive -th root of .

In particular, each nodal point of is sent to a nodal point of .

Proof.

The first statement follows from the definition of an essential action and the invariance of the arrow .

As for (1), observe that if the stabilizer of did not preserve the branches of , then the quotient , which is étale at the point over the fiber , would be smooth over at , so would be in the inverse image of . From the first part of the proposition it would follow that is trivial, a contradiction.

So 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 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 , the two roots of 1 describing the action of a generator of the stabilizer on the tangent spaces to each branch of are inverse to each other.

3.3. The transition scheme

Let be a generic object over a nodal curve , and , two charts; call the -th projection. Consider the scheme

over representing the functor of isomorphisms of the two objects and .

There is a section of over the inverse image of in which corresponds to the isomorphism coming from the fact that both and are pullbacks to of . We will call the scheme-theoretic closure of this section in the transition scheme from to ; it comes equipped with two projections and .

There is also an action of on , defined as follows. Let , and let be an isomorphism over ; then define . This action of on is compatible with the action of on , and leaves invariant. It follows from the definition of an essential action that the action of and on is free.

3.4. Compatibility of charts

Definition 3.4.1.

Two charts and are compatible if their transition scheme is étale over and .

Let us analyze this definition. First of all, is obviously étale over and . Also, since the maps are strict, it is clear that the inverse image of in is set-theoretically equal to the inverse image of . If the two charts are compatible, this also holds scheme-theoretically.

Now, start from two charts and . Fix two geometric points

mapping to the same geometric point , and call the stabilizer of . Also call , and the spectra of the strict henselizations of , and at the points and respectively. The action of on induces an action of on . Also call the pullback of to ; there is an action of on compatible with the action of on . The following essentially says that two charts are compatible if and only if for any choice of and 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 and as above there exist an isomorphism of groups , a -equivariant isomorphism of schemes over , and a compatible -equivariant isomorphism .

Proof.

Consider the spectrum of the strict henselization of at the point , and call the pullback of to . Assume that the two charts are compatible. The action of on described above induces an action of on , compatible with the action of on . The action of on the inverse image of in is free, and its quotient is the inverse image of in ; but is finite and étale over , so the action of on all of is free, and . Analogously the action of on is free, and .

Now, each of the connected components of maps isomorphically onto both and , because is the spectrum of a strictly henselian ring and the projection is étale; this implies in particular that the order of is the same as the number of connected components, and likewise for . Fix one of these components, call it ; then we get isomorphisms , which yield an isomorphism .

Call the stabilizer of the component inside ; the order of is at least . But the action of on is free, and so ; this implies that the order of is , and the projection is an isomorphism. Likewise the projection is an isomorphism, so from these we get an isomorphism , and it is easy to check that the isomorphism of schemes is -equivariant.

There is also an isomorphism of the pullbacks of and to , coming from the natural morphism , which induces an isomorphism . This isomorphism is compatible with , and it is also -equivariant.

Let us prove the converse. Suppose that there exist , and as above. Then there is a morphism which sends a point of into the point of lying over the point corresponding to the isomorphism of the pullback of to with the pullback of to . The morphism is an isomorphism of with in the inverse image of ; also, from the fact that the action of on is essential, it follows that is injective. Since the inverse image of is scheme-theoretically dense in and is unramified over we see that is an isomorphism of with . It follows that is étale over ; analogously it is étale over . So is étale over and at the points and ; since this holds for all and mapping to the same point of the conclusion follows.

Compatibility of charts is stable under base change:

Proposition 3.4.3.
(1)

Let , be two compatible charts for a generic object on . If is an arbitrary morphism, then

and

where and are the pullbacks of and to and , are compatible charts for the pullback of to .

(2)

If 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 , , set in . There is an action of , coming from pulling back the action of on ; also the tautological isomorphism induces an action of on . These two actions commute, and therefore define an action of on , compatible with the action of on . Also, has a structure of an -marked curve, by defining to be the inverse image of , and the map is strict. Then

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 and . An -pointed twisted object of genus consists of

(1)

a proper, -pointed curve of finite presentation, with geometrically connected fibers of genus ;

(2)

a generic object over ; and

(3)

a collection of mutually compatible charts, such that the images of the cover .

A collection of charts 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 to consists of a cartesian diagram

and an arrow lying over the restriction , with the property that the pullback of the charts in are all compatible with the charts in .

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

Let be a twisted object, and a morphism. Then, using Proposition 3.4.3 one can define the pullback of to in the obvious way.

3.7. Stability

Lemma 3.7.1.

Let be a twisted object. Then the morphism induced by extends uniquely to a morphism .

Proof.

The unicity is clear from the fact that is separated and is scheme-theoretically dense in . To prove the existence of an extension is a local question in the étale topology; but if , then the objects induce morphisms , which are -equivariant, yielding morphisms . These morphisms are extensions of the pullback to of the morphism . Therefore they descend to .

We can now define the main object of this section:

Definition 3.7.2.

A twisted object is stable if the associated map is Kontsevich stable.

3.8. The stack of stable twisted objects

Fix an ample line bundle over . We define a category as follows. The objects are stable twisted objects , where is a nodal -pointed curve of genus , such that for the associated morphism the degree of the line bundle is . 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 , admitting a projective coarse moduli space . The proof of the theorem will begin in Section 5.

We shall also consider the full subcategory of balanced twisted objects. It will be shown in Proposition 8.1.1 that this is an open and closed substack in , whose moduli space is open and closed in .

4. Twisted curves and twisted stable maps

In this section we give a stack-theoretic description of the category 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 . It is also convenient for studying deformation and obstruction theory for .

4.1. Nodal stacks

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

Proposition 4.1.1.

The morphism is a proper flat nodal curve of finite presentation, with geometrically connected fibers.

Proof.

First of all let us show that is flat over . We may assume that is affine; let be its coordinate ring. Fix a geometric point , and call the strict henselization of at . Let be an étale cover of , and a geometric point of lying over ; denote by the strict henselization of at . If is the automorphism group of the object of corresponding to , then acts on , and is the quotient . Since is tame, the order of is prime to the residue characteristic of , therefore the coordinate ring of is a direct summand, as an -module, of the coordinate ring of , so it is flat over .

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 is surjective implies that the fibers are geometrically connected. The fact that is of finite presentation is an easy consequence of the fact that is of finite presentation.

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

Definition 4.1.2.

A twisted nodal -pointed curve over is a diagram

where

(1)

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

(2)

are disjoint closed substacks in the smooth locus of ;

(3)

are étale gerbes;

(4)

the morphism exhibits as the coarse moduli scheme of ; and

(5)

is an isomorphism over .

Note that if we let be the coarse moduli spaces of , then, since is tame, the schemes embed in (they are the images of ), and becomes a usual nodal pointed curve. We say that is a twisted pointed curve of genus if is a pointed curve of genus .

4.2. Morphisms of twisted -pointed nodal curves

Definition 4.2.1.

Let and be twisted -pointed nodal curves. A -morphism (or just a morphism) is a cartesian diagram

such that .

If are morphisms, then we define a 2-morphism to be a base-preserving natural transformation. (This implies that it is an isomorphism.)

In this way, twisted pointed curves form a 2-category. However, we have the following:

Proposition 4.2.2.

The -category of twisted pointed curves is equivalent (in the lax sense, Reference K-S) to a category.

We call the resulting category the category of twisted pointed curves. A morphism in this category is an isomorphism class of 1-morphisms in the 2-category of twisted pointed curves.

Proof.

Since all 2-morphisms are invertible, this claim is the same as saying that a 1-arrow in the 2-category cannot have nontrivial automorphisms. The point here is that the stack has a dense open representable substack, which is sufficient by the following lemma.

Lemma 4.2.3.

Let be a representable morphism of Deligne-Mumford stacks over a scheme . Assume that there exists a dense open representable substack (i.e. an algebraic space) and an open representable substack such that maps into . Then any automorphism of is trivial.

Proof.
(1)

First note that the lemma holds if is an algebraic space: denote by the object of over corresponding to . The fact that the diagonal of the algebraic stack is separated implies that the isomorphism scheme is separated. Since is representable, we have that is an isomorphism. Thus the unique section over the given open set has at most one extension to , which gives the assertion in this case.

We will now reduce the general case to this one by descent. We start with some observations.

(2)

Let be an automorphism of ; for each object of over a scheme we are given an automorphism of over , satisfying the usual condition for being a natural transformation. We are going to need the following two facts.

(a)

If is a morphism in , then if is trivial then also is trivial. This follows immediately from the fact that is a category fibered in groupoids (see 2.1 (1b)).

(b)

If is an étale surjective map of schemes and is the pullback of to , then if is trivial then also is trivial. This follows from the fact that the isomorphism functors of are sheaves in the étale topology (see 2.1 (2a)).

(3)

Let be an étale cover, and call the corresponding object of over . The restriction of to the open subscheme is trivial, and is scheme-theoretically dense in ; applying this lemma in the case of algebraic spaces it follows that is trivial.

(4)

Take an arbitrary object of over a scheme ; then there is an étale cover such that the pullback of to admits a morphism . Applying the two facts mentioned above, we get the result.

4.3. Twisted stable maps into a stack

As before, we consider a proper tame Deligne-Mumford stack admitting a projective coarse moduli scheme . We fix an ample invertible sheaf on .

Definition 4.3.1.

A twisted stable -pointed map of genus and degree over

consists of a commutative diagram

along with closed substacks , satisfying:

(1)

along with is a twisted nodal -pointed curve over ;

(2)

the morphism is representable; and

(3)

is a stable -pointed map of degree .

Definition 4.3.2.

A -morphism (or just a morphism) of twisted stable maps

consists of data , where is a morphism of twisted pointed curves, and is an isomorphism.

Twisted stable maps naturally form a 2-category. But by Proposition 4.2.2, this 2-category is equivalent to a category. We call this category the category of twisted stable maps.

