In this paper we study a notion of twisted stable map, from a curve to a tame Deligne-Mumford stack, which generalizes the well-known notion of stable map to a projective variety.
We fix a noetherian base scheme .
Consider a Deligne-Mumford stack (definition in Section 2.1) admitting a projective coarse moduli scheme Given a curve . it is often natural to consider morphisms , (or equivalently, objects in case ); is the moduli stack of geometric objects, these morphisms correspond to families over For example, if . the stack of stable curves of genus , then morphisms , correspond to families of stable curves of genus over and if ; the classifying stack of a finite group , we get principal , over -bundles It is interesting to study moduli of such objects; moreover, it is natural to study such moduli as . varies and to find a natural compactification for such moduli.
One approach is suggested by Kontsevich’s moduli of stable maps.
First consider a projective scheme with a fixed ample sheaf Given integers . it is known that there exists a proper algebraic stack, which here we denote by , of stable, , maps of genus -pointed and degree into (See .ReferenceKo, ReferenceB-M, ReferenceF-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. ,
Now fix a proper Deligne-Mumford stack admitting a projective coarse moduli space on which we fix an ample sheaf as above. We further assume that is tame, that is, for any geometric point the group , has order prime to the characteristic of the algebraically closed field .
It is tempting to mimic Kontsevich’s construction as follows: Let be a nodal projective connected curve; then a morphism is said to be a stable map of degree if the associated morphism to the coarse moduli scheme is a stable map of degree .
It follows from our results below that the category of stable maps into is a Deligne-Mumford stack. A somewhat surprising point is that it is not complete.
To see this, we fix and consider the specific case of with Any smooth curve . of genus admits a connected principal corresponding to a surjection -bundle, thus giving a map , If we let . degenerate to a nodal curve of geometric genus then , and since there is no surjection , there is no connected principal , over -bundle This means that there can be no limiting stable map . as a degeneration of .
Our main goal here is to correct this deficiency. In order to do so, we will enlarge the category of stable maps into The source curve . of a new stable map will acquire an orbispace structure at its nodes. Specifically, we endow it with the structure of a Deligne-Mumford stack.
It is not hard to see how these orbispace structures come about. Let be the spectrum of a discrete valuation ring of pure characteristic 0, with quotient field and let , be a nodal curve over the generic point, together with a map of degree whose associated map , is stable. We can exploit the fact that is complete; after a ramified base change on the induced map will extend to a stable map over Let . be the smooth locus of the morphism Abhyankar’s lemma, plus a fundamental purity lemma (see Lemma ;2.4.1 below) shows that after a suitable base change we can extend the map to a map in fact the purity lemma fails to apply only at the “new” nodes of the central fiber, namely those which are not in the closure of nodes in the generic fiber. On the other hand, if ; is such a node, then on an étale neighborhood of the curve , looks like
where is the parameter on the base. By taking roots, -th
we have a nonsingular cover where is defined by The purity lemma applies to . so the composition , extends over all of There is a minimal intermediate cover . such that the family extends already over this ; will be of the form and the map , is given by , Furthermore, there is an action of the group . of roots of 1, under which sends to and to and , This gives the orbispace structure . over and the map , extends to a map .
This gives the flavor of our definition.
We define a category fibered over , of twisted stable , maps -pointed of genus and degree . This category is given in two equivalent realizations: one as a category of stable twisted objects over nodal pointed curves endowed with atlases of orbispace charts (see Definition -valued3.7.2); the other as a category of representable maps from pointed nodal Deligne-Mumford stacks into such that the map on coarse moduli spaces is stable (see Definition ,4.3.1). In our treatment, both realizations are used in proving our main theorem:
The category is a proper algebraic stack.
The coarse moduli space of is projective.
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 .
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 ,ReferenceLa.
The case where is the classifying stack of a finite group allows one to improve on the spaces of admissible covers, give moduli compactifications of spaces of curves with abelian and nonabelian level structures, and, with a suitable choice of the group show that there is a smooth, fine moduli space for admissible , which is a finite covering of -covers, This is the subject of our preprint .Reference-C-V with Alessio Corti, with some ideas contributed by Johan de Jong. Some of these applications were indicated in our announcement Reference-V1. This approach to admissible covers is closely related to the work of Wewers ReferenceWe.
