DEGENERATIONS OF TWISTED MAPS TO ALGEBRAIC STACKS

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ANDREA DI LORENZO AND GIOVANNI INCHIOSTRO

Abstract. We give a deﬁnition of twisted map to a quotient stack with projective good moduli space, and

we show that the resulting functor satisﬁes the existence part of the valuative criterion for properness.

1. Introduction

Given a projective variety X, a classic problem is to study the geometry of curves in X. How many

rational curves in P2of degree dpass through 3d−1 points? Or how many rational curves are there on a

quintic threefold? These are two examples of famous questions, which were answered by Kontsevich [Kon95]

by studying degenerations of maps from curves to X. Indeed, Kontsevich constructed the so called moduli

stack of stable maps, which since then proved to be one the main tool one can use to solve enumerative

problems as the ones above ([Kon95,FP97]).

With the recent developments in the theory of algebraic stacks [Alp13,HL15,AHLH18,AHR20], it is

natural to wonder what happens if one replaces the target projective variety Xwith an algebraic stack

M. For example, if the target algebraic stack is Deligne-Mumford and admits a projective coarse moduli

space, the situation is similar to the case where Xis a projective variety. Abramovich and Vistoli (see

[AV02,ACV03,AOV11]) constructed a space of so-called twisted stable maps, that one can use to compactify

the space of stable maps from a curve to M.

If the target is BGminstead, studying degenerations of maps from a curve to BGmis strongly related to

the problem of ﬁnding a compactiﬁcation of the universal Jacobian [Cap94] or, if the target stack is BGLn,

a compactiﬁcation of the universal moduli space of vector bundles over Mg[Pan96].

The goal for this paper is to study degenerations of maps from a family of curves to an algebraic stack

Madmitting a projective good moduli space M → M, with the idea of interpolating between the stable

maps of Kontsevich and Caporaso’s compactiﬁed Jacobian. We introduce in Section 2.1 a notion of family

of twisted maps. Our main result can be formulated informally as follows.

Theorem 1.1 (Theorem 5.1).Let M:= [X/G]be a quotient stack by a linear group Gwith a projective

good moduli space, and assume we are given a family of twisted maps φ:C∗→ M over the punctured disc

D∗. Then, up to replacing D∗with a ramiﬁed cover, we can extend φto a family of twisted maps C → M

over the whole disk D.

More precisely, we prove that the functor of twisted maps to an algebraic stack having a projective good

moduli space satisﬁes the existence part of the valuative criterion for properness.

As already said, our deﬁnition of twisted map interpolates between the quasistable curves of Caporaso

and the twisted stable curves of Abramovich and Vistoli. We show in 2.2.1 and 2.2.2 that our deﬁnition

is minimal: one cannot drop neither the stacky structure nor the destabilizing P1’s and still hope for an

analogue of Theorem 1.1 to be true.

The main contribution of this work, beyond Theorem 1.1, is the explicit nature of our algorithm for

constructing the degeneration of Theorem 1.1 around each destabilizing P1(see Lemma 4.2). For example,

in Section 6we study some limits obtained from our algorithm in the case where target stack is [A1/Gm]

arXiv:2210.03806v2 [math.AG] 9 Jan 2023

2 A. DI LORENZO AND G. INCHIOSTRO

(namely, the stack parametrizing line bundles with section). In our example, we argue that one can assume

the chain of destabilizing stacky P1’s to have length at most 1, to map to the closed point of [A1/Gm], and

we explicitly describe the line bundle on the family of curves. We expect a similarly accessible description

to be available in general.

Where are we headed. As it is, our deﬁnition of twisted maps does not lead to an algebraic stack,

essentially because we are imposing no numerical stability conditions. For instance, when the target stack

is BGm, we can ask for the line bundles to satisfy Caporaso’s basic inequality [Cap94, 0.3], a numerical

condition involving the degrees of the line bundles on the irreducible components of the curve. This is the

key ingredient for making the moduli problem an algebraic stack with a projective good moduli space.

However, our deﬁnition of twisted maps can be regarded as a ﬁrst step towards the construction of a stack

of twisted stable maps to a quotient stack having a projective good moduli space. The next step in our

project is to tackle the case M= [X/Gn

m], leveraging our understanding of the behaviour of degenerations

of twisted maps. In particular, we aim to show that, after imposing some numerical conditions, we actually

get an algebraic stack which is Θ-reductive and has unpunctured inertia: applying the results of Alper,

Halpern-Leistner and Heinloth [AHLH18, Theorem A], we would get a good moduli space of twisted stable

maps to [X/Gn

m].

Another interesting direction to study is when M=BGfor a linear algebraic group G; in this case

our functor of twisted maps can be regarded as the functor of G-torsors over families of curves, and our

main theorem shows how to extend these families of G-torsors over curves. Hopefully, after imposing some

numerical stability conditions, one can extract a proper algebraic stack out of our functor; we are currently

investigating this path too.