4.4. Equivalence of stable twisted objects and twisted stable maps

Theorem 4.4.1.

The category of twisted stable maps is equivalent to the category of stable twisted objects, via an equivalence which preserves base schemes.

Proof.

First part: construction of a functor from stable twisted objects to the category of twisted stable maps.

Step 1: construction of given . Consider a stable twisted object with . For each pair of indices let be the transition scheme from to . Let be the disjoint union of the , and let be the disjoint union of the . The definition of via isomorphisms (see 3.3) implies that these have the following structure:

there are two projections , which are étale;

there is a natural diagonal morphism which sends each to ; and

there is a product , sending each to via composition of isomorphisms.

These maps give the structure of an étale groupoid, which defines a quotient Deligne-Mumford stack, which we denote by . This is obviously a nodal stack on , and its moduli space is . It is clear that is an isomorphism over . Also note that, étale locally over , the stack is isomorphic to .

This construction depends on the atlas chosen, however we have:

Lemma 4.4.2.

Let and be two compatible atlases on a generic object . Then the stacks associated to the corresponding twisted objects are canonically isomorphic.

Proof.

In fact, let be the union of and . Let , and be the groupoids constructed from these three atlases, and let , and be the quotient stacks. There are obvious embeddings of and into inducing isomorphisms of and with ; by composing the isomorphism with the inverse of we obtain the desired canonical isomorphism .

Step 2: construction of . Since, given two indices , the inverse image of in coincides with the inverse image of , the collection of the defines a closed substack of . Since the are étale over , it follows that is étale over ; furthermore, the moduli space of is , so for any algebraically closed field the induced functor induces a bijection of the set of isomorphism classes in and . This means that is an étale gerbe over , and has the structure of a twisted -pointed curve.

If we start from a different, but compatible, atlas, the isomorphism between the two twisted curves constructed above preserves these markings.

Step 3: construction of . Putting together the objects we have an object on , and the tautological isomorphism between the two pullbacks of and to yield an isomorphism of the two pullbacks of to . Fix a morphism , where is a scheme, and set , . Denote by the pullback of to ; then induces an isomorphism of the two pullbacks of to satisfying the cocycle condition. By descent for objects of (see 2.1 (2b)), this isomorphism allows to descend to an object of , or equivalently a morphism . Thus we have associated to an object of an object of .

Step 4: construction of . We need to associate to each arrow in an arrow in in such a manner that we get a morphism of stacks .

Consider an arrow in over a morphism of schemes , which corresponds to the following commutative diagram:

Set , . Denote by the pullback of to , the object obtained by descent to ; there is an arrow , together with descent data, inducing an arrow .

Thus we have a morphism of stacks .

If we start from two different but compatible atlases and on the same generic object, we obtain two morphisms and . It is easily seen that there is a canonical isomorphism between this morphism and the morphism obtained by composing the isomorphism constructed above with the morphism .

Step 5: the morphism is representable. This is a consequence of the fact that the action of the finite groups appearing in the charts of the atlas is essential, because of the following well-known lemma.

Lemma 4.4.3.

Let be a morphism of Deligne-Mumford stacks. The following two conditions are equivalent:

(1)

The morphism is representable.

(2)

For any algebraically closed field and any , the natural group homomorphism is a monomorphism.

Proof.

By definition, the morphism is representable if and only if the following condition holds:

For any algebraic space and any morphism , the stack is equivalent to an algebraic space.

For fixed the latter condition is equivalent to the following:

The diagonal morphism is a monomorphism.

This means:

Given an algebraically closed field , and an element and two elements , with isomorphisms , there exists a unique isomorphism such that .

We can write

where , and . Similarly as above. The existence of implies . Also, composing with the given isomorphisms we may reduce to the case that, in fact, the same is true for the and . Thus the condition above is equivalent to saying that for any there is a unique such that , which is what we wanted.

Now, let , where is an algebraically closed field, and lift to a geometric point ; let be such that . The automorphism group of in is the stabilizer of inside ; the fact that this acts faithfully on the fiber of over means exactly that this automorphism group injects inside the automorphism group of the image of in .

Step 6: stability. The map induced by this morphism is the map induced by the twisted object , and so is stable; this way we get a twisted stable map .

Step 7: construction of the morphism of twisted stable maps induced by a morphism of twisted objects. Let be a morphism of twisted objects, and let and be the corresponding twisted stable maps. Denote also by the twisted stable map associated to the twisted object . Since, as we have seen, there is a canonical isomorphism between and , we are reduced to the case . In this case there is a morphism of groupoids to , which induces a morphism of stacks .

This completes the definition of the functor from twisted objects to twisted stable maps.

Second part: construction of a functor from twisted stable maps to twisted objects.

Step 1: construction of . Let us take a twisted stable map , with moduli space . Since is isomorphic to its inverse image in , the map induces a map , and correspondingly a generic object on .

By Lemma 2.2.3, 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 .

For each we define as the pullback of to ; then the action of the groups preserves the subschemes . These subschemes are obviously disjoint and contained in the smooth locus of the projection , so is an -marked curve.

If we call the object of corresponding to the morphism , the action of on lifts to an action of on . We get charts for the generic object . The fiber product is étale over and . Since the given morphism is representable and separated we have that is a closed subspace of

where and denote the projections of onto the two factors. This implies that is the transition scheme of the two charts, which are therefore compatible. It follows that we have defined an atlas on , and so a twisted object. The induced morphism is the one induced by , so it is stable.

If we start from a different covering of by we get a canonically isomorphic twisted object. In fact, we essentially get the same twisted object with a different atlas.

Step 2: construction of morphisms. Given a morphism from to , we automatically get a morphism . This association is obviously compatible with composition of morphisms.

Since and are tame, the formation of their moduli spaces and commute with base change (Lemma 2.3.3), so that . Since is representable, we have that is representable. Since this is an isomorphism over and the geometric fibers of are seminormal, we have that is isomorphic to , and the morphism is isomorphic to the composite . From this it follows immediately that the pullback of a chart is compatible with all charts in .

Third part: proving that the two functors defined above are inverse to each other.

First of all start from a twisted map , and call the associated twisted object. This is obtained by taking a covering of via as in part 2; if we call the disjoint union of the , and let , then the twisted curve associated with is the quotient of the groupoid , which is (canonically isomorphic to) . It is straightforward to check that the morphism obtained from is canonically isomorphic to the given one.

Now start from a stable twisted object , and consider the associated twisted stable map . The atlas yields a covering of with stacks ; the associated twisted object is canonically isomorphic to .

Remark 4.4.4.

We draw attention to the following counter-intuitive phenomenon: given a twisted stable map , we can look at the automorphisms in . Since , such an automorphism is a local object, dictated by the structure of along the special locus of . However, if such an automorphism comes from an automorphism of , then it is determined by the associated automorphism of the generic object on . The structure of is detailed in Reference -C-V.

5. The category is an algebraic stack

5.1. The stack axioms

Proposition 5.1.1.

The category is a limit-preserving stack, fibered by groupoids over .

Proof.

Condition 2.1 (1): By definition is fibered by groupoids over , since there are pullbacks, and since all the morphisms of objects are given by fiber diagrams.

Condition 2.1 (3A): It is also not difficult to see that is limit preserving: given a twisted object over , where , we may replace by an atlas with finitely many charts if necessary; then the schemes are of finite presentation, and therefore they come from some ; since the stack is limit preserving, the objects come from for some .

Condition 2.1 (2b): We need to show that has effective étale descent for objects.

Given a scheme , an étale cover , and twisted objects together with isomorphisms between the pullbacks of and to satisfying the cocycle condition, we claim that these descend to a twisted object on . The existence of the projective curve is immediate from the sheaf axiom for . Now, any chart for is also automatically a chart for . The charts coming from one are obviously compatible. Compatibility of the charts coming from and follows since they are compatible when pulled back to .

Condition 2.1 (2a) and (3a): The other sheaf axiom requires each functor to be a sheaf in the étale topology. Since we would like to show that is an algebraic stack, we might as well prove that the functor is representable:

Proposition 5.1.2.

For any pair of stable twisted objects

over the same scheme , the functor of isomorphisms of twisted objects is representable by a separated scheme of finite type over .

Proof.

Step 1: reduction to the case . Consider the associated maps ; there is a natural transformation , where the second functor is the one associated with the diagonal in the stack . Since the second functor is known to be representable by separated schemes of finite type, it is enough to prove that this natural transformation is representable by separated schemes of finite type. This means that we can assume that is equal to , and reformulate the problem as follows: if and are stable twisted objects over the same nodal curve , then the functor of isomorphisms of the two twisted objects inducing the identity on is representable by a separated scheme of finite type over .

Step 2: reduction to the case where . Consider the scheme , which is finite over ; it can be extended, to a scheme finite over , as follows.

Let and be the twisted curves underlying the twisted stable maps associated to . Then the scheme

is open inside the stack

Let be the coarse moduli space of the latter stack. Then is finite over , and

is open.

By definition, each morphism of twisted objects induces a section ; this defines a morphism from to the Weil restriction , which is quasiprojective over since is projective and is finite (see Reference G-FGA). It is enough to prove that the morphism is representable, separated and of finite type, so we may assume that is equal to .

Step 3: the case . We see that the following lemma implies the thesis.

Lemma 5.1.3.

Let be a generic object with two atlases and . Then there exists a closed subscheme such that given a morphism , the pullbacks of and to are compatible if and only if factors through .

Proof.

We will prove a local version of the fact above.

Lemma 5.1.4.

Let be a generic object with two charts and . Let be a geometric point of ; assume that the fiber of over has a unique special point , and that there are two unique geometric points and over . Then there exist an étale neighborhood of and a closed subscheme such that given a morphism , the pullbacks of the two charts to are compatible if and only if factors through .