A similar reasoning applies to curves with structures, e.g. theta characteristics. This is studied in -spinReference-J.
The recursive nature of the theorem allows one to construct both minimal models and stable reduction for pluri-fibered varieties. This is related to recent work of Mochizuki ReferenceMo and deserves further study.
In ReferenceC-R, W. Chen and Y. Ruan introduce Gromov-Witten invariants of an orbifold, using a differential geometric counterpart of our stack of twisted stable maps. In our joint work Reference-G-V with Tom Graber we give an algebraic treatment of these Gromov-Witten invariants, and in the special case of 3-pointed genus 0 maps of degree 0, construct the Chen-Ruan product with integer coefficients. See also ReferenceF-G for an algebraic treatment of global quotients.
In this paper we verify that is a proper stack by going through the conditions one by one. It may be worthwhile to develop a theory of Grothendieck Quot-stacks and deduce our results from such a theory. It seems likely that some of our methods could be useful for developing such a theory.
While our paper was circulating, we were told in 1999 by Maxim Kontsevich that he had also discovered the stack of twisted stable maps, but had not written down the theory. His motivation was in the direction of Gromov-Witten invariants of stacks.
We would like to thank Kai Behrend, Larry Breen, Barbara Fantechi, Ofer Gabber, Johan de Jong, Maxim Kontsevich, and Rahul Pandharipande, for helpful discussions. We are grateful to Laurent Moret-Bailly for providing us with a preprint of the book ReferenceL-MB before it appeared. The first author thanks the Max Planck Institute für Mathematik in Bonn for a visiting period which helped in putting this paper together.
We refer the reader to ReferenceAr and ReferenceL-MB for a general discussion of algebraic stacks (generalizing ReferenceD-M), and to the appendix in ReferenceVi for an introduction. We spell out the conditions here, as we follow them closely in the paper. We are given a category along with a functor We assume .
for any morphism of schemes and any object there is an object and an arrow over and ;
for any diagram of schemes
and any objects sitting in a compatible diagram
there is a unique arrow over making the diagram commutative.
We remark that this condition is automatic for moduli problems, where is a category of families with morphisms given by fiber diagrams.
is a stack, namely:
the functors are sheaves in the étale topology; and
any étale descent datum for objects of is effective.
The stack is algebraic, namely:
the functors are representable by separated algebraic spaces of finite type; and
there is a scheme locally of finite type, and a smooth and surjective morphism ,.
These last two conditions are often the most difficult to verify. For the last one, M. Artin has devised a set of criteria for constructing by algebraization of formal deformation spaces (see ReferenceAr, Corollary 5.2). Thus, in case is of finite type over a field or an excellent Dedekind domain, condition (3b) holds if
is limit preserving (see ReferenceAr, §1);
is compatible with formal completions (see ReferenceAr, 5.2 (3));
Schlessinger’s conditions for pro-representability of the deformation functors hold (see ReferenceAr, (2.2) and (2.5)); and
there exists an obstruction theory for (see ReferenceAr, (2.6)) such that
Furthermore, we say that is a Deligne-Mumford stack if we can choose as in (3b) to be étale. This holds if and only if the diagonal is unramified (ReferenceL-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 ReferenceL-MB, Chapter 7. Thus a stack is proper if it is separated, of finite type and universally closed. In ReferenceL-MB, Remarque 7.11.2, it is noted that the weak valuative criterion for properness using traits might be insufficient for properness. However, in case has finite diagonal, it is shown in ReferenceE-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 ( .ReferenceL-MB, Proposition 7.12).
Recall the following result:
Theorem 2.2.1 (Keel-Mori ReferenceK-M).
Let be an algebraic stack with finite diagonal over a scheme There exists an algebraic space . and a morphism such that
is proper and quasifinite;
if is an algebraically closed field, then is a bijection;
whenever is an algebraic space and is a morphism, then the morphism factors uniquely as more generally ;
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 ;
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:
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 .