Proof of the global Lemma 5.1.3 given the local Lemma 5.1.4.

First fix a geometric point of , and let be the special points above it. For each let and be charts from and . Refining the charts if necessary, we may assume that there are unique geometric points and in and over . By Lemma 5.1.4 there exists an étale neighborhood of and a closed subscheme for compatibility of and . We can choose a common refinement of such that the pullbacks of and cover all special points over . Then the intersection is a closed subscheme of such that given a morphism , the pullbacks of the two atlases to are compatible if and only if factors through .

Now choose a finite number of geometric points and étale neighborhoods which cover . Applying Proposition 3.4.3, the union of the images of the closed subschemes is the closed subscheme required.

Proof of the local lemma.

By passing to the fiber product we may assume that . By refining we may assume that and are affine, and that there exists an invariant effective Cartier divisor on containing the locus where the projection is not smooth, but none of the fibers of . Using the local criterion for flatness, it is easy to see that such a divisor is flat.

By passing to an étale neighborhood of we can split into a number of connected components, so that the component containing is finite over ; then by deleting the other connected components we see that we may assume that is finite over .

Consider the transition scheme of the two charts. There is a free action of the group over ; set . The projection is an isomorphism over the smooth locus of , and is an isomorphism if and only if the projection is étale. Take a morphism ; then the transition schemes of the pullbacks of the two charts to is the scheme-theoretic closure of the inverse image of the smooth locus of in , so is étale over if and only if the projection has a section.

Set , , , and call the ideal of in . The coordinate ring of the complement of inside is , and there is a natural homomorphism ; given a -algebra the coordinate ring of the quotient of the transition scheme of the pullbacks of the two charts to is the image of in , so has a section if and only if the image of in is equal to . Take a set of generators of as an -algebra, and call their images inside ; the condition that the image of be equal to is equivalent to the condition that the images of the in be zero. Fix an integer such that the are all contained in ; then the sequence

is exact and is flat over , so the sequence stays exact after tensoring with . The conclusion is that is étale over if and only if the images of the in are zero. But is finite and flat as a -module, so it is projective, and can be embedded as a direct summand of a free -module . If is the ideal generated by the coefficients of the with respect to a basis of , it is clear that the closed subscheme has the desired property.

This completes the proof of Proposition 5.1.2, and also of Proposition 5.1.1.

5.2. Base change

Artin’s criteria for an algebraic stack work over a scheme of finite type over a field or an excellent Dedekind domain. Since is of finite presentation, it is obtained by base change from a scheme of finite type over . We need a similar statement of , which follows from the following result:

Proposition 5.2.1.

Let be a morphism of noetherian schemes and let be as in the main theorem. Then .

Proof.

Since is tame, by Lemma 2.3.3 we have that is the moduli space of . From this and Definition 4.3.1 the result is immediate.

5.3. Deformations and obstructions

So far we have seen that is a limit-preserving stack, with separated representable diagonal of finite type. In order to show that it is algebraic, we need to produce an étale covering by a scheme . We follow Artin’s method, in which one starts from a deformation and obstruction theory, one constructs formal deformation spaces, and one shows they are algebraizable. We start by constructing a deformation and obstruction theory for . Here it is convenient to realize as the category of twisted stable maps, rather than stable twisted objects.

Proposition 5.3.1.

The category has a deformation and obstruction theory satisfying conditions 2.1(3D) (i) through (iii).

Let be a reduced -algebra of finite type, a twisted stable map over . For any -module of finite type, let us call the set of isomorphism classes of twisted stable maps , with a given isomorphism of the restriction of to with . Recall that the set has a natural -module structure. Denote by the cotangent complex (see Reference Il, II, 1.2.7, Reference L-MB, 17.3) and by the normal sheaf of in .

Lemma 5.3.2.

There exists a canonical exact sequence of -modules

which is functorial in .

Proof.

By Reference Il, III, Proposition 2.1.2.3, the group classifies extensions of the twisted map , with the markings ignored. The natural map is surjective, because is unobstructed inside , as . The group classifies extensions of inside the trivial extension , so surjects onto the kernel of the map . Finally, is the group of -derivations from into , therefore it is the group of infinitesimal automorphisms of over fixing , so it surjects onto the kernel of .

Lemma 5.3.3.

Let be infinitesimal extensions as in Reference Ar, so that is an -module, and let be a twisted pointed map. The obstruction to lifting to lies in the group .

Proof.

If there is an extension of , then lifts to a subgerbe of , as indicated above. So we see that the obstruction to extending the twisted stable map coincides with the obstruction to extending . According to Reference Il, III, Proposition 2.1.2.3, this obstruction is an element of .

Lemma 5.3.4.

Let be an infinitesimal extension of , and let be a tame proper Deligne-Mumford stack. Let be a complex bounded above, of sheaves of -modules, with coherent cohomology. For each étale morphism and each finite -module , set

Let be a finite -module. Then:

(1)

If is étale, then

(2)

If is a maximal ideal of , then

(3)

There is an open dense subset such that

for all .

Proof.

Let . Since is bounded above, and locally quasi-isomorphic to a complex of locally free sheaves, we have

By Lemma 2.3.4, the hypercohomology of the complex is isomorphic to the hypercohomology of the complex of sheaves on .

Statement (1) follows from the fact that formation of hypercohomology commutes with flat base change, for separated algebraic spaces; the case of a single sheaf is in Reference L-MB, 13.1.9. From this one can deduce the general case by the usual spectral sequence argument.

Statement (2) can be deduced from the theorem on formal functions (see Reference Kn, V, 3.1). Note that each term in the complexes on the right is finite over the artinian ring , and therefore the complexes and their cohomologies satisfy the Mittag-Leffler condition. A standard spectral sequence argument (see Reference G-EGA, 0, 13.2.3) gives the statement.

Statement (3) is proved as follows. Since is reduced, by generic flatness we can localize, and assume that the cohomology sheaves of are flat over . By localizing further, we can also assume that the hypercohomology groups of are projective over . Then the statement follows from the standard base change theorem (see Reference B-G-I, IV, 3.10, and proof of Reference L-MB, 13.1.9).

We now come to our proof of Proposition 5.3.1.

Condition (i) follows from (1) in the lemma: for the obstructions it is immediate, and for the deformations it follows using the exact sequence .

For (ii), first notice that is still exact, because each of the terms of is an artinian -module, and so they satisfy the Mittag-Leffler condition. The sequence is also exact, because is flat over , so the conclusion follows from (2) in the lemma above, together with the five lemma.

For (iii), note that there is a dense open set such that is exact for all in . Then the result follows from part (3) of the lemma, and the five lemma.

5.4. Algebraization

To show that is an algebraic stack, we need to verify conditions 2.1 (3B) and (3C).

Condition (3C) calls for Schlessinger’s conditions as in Reference Ar, section 2, (S 1) and (S 2).

Condition (S 1), and even the stronger condition (S 1) (see Reference Ar, (2.3)), follows from standard principles. Indeed, given an infinitesimal extension and a ring homomorphism such that is surjective, and given stable twisted objects and there is a gluing , along with gluings of the schemes underlying the charts. The objects can be glued since the condition applies to .

The finiteness condition (S 2) follows from the properness of and the exactness of the pushforward along (Lemma 2.3.4), because of the cohomological description of .

We now come to condition (3B) – that formal deformations are algebraizable.

Proposition 5.4.1.

Let be a complete noetherian local ring over our base scheme , with maximal ideal . Set . Let be a twisted stable map over , together with isomorphisms . Then there exist a twisted stable map over together with isomorphisms , compatible with the .

Proof.

Step 1: construction of . Let be the moduli space of . The fact that the stack is algebraic insures the existence of a projective curve over , together with isomorphisms compatible with the isomorphisms , and a map extending the maps .

Step 2: construction of . The stack is proper and tame over , and its restriction to is . Consider the maps

The morphisms are representable and finite, so they define a sheaf of finite algebras over . By the existence theorem A.1.1 given in the appendix, there is a sheaf of finite -algebras with compatible isomorphisms ; by Reference L-MB, Proposition 14.2.4, there is a finite representable map such that .

By the local criterion of flatness (Reference Ma, Theorem 49, condition 5) is flat over . By deformation theory, is a nodal stack over .

Let us show that the coarse moduli space of is indeed . Call the coarse moduli space of , with the induced map . This is finite. Because of the local criterion of flatness, it is flat (see Reference Ma, 20.G). Furthermore since the map is an isomorphism outside the special locus of , the map is finite, flat and of degree 1, so it is an isomorphism.

Step 3: construction of . Let be the union of the markings. Again by the existence theorem, there is a unique substack whose intersection with the is . The stack is flat over , and is étale over the closed point, so it is étale. By the same argument, the diagonal is also étale. Therefore is a gerbe over its moduli space, which is a union of disjoint sections of .

Step 4: structure of . We only have left to prove that the morphism is an isomorphism outside the union of and the singular locus of the map . There exists a closed substack of where the inertia groups are nontrivial; this substack meets at most at points of and . We need to analyze the structure of near a point of and near a point of .

Take a geometric point , and let be the completion of the strict henselization of at . Call the automorphism group of ; this is a cyclic group of order prime to the characteristic of . Denote .

Case 1: .

Now is isomorphic to , where is an indeterminate which is a semi-invariant for . The ideal generated by defines at . Lift to a semi-invariant element . Then ; we only need to check that the ideal generated by defines . Denote the reduction of in by . The ideal of in is generated by a semi-invariant , and it is easy to see that , where is an invariant unit. So also generates the ideal of in , and this implies that generates the ideal of in . This proves that is isomorphic to outside in a neighborhood of each point of .

Case 2: .

This time is isomorphic to . We can choose and semi-invariants. By deformation theory, there exists a lifting of and of , such that ; in other words, . Let be any semi-invariant lifting of . Then for some unit . If we denote , then we have . Denote by and the characters by which acts on and . Write

Clearly is a semi-invariant lifting of . If , then clearly and we are done. If, on the other hand, , then , and we have

Again, this proves that is isomorphic to outside in a neighborhood of each point of .

So the pair , with the map gives a twisted stable map. This map is stable, since is stable.

Thus is an algebraic stack. Since an automorphism of a twisted object fixing is determined by its action on the generic object , and since is a tame Deligne-Mumford stack, we obtain that is of Deligne-Mumford type and tame.

Remark 5.4.2.

The proof of Case 2 above implies the following lemma, which will be used later (Proposition 8.1.1).

Lemma 5.4.3.

Let be a twisted object. Then there is an open and closed subscheme such that if is a geometric point of , then the pullback of to is balanced if and only if is in .

6. The weak valuative criterion

We wish to show that is proper and is projective. We start by verifying the weak valuative criterion for .

Let be a discrete valuation ring, . Let be the generic point, the special point. For an injection of discrete valuation rings , we denote by the corresponding schemes.

Proposition 6.0.1.

Let be a stable twisted object. Then there is an injection of discrete valuation rings inducing a finite extension of fraction fields, and an extension

that is, is a stable twisted object and its pullback to is isomorphic to the pullback of to . The extension is unique up to a unique isomorphism, and its formation commutes with further injections of discrete valuation rings. If is balanced, then is balanced as well.

Proof.

We will proceed in steps.

Step 1: extension of . Let be the coarse moduli morphism. By Reference F-P and Reference B-M, we know that is a proper algebraic stack. By the weak valuative criterion for , it follows that there is an injection of discrete valuation rings , inducing a finite extension of fraction fields, and an extension

such that is a family of Kontsevich stable maps. The extension (Equation 3) is unique up to a unique isomorphism and commutes with further base changes.

We now replace by . Let be a uniformizer.

Step 2: extension of over . We may assume that over we have a chart with and trivial : such a chart is compatible with any other chart, so we may add it to the atlas. We may extend this via . However the map does not necessarily extend over , so we may need a further base change.

Let be the components of the special fiber , and let be the generic points. Consider the localization . This is the spectrum of a discrete valuation ring. By the weak valuative criterion for , there is a tamely ramified quasifinite surjective cover and a map lifting the map on .

We proceed to simplify these schemes . Denote . By Abhyankar’s lemma (Reference G-SGA1, exp. XIII, section 5) we may assume that is étale over . By descent we already have a lifting . Taking divisible by all the , we have an extension .

We now replace by . Thus there is an extension . Note that this extension is unique, since is separated. For the same reason it commutes with further base changes.

By Reference dJ-O, there is a maximal open set with an extension of the morphism, . Since contains both and , we have for a finite set of closed points . By the purity lemma (Lemma 2.4.1) we have that , and there is an extension .

The uniqueness of the lifting in the purity lemma guarantees that this extension is unique up to a unique isomorphism, and commutes with further base changes.

Step 3: extension of over nongeneric nodes. Let be a node, and assume is not in the closure of .

To build up a chart near , we first choose an étale neighborhood of , as follows. Let be a Zariski neighborhood such that . We already have a morphism . We take an étale neighborhood of over so that is a split node on . Thus we can find elements in the maximal ideal of , such that , satisfying the equation for some . In other words, is also an étale neighborhood of the closed point in .

Write , where is the maximal power of the residue characteristic in , and thus is prime to the characteristic.

We now find a nodal curve , which is Galois over , with an equivariant extension . Let be defined as follows:

and . There is an obvious balanced action of the group of -th roots of unity on via , and is the associated quotient morphism. Let . The action of clearly lifts to . We write for the quotient map.

Notice that , and therefore , satisfies Serre’s condition . Also, (and therefore also ) has a homeomorphic nonsingular cover given by taking roots of order of and . Thus the local fundamental group is trivial, and the purity lemma applies. By the purity lemma we have a lifting of the morphism . Notice that for we have . Thus by the uniqueness of the lifting, the map commutes with the action of .

Let be the associated object and let be the fiber over . The group acts on , stabilizing . Let be the subgroup acting trivially on . Denote by and the quotients. By Lemma 2.5.1 there is a -equivariant object , and the action is essential.

Thus we have a chart near . This is easily seen to be unique and to commute with further base changes. Note that the chart we have constructed is automatically balanced.

Step 4: extension of charts over generic nodes. Let , and let be in the closure of . The construction of a chart here is similar to step 2. We choose a Zariski neighborhood of such that . As before, we may choose an étale neighborhood over such that is a split node, thus is étale over . We may assume that is given by where for an appropriate integer prime to the residue characteristic, with the group . Otherwise we can add a compatible chart with such a to our atlas. There is an obvious extension via the same equation , and the action of extends automatically. We already have a lifting . Since is normal crossings, the purity lemma applies, and guarantees that there is a unique equivariant extension to . It is easy to check that this gives an extension of the chart, which is essential (as the scheme of is finite unramified). Also, if the chart on is balanced at , then the chart we have constructed is evidently balanced as well.

Step 5: extension of over . This step is identical to the previous one.

The uniqueness and base change properties are straightforward.

7. Boundedness

To complete the proof of properness of , we need to show it is of finite type. Since is of finite type, it suffices to show that the morphism is of finite type. As a first step, we show in Theorem 7.1.1 that for a family of stable maps over a scheme , there is a family of twisted stable maps over a scheme of finite type over which on geometric points exhausts all possible liftings of the given stable maps. The existence of algebraic deformation spaces allows one to capture the infinitesimal structure as well, showing that is of finite type.

As it turns out, the difficult step in Theorem 7.1.1 is in showing that there is an exhaustive family of generic objects for a stable map with smooth source curve (Lemma 7.2.5). We show this in a sequence of lemmas by “explicitly” giving descent data for all such generic objects.

7.1. The statement

The bulk of this section is devoted to proving the following result:

Theorem 7.1.1.

Given a morphism , where is a scheme of finite type over , there exists a morphism of finite type, and a lifting , such that a geometric point of is in the image of if its image in is in the image of .

Let be the corresponding underlying family of curves, with stable map .

7.2. The smooth case

Proposition 7.2.1.

The theorem holds when is smooth.

By noetherian induction, it suffices to prove the proposition after a dominant base change of finite type on . In particular we may assume irreducible. Consider , the closure of the image of in ; by Reference L-MB, 16.6, there exists a finite generically étale map . Then by Lemma 2.3.3, the scheme is the moduli space of , so, by Reference L-MB, 11.5, the stack is generically étale over . So it follows that is also generically étale over .

Let be the normalization of an irreducible component of mapping surjectively onto . By refining , we may assume that

(1)

is smooth;

(2)

there exists an open subset such that

(a)

is a union of disjoint sections containing ,

(b)

if we denote by , then is étale, and

(c)

the scheme of automorphisms of obtained by composing is étale over of constant degree .

Lemma 7.2.2.

There exists a morphism of finite type, and a morphism such that is smooth of relative dimension , is étale over of degree , and tame over , with the following property.

For every geometric point and every tame cover of order which is étale over , there exists a lifting , such that is isomorphic to as a covering of .

Proof.

Since is tame of degree , the genus of every connected component of is bounded by some integer . It is sufficient to consider one connected component of at a time; call its degree over . Consider the stack of maps ; there is a locally closed substack consisting of maps which are étale over and with smooth source curve. There exists a scheme surjective and of finite type; let be the corresponding universal family. We use the following well-known lemma:

Lemma 7.2.3.

There exists a constructible subset such that for a geometric point the fiber is a tame covering if and only if is in .

Proof.

By noetherian induction it suffices to show that there is an open set where the lemma holds, and thus we may replace by a dominant generically finite scheme over it. Let be the geometric generic point of , and let be the normalized cover associated with , where is the symmetric group. There is a finite extension such that the cover comes from . There is a dominant generically finite such that is the generic point of and comes from and the latter is the normalized cover associated with . The set of fixed points of elements of order in is closed in , and the image in is constructible.

To conclude the proof of Lemma 7.2.2, we take to be a disjoint union of locally closed subschemes of covering .

Consider , . By Abhyankar’s Lemma, are smooth.

We have two morphisms obtained by composing the map with the two projections . Consider the scheme . The morphism is finite, and therefore is projective over . So there exists a scheme , quasiprojective over , parametrizing sections of . In other words, there exists a diagram

denote the universal section .

Considering the three natural projections , we have a composition map . There exists a closed subscheme where the composition

Lemma 7.2.4.

There exists a lifting of , such that the following holds: whenever is a geometric point and is a lifting of , there exists a lifting and an isomorphism .

Proof.

The section , when restricted to the open set the inverse image of , gives descent data for constructing , which is effective since is a stack. Given a geometric point and a lifting of , we have a diagram

giving two morphisms which induce the same morphism . Therefore the scheme is étale over , as it is a torsor under . For the same reason, it is tame along . Thus there exists a lifting such that is isomorphic to the normalization of in . In particular, we have an isomorphism of with ; by conjugation this gives a section , which extends to , since is a smooth curve. This extended morphism is easily seen to satisfy the cocycle condition , so it gives a point , and the corresponding descent data give the lifting .

Replacing by a regular stratification, this proves the following:

Lemma 7.2.5.

There exist a morphism of finite type, with regular, and a lifting , such that whenever is a geometric point and is a lifting of , there exists a lifting and an isomorphism .

This gives an exhaustive family of generic objects over . We can now consider extensions of these to twisted objects.

Lemma 7.2.6.

Let be a smooth curve over an algebraically closed field, an open subset, a morphism. Then there exists a twisted pointed curve , unique up to a unique isomorphism, with moduli space and a representable morphism extending .

Proof.