If is the moduli space of then , is the moduli space of .
If is also surjective and is the moduli space of then , is the moduli space of .
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 ;ReferenceL-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. .
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. .■
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 .
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:
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:
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 .
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 to sheaves of -modules-modules.
The functor carries quasicoherent sheaves to quasicoherent sheaves, coherent sheaves to coherent sheaves, and is exact.
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 sheaf -equivariant on then , which, by the tameness assumption, is a direct summand in , From this the statement follows. .■
We recall the following purity lemma from Reference-V2:
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.
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.■
The reader can verify that the statement and proof work in higher dimension. See also related lemmas in ReferenceMo.
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 .
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 .
In both cases satisfies condition and the local fundamental group around is trivial, hence the purity lemma applies.■
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 , lifting -invariant 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 isomorphisms. Then the given arrow -equivariant and its composition with both satisfy the conditions of the lemma, so there is an induced isomorphism .
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 object -equivariant on the quotient and a , arrow -equivariant 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 , and -equivariant morphism -invariant 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 .
Our goal in this section is to introduce the notion of a stable twisted object. This is a representable object on a suitable atlas of orbispace charts on a nodal curve. Basic charts have the form -valued at a node, or along a marking, where acts freely away from the origin; for completeness of the picture we allow more general charts. The category of stable twisted objects is a concrete, yet somewhat technical, incarnation of our stack which is convenient in many steps of our proof of Theorem ,1.4.1.
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.
A divisorially nodal curve, or simply -marked curve -marked 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 curve -pointed is considered an curve by taking as the -marked the images of the sections .
If is an nodal curve, we define the special locus of -marked 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 ,.
If is a marked curve, and is an arbitrary morphism, we define the pullback to be where , and .
If and are curves, a morphism of -marked curves -marked is a morphism of which sends each -schemes into .
A morphism of curves -marked 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.
Let be an curve and -marked a finite group. An action of on is an action of on as an such that each element on -scheme, 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.
Let be an curve, with an action of a finite group -marked 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.
Let be an nodal curve. A generic object on -pointed is an object of .
We will often write for a generic object on a curve .
Let be an nodal curve and -pointed a generic object on A chart . for consists of the following collection of data.
An curve -marked and a strict morphism .
An object of .
An arrow in compatible with the restriction .
A finite group .
A tame, essential action of on .
Furthermore, we require that the following conditions be satisfied.
The actions of leave the morphism and the arrow invariant.
The induced morphism is étale.
The following gives a local description of the action of .
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 ,
the stabilizer of is a cyclic group which sends each of the branches of to itself;
if is the order of then a generator of , acts on the tangent space of each branch by multiplication with a primitive root of -th.
In particular, each nodal point of is sent to a nodal point of .
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.■
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.
Let be a generic object over a nodal curve and , , two charts; call the projection. Consider the scheme -th
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.
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.
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 , isomorphism -equivariant of schemes over and a compatible , isomorphism -equivariant.
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:
Let , be two compatible charts for a generic object on If . is an arbitrary morphism, then
where and are the pullbacks of and to and are compatible charts for the pullback of , to .
If is étale and surjective, then the converse holds.
The proof is immediate from Proposition 3.4.2.
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 curve, by defining -marked 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.
Fix two nonnegative integers and An . twisted object -pointed of genus consists of
a proper, curve -pointed of finite presentation, with geometrically connected fibers of genus ;
a generic object over and ;
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).
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.
Both statements are immediate from the local characterization of compatibility in Proposition 3.4.2.■
The lemma above allows one to define a twisted object using a maximal atlas, if one prefers.
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.
Let be a twisted object. Then the morphism induced by extends uniquely to a morphism .
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 , yielding morphisms -equivariant, 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:
A twisted object is stable if the associated map is Kontsevich stable.
Fix an ample line bundle over We define a category . as follows. The objects are stable twisted objects where , is a nodal curve of genus -pointed 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 ,.
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 ..
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 ;ReferenceK-M this exists as an algebraic space.
The morphism is a proper flat nodal curve of finite presentation, with geometrically connected fibers.
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 of the coordinate ring of -module, 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. ”,