Choose an étale neighborhood of the inverse image in of , such that , where is a finite group. Consider

Passing to an étale neighborhood of , we may assume that the normalization of some irreducible component of has exactly one point over . Let be the stabilizer of . Note that fixes . Consider the subgroup stabilizing the object . Denote , , and the image of . Refining if necessary, we may assume that is the unique fixed point of in . By Corollary 2.5.2 we have that

forms a chart at .

Uniqueness follows from the separatedness of .

We return to the proof of the proposition. Let be the geometric generic point. By the lemma, there exists a representable lifting

over a twisted curve .

There exists a dominant quasifinite morphism such that the lifting above descends to . We claim that there exists an open dense subscheme such that is representable. This follows from the following well-known fact:

Lemma 7.2.7.

Let be a morphism of Deligne-Mumford stacks, and let be proper. Then there is an open subset such that if is a geometric point, the restricted morphism is representable if and only if factors through .

Proof.

Consider the morphism of inertia stacks . The kernel is a finite unramified representable group stack , therefore the complement of the identity is also finite over . Then is the complement of the image of in , which is closed.

We may now assume that we have a lifting over . To conclude the proof of the proposition, there exists an open and closed subscheme , such that for any section in the inertia group of a geometric point in is trivial if and only if the point lies over .

7.3. Proof of Theorem 7.1.1

By taking a surjective map of finite type we may assume that:

(1)

is of locally constant topological type.

(2)

All irreducible components of have geometrically irreducible fibers.

(3)

Any irreducible component of the singular locus of maps isomorphically to its image.

This implies that the normalization can be viewed as a union of families of smooth pointed curves with a stable map to . By Proposition 7.2.1 we may assume that there is a dominant morphism of finite type, and a lifting

For each , let be the two sections over , and the gerbes in over them. Replacing by an étale cover we may assume that and have sections and . Refining we may assume that there are charts

for along and and that the sections lift to . Also we may assume that surjects onto the inverse image of in . This implies that the groups are cyclic, and isomorphic to .

Denote . We have a canonical isomorphism , thus there is an embedding of . Consider the subscheme of the scheme of isomorphisms corresponding to isomorphisms which identify with . Refining , we may assume that is a union of disjoint copies of . Consider a section .

We choose:

(1)

a geometric point ,

(2)

a universal deformation space for at , equivariant under the action of .

By refining and , we may assume that there is a lifting of .

Now the chosen isomorphism of with and the universal property of the deformation space imply that, after refining , there is a lifting of such that the two morphisms coincide.

We thus obtain a morphism

The identification of with given by defines an action of on both . It acts on via the embedding . The morphism is -equivariant, therefore

is a chart for a twisted map .

Taking the union over all , this defines

This clearly exhausts all liftings over all geometric points. Setting completes the proof of Theorem 7.1.1.

7.4. Finiteness of fibers of

To show that has finite fibers, we need to show that given a stable map over an algebraically closed field, there are only finitely many twisted stable maps over it. It is easy to see that there are only finitely many possible choices for , since it admits a finite projective morphism to , in other words, the “twisting” of is bounded by the order of stabilizers in . Thus the following lemma suffices:

Lemma 7.4.1.

Let be a twisted curve over an algebraically closed field with coarse moduli space . A morphism can come from at most finitely many representable maps , up to isomorphism.

Proof.

First of all observe that given an object there are at most finitely many twisted objects on whose restriction to is isomorphic to ; so it is enough to prove that any given map , where is a smooth curve, comes from at most finitely many objects on .

Over , it is easy to deduce this finiteness result analytically: given a base point and a fixed lifting , it is easy to associate, in a one-to-one manner, to any lifting a monodromy representation . Since is finitely generated there are only finitely many such representations.

We now give a short algebraic argument using the language of stacks.

Call the normalization of the pullback of to ; if there is at least one map giving rise to , then this induces a section . It is proved in Reference Vi that this section is étale; the argument is similar to that of the purity lemma: by passing to the strict henselization one can lift to the universal deformation space. The image of this lifting is then evidently étale, and therefore is étale. This strongly restricts the structure of .

Call the fiber product ; then the étale finite groupoid is a presentation of . Each lifting of comes from a section , so it is enough to show that up to isomorphisms there are only finitely many such sections. But such a section is determined by the pullback of to via , and a morphism of groupoids from to over . But up to isomorphism there are only finitely many étale coverings of bounded degree, because the tame fundamental group of is topologically finitely generated. For each there are only finitely many liftings .

7.5. Conclusion of the proof of the main theorem

Theorem 7.1.1 and the fact that the stack is of finite type (see, e.g., Reference B-M) imply that the stack is of finite type. Since the stack has finite diagonal and satisfies the weak valuative criterion for properness, it is proper. The induced morphism of coarse moduli spaces is proper with finite fibers. Since is projective, we have that is projective as well. We have already seen that is of Deligne-Mumford type and tame.

8. Some properties and generalizations

8.1. Balanced maps

Proposition 8.1.1.

The stack of balanced twisted stable maps is an open and closed substack in .

This is immediate from Lemma 5.4.3.

8.2. Fixing classes in Chow groups modulo algebraic equivalence

Assume is the spectrum of a field. Given a class in the Chow group of cycles of dimension 1 on modulo algebraic equivalence, such that its intersection number with the Chern class of the chosen ample sheaf is , we have an open and closed substack . Pulling back, we have an open and closed substack . A similar construction can be obtained using classes on , whenever one can construct them with integral coefficients; see, e.g., Reference E-G, Reference Kr.

8.3. Stable maps over a base stack

Suppose we are given a tame morphism of algebraic stacks, where is noetherian. Suppose further that it factors as , where is proper and quasifinite, and is projective (but we do not assume that is representable). We can define a category of twisted stable maps exactly as before. We claim that the analogue of the main theorem still holds, which may be of use in some applications.

The point is the following: most arguments in this paper go through word for word. To produce a smooth parametrization take a smooth surjective morphism , and denote by the pullback of to . Then

is representable, smooth, surjective and of finite type. Composing with a smooth parametrization of the stack we get what we want. To show that is proper, note that its pullback after a faithfully flat base change to a scheme is proper.

8.4. When is only a proper algebraic space

So far we have assumed that is projective. In case is only proper, one can proceed as follows.

First, it was pointed out by Johan de Jong that in this case is an algebraic stack, locally of finite type, such that each irreducible component is proper over . The only part which is not evident is the properness of the irreducible components. This can be seen as follows: let be a generic point of such an irreducible component, with a stable map . An easy gluing argument allows us to reduce to the case where is irreducible. We then replace by the closure of the image of . There is a birational morphism such that is projective. There is a lifting , which has a certain degree with respect to some ample sheaf on . The map induces . Take the closure of the image, which is a proper stack. There is a universal stable map . The composite map may be unstable, but using Knudsen’s contraction procedure we obtain an associated map , giving a morphism whose image, which is the component with which we started, is proper.

Now considering a tame stack proper over , the arguments of the main theorem along with the one above show that is again an algebraic stack, locally of finite type, such that each irreducible component is proper over .

It can be shown that a similar result holds when is only tame, separated and of finite type, without properness. The crucial point is that exists in such a situation (though is not proper, not even of finite type), and then one can apply our arguments, or simply apply our main theorem working with schemes over rather than over , with target stack with the universal curve, rather than . Unfortunately we are not aware of a treatment of in this general case in the literature. A particular open and closed substack of whose existence is clear is , the stack of maps which are constant on moduli spaces. In this case . This case is considered in Reference -G-V.

8.5. Other open and closed loci

Consider a twisted object , and take a geometric point of the support of the -th section . If is a chart in the atlas, and is a geometric point of lying over , the stabilizer of in has an order which is independent of the chart. We call this order the local index of the twisted object at ; we can think of it as a measure of how twisted the object is around . One immediately checks that the local index only depends on the image of in ; this way we get a function .

Proposition 8.5.1.

The function is locally constant.

Proof.

Let be a geometric point of , a chart, a geometric point of lying over . By refining the chart we may assume that is a fixed point of . Since the morphism is étale, it follows that leaves the whole component of containing fixed, and the thesis follows easily from this.

Definition 8.5.2.

Let be a sequence of positive integers. A twisted object has global index equal to if the value of the associated function on is constant equal to for all .

Sometimes the local index can be further refined. For example, if is a finite group, and is the classifying stack of over , that is, the stack whose objects are Galois étale covers with group , then the stabilizer of a geometric point of is a cyclic subgroup of the automorphism group of an étale Galois cover of the spectrum of an algebraically closed field. This group is isomorphic to , and the isomorphism is well defined up to conjugation, so from each geometric point of one gets a conjugacy class of cyclic subgroups of , which only depends on the index and on the image of the geometric point in . It is easy to check that this conjugacy class is locally constant on .

9. Functoriality of the stack of twisted stable maps

9.1. Statements

Let be a proper -pointed twisted curve of genus and let be a morphism of stacks. Consider the corresponding morphism of coarse moduli spaces . Assume that either is nonconstant or . It is well known (see Reference B-M, Proposition 3.10) that is stabilizable: there exists a canonical proper surjective morphism of pointed curves which gives a factorization such that is a stable pointed map and the fibers are the nonstable trees of rational curves.

Proposition 9.1.1.

There exists a factorization such that

(1)

is a twisted stable map, and

(2)

on the level of coarse moduli spaces this induces the factorization .

The factorization is unique up to a unique isomorphism. The formation of commutes with base change on . Moreover, if is balanced, then so is .

We denote . The following corollaries follow immediately from the proposition:

Corollary 9.1.2.

Let be a morphism of proper tame stacks over having projective coarse moduli spaces. Denote by the open and closed locus where is stabilizable: if , then . Otherwise it is the locus where the composite map of moduli spaces is nonconstant. Then there exists a morphism , sending balanced maps to balanced maps, which makes the following diagram commutative:

where the vertical arrows are the canonical maps described in Theorem 1.4.1 and the morphism is the one described by Knudsen and Behrend-Manin, induced by the contraction of rational trees which become nonstable over .

Corollary 9.1.3.

Let be the open and closed substack where the last marking is representable. Assume either or . Then there exists a morphism , sending balanced maps to balanced maps, which makes the following diagram commutative:

where the vertical arrows are the canonical maps described in Theorem 1.4.1 and the morphism is the one described by Knudsen and Behrend-Manin, induced by the contraction of rational components which become nonstable when forgetting the last marking .

Remark 9.1.4.

Given a -scheme of finite type , not necessarily proper, one defines a stable divisorially -marked map over to be a nodal divisorially -marked curve and a proper morphism such that, for each geometric point of , the group is finite. Ideally, one should prove the existence of the contraction of nonstable trees for divisorially -marked maps even in the nonproper case, and then deduce our proposition using a presentation of . There are some technical difficulties in carrying out such a local construction. We will use the uniqueness of the local construction to deduce the uniqueness in the twisted case, but we will use a global approach for proving existence.

9.2. Twisted nonstable trees

Before proving the proposition, we state two lemmas which illuminate the geometric picture underlying the proposition. Consider a twisted curve over an algebraically closed field, and a representable morphism which is constant on moduli spaces. If is the image point in , then the reduction of its inverse image in is the classifying stack of the automorphism group of a geometric point of lying over . Thus the map factors through .

Lemma 9.2.1.

Let be a tree of rational curves over an algebraically closed field . Let be a smooth closed point. Let be a -pointed twisted curve with associated coarse curve . Let be a finite group with order prime to , and let be a representable morphism. Then is an isomorphism, and the map corresponds to the trivial principal bundle.

Proof.

We proceed by induction on the number of components of , starting from the case where is empty, when the statement is vacuous. Let be a tail component; in case is reducible we may assume , so in any case contains at most one special point . Denote .

Let be the principal -bundle corresponding to . Now restricting to , we obtain a tame principal bundle , which is trivial, since .

Define to be the reduction of the inverse image of . The principal bundle has a section outside , which automatically extends to . Therefore is trivial, which means that factors through . Since is representable, we have that is representable.

By induction, the lemma holds over . It follows that is representable, and since , the principal bundle is trivial.

Lemma 9.2.2.

Let be a tree of rational curves over an algebraically closed field . Let be two distinct smooth closed points. Let be a -pointed twisted curve with associated coarse curve . Let be a finite group with order prime to , and let be a representable morphism. Denote by the unique chain of components connecting with . Then:

(1)

is an isomorphism away from the special points of , namely and the nodes of .

(2)

There exists a cyclic subgroup such that the image of the automorphism group of a geometric point lying over any of the special points of maps isomorphically to .

(3)

Let be the principal -bundle associated to . Then each connected component of is a tree of rational curves.

(4)

For a geometric point lying over a special point of , we let act on the tangent space of at via the isomorphism in 2 above. Assuming is balanced, then the character of the action of on the tangent space to at is opposite to the character of the action at .

Proof.

Consider a connected component . Then is a tree of rational curves which is attached to at a unique point . Let be the reduction of the inverse image of and let be the reduction of the inverse image of in . Then we can apply the previous lemma to and conclude that is an isomorphism. Thus we may replace by and assume that .

Let be the principal -bundle associated with the morphism . Then is a tamely ramified covering which is étale outside of the special points. Let be a connected component and let be its stabilizer. Note that , and since the action of on the set of connected components of is transitive, we have that is an isomorphism. Let be an irreducible component, and its image. Then, since the tame fundamental group of is cyclic, we know that is a smooth rational curve, and the map is totally ramified at exactly the two special points of on . This implies that the map of dual graphs is étale, and since is simply connected this implies that this map of dual graphs is an isomorphism. From this it follows that there is exactly one irreducible component of over each irreducible component of and the results are clear.

9.3. Proof of the proposition

We may assume that is noetherian, since and are of finite presentation.

Step 1: reduction to the representable case. First we reduce to the case where is representable, as follows: let be a presentation of . Consider the pullbacks of to and , respectively. These are divisorially -marked twisted curves. Let and be the respective coarse moduli spaces, which are divisorially -marked curves. By Lemma 2.2.2 there is an induced étale groupoid whose quotient is a twisted pointed curve with a representable morphism . The coarse moduli space of is canonically isomorphic to . Note that, since is tame, the formation of commutes with base change on , so the formation of also commutes with base change on .

Let us assume that is balanced. In order to show that is balanced as well, it is enough to consider the case where is the spectrum of an algebraically closed field. Let be a chart for . Without loss of generality we may assume is cyclic and has a unique fixed geometric point , and the action of is free outside . The morphism induces a morphism . Denote the kernel of this morphism by and the image by . Then is a chart for , which is evidently balanced.

Thus from here on we will assume is representable.

Step 2: uniqueness. Let be a contraction as in the proposition. Consider a presentation of and consider the following fiber diagram:

Note that , , , and are representable. Moreover, it follows from Lemmas 9.2.1 and 9.2.2 that the fibers of and are precisely the nonstable trees in and .

Lemma 9.3.1 below implies that the groupoid is uniquely determined as a local contraction of nonstable trees in , and its formation commutes with base change on . Therefore is unique, and its formation also commutes with base change on .

Lemma 9.3.1.

Let be a scheme of finite type, a divisorially marked curve, and a proper morphism. If is a contraction of the nonstable trees, then it is unique up to a canonical isomorphism and its formation commutes with base change.

Proof of the lemma.

Note that the last statement follows from unicity. Now is uniquely determined as a topological space, since the fibers are uniquely determined and the topology is the quotient topology since the map is proper. Therefore it is enough to check that the morphism is an isomorphism.

Note that this morphism is injective since the kernel is supported in the locus of special points of , which contains none of the associated points of .

Denote by the quotient sheaf

We need to show . We may assume that is the spectrum of a local ring .

(1)

Consider the case where is a field. We may pass to the algebraic closure, and then the statement follows from the fact that nodal curves are seminormal.

(2)

Consider the case . In this case the ideal of the support of has depth . This implies that the sequence splits locally on , by the characterization of depth (see Reference Ma, Theorem 16.7). Let be a local splitting. It is enough to prove that this is injective. Let be a nonzero section of the kernel of this splitting. By restriction to the central fiber we see that , where is the maximal ideal of . Let be the maximal integer such that . We note that as a section of , the element vanishes outside the nonstable trees. Since is a constant vector bundle on , we have that the restriction of to the central fiber is again zero, therefore , contradicting the maximality of .

(3)

In the general case, denote by the largest proper ideal of finite length (so is either a field or of positive depth). Let be an ideal of length 1. By induction on the length of , we may assume the statement holds true over .

The exact sequence

induces an exact sequence

Note that is isomorphic to , where is the central fiber. Applying we get an exact sequence

Note that the last term is zero. This follows by considering the inverse image of an affine neighborhood of a point on containing the image of a contracted tree and computing the cohomology long exact sequence of

Now we get a commutative diagram

where the left and right columns are isomorphisms. This implies that the center column is also an isomorphism, which concludes the proof the lemma.

Step 3: construction of . Consider the diagram

where is the contraction constructed by Behrend-Manin. Let be a sufficiently relatively ample invertible sheaf on , and let be its pullback to . Here “sufficiently ample” means that the degrees of on the components of the fibers of are larger than a certain integer to be specified later.

We define a quasicoherent sheaf of graded algebras

Claim.
(1)

For any quasicoherent sheaf of -modules , we have

for all .

(2)

For all , the formation of commutes with base change on , and the sheaf is flat over .

(3)

The algebra is generated over in degree 1; in particular it is locally finitely generated.

Proof.

Note that for any quasicoherent sheaf of -modules we have

therefore the formation of commutes with base change on (see Reference G-EGA, 7.3.1).

Let be an affine scheme and a surjective étale morphism of finite type. Denote by the pullback of to . Let be the subcurve consisting of components of fibers mapping to a point in . Note that, since is noetherian, the number of topological types occurring in the fibers of is finite. Moreover, it follows from Lemma 9.2.1 and statement (3) in Lemma 9.2.2 that the inverse images of the nonstable trees of in are disjoint unions of trees of rational curves.

To check that , it is enough to check the case where is the structure sheaf of a geometric point of . This is a consequence of the following elementary lemma, whose proof is left to the reader:

Lemma 9.3.2.

Let be a quasiprojective nodal curve, and let be the maximal proper subcurve. Then there exists a number , depending only on the topological type of , such that the following holds: Let be an invertible sheaf satisfying the following properties:

(1)

for any component , the degree of on is either or , and

(2)

the union of the components such that is a disjoint union of trees of rational curves.

Then

, and

the algebra is generated in degree .

Let . Then is the seminormal curve obtained by contracting the components of on which has degree .

This concludes the proof of part (1) of the Claim. Also, once we prove part (2), then Lemma 9.3.2 implies part (3).

For part (2), note that since , the formation of commutes with base change on . This means that for any quasicoherent sheaf of -modules we have the equality . But the functor sending to is evidently left exact, which means that the sheaf is flat. This concludes the proof of the Claim.

Define .

Step 4: verification that is a twisted stable pointed map.

Since is projective, it is representable. Since is flat over for all , we have that is flat.

Let be one of the markings. We claim that the composition is an embedding, and we define to be its image in . This, as well as the fact that satisfies statement (1) of the proposition, can be checked on geometric fibers.

We claim that statement (2) can also be checked on the level of geometric fibers. First, the local criterion of flatness implies that a proper morphism of flat -schemes which is an isomorphism on the geometric fibers of is an isomorphism. Also, the formation of coarse moduli spaces of tame stacks commutes with arbitrary base change (see Lemma 2.3.3). Since the moduli space of a family of twisted nodal curves is flat, the claim follows.

So we assume that is the spectrum of an algebraically closed field.

The formation of the algebra commutes with base change along étale covers . It follows from Lemma 9.3.2 that over , the curve is the contraction of the nonstable trees in . Using descent and the uniqueness of the contraction, we have that satisfies statement (1) in the proposition. Denote by the moduli space of . The morphism is a bijection on geometric points and is an isomorphism away from the special points, and since is seminormal, the morphism is an isomorphism.

Appendix A. Grothendieck’s existence theorem for tame stacks

A.1. Statement of the existence theorem

Consider a proper tame stack of finite type over a noetherian complete local ring with maximal ideal . Then has a moduli space , which is a proper algebraic space over .

For each nonnegative integer set . We define the category of formal coherent sheaves on in a rather simple-minded way. A formal coherent sheaf on is a collection of coherent sheaves on , together with isomorphisms of sheaves over . A homomorphism of formal coherent sheaves is a compatible sequence of morphisms , in the obvious way. We shall denote the category of formal coherent sheaves on by . From this description it is not even clear that is abelian.

There is an obvious functor from the category of coherent sheaves on to , sending a coherent sheaf to the compatible system of restrictions .

Theorem A.1.1.

If is proper and tame over , the functor is an equivalence of categories.

This is of course well known if is an algebraic space (see Reference Kn, V, 6.3). The proof for stacks is not too different from the one in Reference Kn.

A.2. Restatement in terms of -modules

For the proof, we will need a more manageable description of the category of formal coherent sheaves. Consider the étale site of , where the objects are the étale morphisms , where is a scheme, the arrows are maps over , and coverings are defined as usual (Reference L-MB). On this site we have two sheaves of rings; the usual structure sheaf, which we denote by , which sends each étale map to , and the completed structure sheaf , which sends an étale map to the completion of with respect to the inverse image of the maximal ideal of . We denote by the ringed site , equipped with the latter sheaf of rings .

Notice that in the case that is a scheme, the site is not the étale site of the corresponding formal scheme; however, since the structure sheaf of the formal scheme is supported on the closed fiber, the two categories of sheaves of modules on and on the étale site of the formal scheme are canonically equivalent, so this does not really make a difference.

Consider a sheaf of -modules . Let be an étale map, where is a scheme, and let be the restriction of to the Zariski site of . The sheaf is actually supported on the fiber , so the restriction of to defines a sheaf on the formal completion of the scheme along . We say that is a coherent sheaf on when is a coherent sheaf on the formal scheme for all étale maps . The category of coherent sheaves of modules over will be denoted by ; it is clearly an abelian category.

There is an obvious functor that sends a coherent sheaf of modules over to the object of . This functor is easily checked to be an equivalence of categories; the inverse is obtained by taking inverse limits in the usual fashion, i.e., one can send an object of into , and this is a coherent sheaf of modules over .

Now, the functor above corresponds to the functor which sends a coherent sheaf to . This induces a natural morphism from the cohomology of on to the cohomology of the completion of on . The first important step of the proof of the theorem consists of proving the following form of Zariski’s theorem on formal functions.

A.3. The theorem on formal functions

In what follows we will use repeatedly the well-known fact that the cohomology of a coherent sheaf, considered as a sheaf on the small étale site of a scheme, is the same as the cohomology of the same sheaf on the Zariski site.

Theorem A.3.1.

If is proper over , the morphism is an isomorphism.

Lemma A.3.2.

Let be a formal sheaf on , and set . Then the natural map is an isomorphism.

Proof.

This is very much the proof of the corresponding fact in Reference Kn, V, 2.19, except that it is, hopefully, correct. Let be a surjective étale map, where is an affine scheme. Then all the fiber products are affine, because is separated. The usual spectral sequence

relating Čech cohomology and usual cohomology degenerates, because the cohomology of over each degenerates, so cohomology is equal to Čech cohomology over the covering . We claim that the same is true for the projective limit

For this it is enough to show that the cohomology of on each is zero in positive degree. This is the content of the lemma that follows.

Lemma A.3.3.

The cohomology of the limit of a strict projective system of coherent sheaves on the small étale site of an affine noetherian scheme is zero in positive degree.

Proof.

Let be the affine scheme, the projective system, and the limit. According to a theorem of M. Artin (see Reference Mi, Theorem 2.17), the Čech cohomology of on the small étale site equals its cohomology; so it is enough to show that, given an étale map of finite type, with affine, the Čech cohomology vanishes in positive degree. Let us denote by the augmented Čech complex

of a sheaf .

By the strictness assumption, each of the maps is surjective, so is surjective for each , because is affine and is coherent. Thus the projective system

is a projective system of acyclic complexes satisfying the Mittag-Leffler condition in each degree, and therefore its projective limit

is acyclic.

To conclude the proof of Lemma A.3.2, it is enough to show that the map

is an isomorphism. The Čech complexes form a projective system of complexes which satisfy a Mittag-Leffler condition in each degree. Furthermore the cohomology groups are artinian modules over , so the corresponding projective systems also satisfy a Mittag-Leffler condition. The statement follows from Reference G-EGA, 0, 13.2.3.

Consider the moduli space .

Lemma A.3.4.

The natural map is an isomorphism.

Proof.

This statement is local in the étale topology, so we may assume that , , . The sheaf is associated with the -equivariant -module , and corresponds with the -module . Then the statement corresponds to the fact that the natural map is an isomorphism. The surjectivity follows immediately from the fact that any invariant in the right-hand side lifts to an invariant in . For the injectivity, take an element in which goes to 0 in ; then we can write , where and . By averaging, we may assume that the are invariant, so that .

To complete the proof of the theorem of formal function A.3.1, notice that from the theorem on formal functions for algebraic spaces (Reference Kn, V, 3.1), we conclude that

A.4. Algebraization of extensions, kernels, and cokernels

Lemma A.4.1.

Let and be two coherent sheaves on . The natural map

is an isomorphism for all .

Proof.

This is a local statement in the étale topology, so it follows from the case that is a scheme.

Lemma A.4.2.

Let and be two coherent sheaves on . The natural map

is an isomorphism for all .

Proof.

This follows from the theorem on formal functions and Lemma A.4.1 above, by considering the local global spectral sequences for and .

This implies that the functor is fully faithful. We have to prove that all formal coherent sheaves on are algebraizable, that is, it is isomorphic to the completion of a coherent sheaf on .

Lemma A.4.3.

The kernel and the cokernel of a morphism of algebraizable formal sheaves are algebraizable.

Proof.

Let and be coherent -modules, and a morphism of -modules. By the previous result, comes from a morphism of -modules . Since completion is an exact functor, the kernel and the cokernel of are the completion of the kernel and cokernel of .

Lemma A.4.4.

Any extension of algebraizable formal sheaves is algebraizable.

Proof.

If and are coherent -modules, the natural map

is an isomorphism, by Lemma A.4.2.

A.5. Proof of the existence theorem

By noetherian induction, in order to prove the existence theorem we may assume that every coherent sheaf on is algebraizable when it is zero on an open nonempty substack of . Let be a surjective finite generically étale map, where is a scheme; this exists by Reference L-MB, 16.6. If is the completion of , then the diagram of ringed Grothendieck topologies

is cartesian. From this we deduce the following lemma.

Lemma A.5.1.

Let be a coherent sheaf on . Then the natural map is an isomorphism.

Lemma A.5.2.

Let be a coherent sheaf on . Then the sheaf is a coherent sheaf of -modules.

Proof.

By restricting to an étale map , where is an affine scheme considered with the Zariski topology, this becomes obvious.

From these two lemmas we see that a formal sheaf on of the form is algebraizable.

Now, we form the morphism . Let be a formal coherent sheaf on ; there are two natural maps and from to induced by the two projections ; let be the kernel of the difference . The adjunction map factors through , and, by flat descent, the induced map is an isomorphism over the open dense substack of where the map is flat. By Lemma A.4.3, is algebraizable; also, the kernel and the cokernel of the map are algebraizable, by the induction hypothesis. So the image of is algebraizable, and therefore by the same lemma is algebraizable.

Table of Contents

  1. Abstract
  2. 1. Introduction
    1. 1.1. The problem of moduli of families
    2. 1.2. Stable maps
    3. 1.3. Stable maps into stacks
    4. 1.4. Twisted stable maps
    5. Theorem 1.4.1.
    6. 1.5. Some applications and directions of further work
    7. 1.6. Acknowledgments
  3. 2. Generalities on stacks
    1. 2.1. Criteria for a Deligne-Mumford stack
    2. 2.2. Coarse moduli spaces
    3. Theorem 2.2.1 (Keel-Mori K-M).
    4. Lemma 2.2.2.
    5. Lemma 2.2.3.
    6. 2.3. Tame stacks and their coarse moduli spaces
    7. Definition 2.3.1.
    8. Definition 2.3.2.
    9. Lemma 2.3.3.
    10. Lemma 2.3.4.
    11. 2.4. Purity lemma
    12. Lemma 2.4.1.
    13. Corollary 2.4.3.
    14. Corollary 2.4.4.
    15. 2.5. Descent of equivariant objects
    16. Lemma 2.5.1.
    17. Corollary 2.5.2.
  4. 3. Twisted objects
    1. 3.1. Divisorially marked curves
    2. Definition 3.1.1.
    3. Definition 3.1.2.
    4. Definition 3.1.3.
    5. Definition 3.1.4.
    6. Definition 3.1.5.
    7. Definition 3.1.6.
    8. 3.2. Generic objects and charts
    9. Definition 3.2.1.
    10. Definition 3.2.2.
    11. Proposition 3.2.3.
    12. Definition 3.2.4.
    13. 3.3. The transition scheme
    14. 3.4. Compatibility of charts
    15. Definition 3.4.1.
    16. Proposition 3.4.2.
    17. Proposition 3.4.3.
    18. 3.5. The product chart
    19. 3.6. Atlases and twisted objects
    20. Definition 3.6.1.
    21. Lemma 3.6.2.
    22. Definition 3.6.4.
    23. 3.7. Stability
    24. Lemma 3.7.1.
    25. Definition 3.7.2.
    26. 3.8. The stack of stable twisted objects
  5. 4. Twisted curves and twisted stable maps
    1. 4.1. Nodal stacks
    2. Proposition 4.1.1.
    3. Definition 4.1.2.
    4. 4.2. Morphisms of twisted -pointed nodal curves
    5. Definition 4.2.1.
    6. Proposition 4.2.2.
    7. Lemma 4.2.3.
    8. 4.3. Twisted stable maps into a stack
    9. Definition 4.3.1.
    10. Definition 4.3.2.
    11. 4.4. Equivalence of stable twisted objects and twisted stable maps
    12. Theorem 4.4.1.
    13. Lemma 4.4.2.
    14. Lemma 4.4.3.
  6. 5. The category is an algebraic stack
    1. 5.1. The stack axioms
    2. Proposition 5.1.1.
    3. Proposition 5.1.2.
    4. Lemma 5.1.3.
    5. Lemma 5.1.4.
    6. 5.2. Base change
    7. Proposition 5.2.1.
    8. 5.3. Deformations and obstructions
    9. Proposition 5.3.1.
    10. Lemma 5.3.2.
    11. Lemma 5.3.3.
    12. Lemma 5.3.4.
    13. 5.4. Algebraization
    14. Proposition 5.4.1.
    15. Lemma 5.4.3.
  7. 6. The weak valuative criterion
    1. Proposition 6.0.1.
  8. 7. Boundedness
    1. 7.1. The statement
    2. Theorem 7.1.1.
    3. 7.2. The smooth case
    4. Proposition 7.2.1.
    5. Lemma 7.2.2.
    6. Lemma 7.2.3.
    7. Lemma 7.2.4.
    8. Lemma 7.2.5.
    9. Lemma 7.2.6.
    10. Lemma 7.2.7.
    11. 7.3. Proof of Theorem 7.1.1
    12. 7.4. Finiteness of fibers of
    13. Lemma 7.4.1.
    14. 7.5. Conclusion of the proof of the main theorem
  9. 8. Some properties and generalizations
    1. 8.1. Balanced maps
    2. Proposition 8.1.1.
    3. 8.2. Fixing classes in Chow groups modulo algebraic equivalence
    4. 8.3. Stable maps over a base stack
    5. 8.4. When is only a proper algebraic space
    6. 8.5. Other open and closed loci
    7. Proposition 8.5.1.
    8. Definition 8.5.2.
  10. 9. Functoriality of the stack of twisted stable maps
    1. 9.1. Statements
    2. Proposition 9.1.1.
    3. Corollary 9.1.2.
    4. Corollary 9.1.3.
    5. 9.2. Twisted nonstable trees
    6. Lemma 9.2.1.
    7. Lemma 9.2.2.
    8. 9.3. Proof of the proposition
    9. Lemma 9.3.1.
    10. Claim.
    11. Lemma 9.3.2.
  11. Appendix A. Grothendieck’s existence theorem for tame stacks
    1. A.1. Statement of the existence theorem
    2. Theorem A.1.1.
    3. A.2. Restatement in terms of -modules
    4. A.3. The theorem on formal functions
    5. Theorem A.3.1.
    6. Lemma A.3.2.
    7. Lemma A.3.3.
    8. Lemma A.3.4.
    9. A.4. Algebraization of extensions, kernels, and cokernels
    10. Lemma A.4.1.
    11. Lemma A.4.2.
    12. Lemma A.4.3.
    13. Lemma A.4.4.
    14. A.5. Proof of the existence theorem
    15. Lemma A.5.1.
    16. Lemma A.5.2.

Mathematical Fragments

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 .

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)

.

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 .

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 .

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 .

Lemma 2.3.4.

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

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.

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 .

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 .

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.

Definition 3.2.4.

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

Proposition 3.4.2.

The two charts are compatible if and only if for any pair of geometric points and as above there exist an isomorphism of groups , a -equivariant isomorphism of schemes over , and a compatible -equivariant isomorphism .

Proposition 3.4.3.
(1)

Let , be two compatible charts for a generic object on . If is an arbitrary morphism, then

and

where and are the pullbacks of and to and , are compatible charts for the pullback of to .

(2)

If is étale and surjective, then the converse holds.

Definition 3.7.2.

A twisted object is stable if the associated map is Kontsevich stable.

Proposition 4.2.2.

The -category of twisted pointed curves is equivalent (in the lax sense, Reference K-S) to a category.

Definition 4.3.1.

A twisted stable -pointed map of genus and degree over

consists of a commutative diagram

along with closed substacks , satisfying:

(1)

along with is a twisted nodal -pointed curve over ;

(2)

the morphism is representable; and

(3)

is a stable -pointed map of degree .

Lemma 4.4.3.

Let be a morphism of Deligne-Mumford stacks. The following two conditions are equivalent:

(1)

The morphism is representable.

(2)

For any algebraically closed field and any , the natural group homomorphism is a monomorphism.

Proposition 5.1.1.

The category is a limit-preserving stack, fibered by groupoids over .

Proposition 5.1.2.

For any pair of stable twisted objects

over the same scheme , the functor of isomorphisms of twisted objects is representable by a separated scheme of finite type over .

Lemma 5.1.3.

Let be a generic object with two atlases and . Then there exists a closed subscheme such that given a morphism , the pullbacks of and to are compatible if and only if factors through .

Lemma 5.1.4.

Let be a generic object with two charts and . Let be a geometric point of ; assume that the fiber of over has a unique special point , and that there are two unique geometric points and over . Then there exist an étale neighborhood of and a closed subscheme such that given a morphism , the pullbacks of the two charts to are compatible if and only if factors through .

Lemma 5.4.3.

Let be a twisted object. Then there is an open and closed subscheme such that if is a geometric point of , then the pullback of to is balanced if and only if is in .

Equation (3)
Theorem 7.1.1.

Given a morphism , where is a scheme of finite type over , there exists a morphism of finite type, and a lifting , such that a geometric point of is in the image of if its image in is in the image of .

Proposition 7.2.1.

The theorem holds when is smooth.

Lemma 7.2.2.

There exists a morphism of finite type, and a morphism such that is smooth of relative dimension , is étale over of degree , and tame over , with the following property.

For every geometric point and every tame cover of order which is étale over , there exists a lifting , such that is isomorphic to as a covering of .

Lemma 7.2.5.

There exist a morphism of finite type, with regular, and a lifting , such that whenever is a geometric point and is a lifting of , there exists a lifting and an isomorphism .

Proposition 8.1.1.

The stack of balanced twisted stable maps is an open and closed substack in .

Lemma 9.2.1.

Let be a tree of rational curves over an algebraically closed field . Let be a smooth closed point. Let be a -pointed twisted curve with associated coarse curve . Let be a finite group with order prime to , and let be a representable morphism. Then is an isomorphism, and the map corresponds to the trivial principal bundle.

Lemma 9.2.2.

Let be a tree of rational curves over an algebraically closed field . Let be two distinct smooth closed points. Let be a -pointed twisted curve with associated coarse curve . Let be a finite group with order prime to , and let be a representable morphism. Denote by the unique chain of components connecting with . Then:

(1)

is an isomorphism away from the special points of , namely and the nodes of .

(2)

There exists a cyclic subgroup such that the image of the automorphism group of a geometric point lying over any of the special points of maps isomorphically to .

(3)

Let be the principal -bundle associated to . Then each connected component of is a tree of rational curves.

(4)

For a geometric point lying over a special point of , we let act on the tangent space of at via the isomorphism in 2 above. Assuming is balanced, then the character of the action of on the tangent space to at is opposite to the character of the action at .

Lemma 9.3.1.

Let be a scheme of finite type, a divisorially marked curve, and a proper morphism. If is a contraction of the nonstable trees, then it is unique up to a canonical isomorphism and its formation commutes with base change.

Lemma 9.3.2.

Let be a quasiprojective nodal curve, and let be the maximal proper subcurve. Then there exists a number , depending only on the topological type of , such that the following holds: Let be an invertible sheaf satisfying the following properties:

(1)

for any component , the degree of on is either or , and

(2)

the union of the components such that is a disjoint union of trees of rational curves.

Then

, and

the algebra is generated in degree .

Let . Then is the seminormal curve obtained by contracting the components of on which has degree .

Theorem A.1.1.

If is proper and tame over , the functor is an equivalence of categories.

Theorem A.3.1.

If is proper over , the morphism is an isomorphism.

Lemma A.3.2.

Let be a formal sheaf on , and set . Then the natural map is an isomorphism.

Lemma A.4.1.

Let and be two coherent sheaves on . The natural map

is an isomorphism for all .

Lemma A.4.2.

Let and be two coherent sheaves on . The natural map

is an isomorphism for all .

Lemma A.4.3.

The kernel and the cokernel of a morphism of algebraizable formal sheaves are algebraizable.

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Article Information

MSC 2000
Primary: 14H10 (Families, moduli), 14D20 (Algebraic moduli problems, moduli of vector bundles)
Author Information
Dan Abramovich
Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
abrmovic@math.bu.edu
ORCID
MathSciNet
Angelo Vistoli
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
vistoli@dm.unibo.it
ORCID
MathSciNet
Additional Notes

The first author’s research was partially supported by National Science Foundation grant DMS-9700520 and by an Alfred P. Sloan research fellowship.

The second author’s research was partially supported by the University of Bologna, funds for selected research topics.

Journal Information
Journal of the American Mathematical Society, Volume 15, Issue 1, ISSN 1088-6834, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on and published on .
Copyright Information
Copyright 2001 American Mathematical Society
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