Organization of the paper. In Section 2we introduce the main object of interest, namely twisted maps to

an algebraic stack, and we present some basic properties of them. For example, we show that our deﬁnition

recovers the one of Abramovich and Vistoli if Mis Deligne-Mumford. In Section 3we show how to extend

maps to codimension one points of the special ﬁber of the family of curves (Theorem 3.9). In Section 4we

introduce the main tools to extend maps to codimension two points: we prove two technical propositions

(Proposition 4.1 and Proposition 4.5) concerning extensions of line bundles over stacky surfaces, that will

be essential for proving the main results of this section. Section 5is divided into two parts. In the ﬁrst part,

we prove our main result (Theorem 1.1), and in the second part (namely Section 5.1) we prove some results

on the birational geometry of stacky surfaces, which are of independent interest: we study when one can

contract certain (stacky) P1s on a surface. Our results can be understood as a generalization of the results

of Artin in [Art62] to stacky surfaces. Finally in Section 6we run a speciﬁc example: the case of twisted

maps to M= [A1/Gm].

Conventions. In what follows, unless otherwise stated, every stack and morphism is assumed to be essen-

tially of ﬁnite type over a ﬁeld kof characteristic zero. If Cis a one dimensional Deligne-Mumford stack

with coarse moduli space C → C, and Lis a line bundle on C, there is a minimal natural number nsuch

that L⊗ndescends to a line bundle Lon C. The degree of Lwill be deg(L)

Acknowledgements. We are thankful to Dan Abramovich, Jarod Alper, Dori Bejleri, Samir Canning,

Andres Fernandez Herrero, Daniel Halpern-Leistner, Ming Hao Quek, Mauro Porta, Zinovy Reichstein,

Minseon Shin and Angelo Vistoli for helpful conversations.

DEGENERATIONS OF TWISTED MAPS TO ALGEBRAIC STACKS 3

2. Twisted maps to algebraic stacks

In this section we ﬁrst deﬁne twisted maps with target an algebraic stack Mhaving a projective good

moduli space (Deﬁnition 2.2). Then we motivate our deﬁnition, showing that it recovers the one for Deligne-

Mumford stacks [AV02]. We ﬁnally provide two examples showing that our deﬁnition is somehow “minimal”,

i.e. both stacky nodes and destabilizing P1s have to be included in the domain curve.

2.1. Twisted maps. We begin with the notion of quasi stable maps.

Deﬁnition 2.1. Let n≥0 be an integer, let (C;p1, ..., pn) be a nodal curve together with nsmooth points

and let Mbe a projective variety with ample line bundle L. We say that f:C→Mis quasi stable if

f∗L⊗ωC(p1+... +pn) is nef, and the irreducible components where f∗L⊗ωC(p1+... +pn) is not positive

are isomorphic to P1. Such components are called destabilizing P1s.

Recall [AV02, Deﬁnition 4.1.2] that a twisted nodal n-pointed curve over Bis the data of three morphisms

Σi→ C → C→Bsuch that

• C is a Deligne-Mumford stack, the map C → Bis proper and ´etale-locally over Bit is a nodal curve;

•Σi⊂ C are disjoint closed substacks contained in the smooth locus of Csuch that Σi→C→Bare

´etale gerbes;

• C → Cis a coarse moduli space and is an isomorphism away from Σiand the singular locus of

C→B.

From now on, we will say twisted (n-pointed) curve to indicate a twisted nodal n-pointed curve.

Deﬁnition 2.2. Let Mbe an algebraic stack with projective good moduli space M → M, let n≥0 be

a non-negative integer. A twisted map to Mover a scheme Bconsists of a triplet (π:C → B, φ :C →

M,{σi: Σi→ C}) such that:

(1) (π:C → B, Σi) is a twisted n-pointed curve over B;

(2) if we denote by C(resp. S) the coarse moduli space of C(resp. ∪Σi), then for every b∈Bthe

pointed map fb: (Cb, Sb)→Mis quasi stable;

(3) the smooth points of Cbthat are not on the image of ∪σiare schematic;

(4) if Dis an irreducible components of Cbwith coarse moduli space which is an fb-destabilizing P1,

then for every dthe map D → M does not factor as D → Bµd→ M.

Remark 2.3. We will mainly deal with the case where n= 0, namely the non-pointed case. The reason

we introduce pointed quasi stable maps is because they arise naturally when taking the normalization of an

unpointed quasi stable map.

We now comment brieﬂy on the last point of Deﬁnition 2.2. We begin by showing that if Mis a Deligne-

Mumford stack, we recover the deﬁnition of twisted stable maps `a la Abramovich-Vistoli. For this, we need

the following two lemmas.

Lemma 2.4. Let f:C→P1be a generically ´etale morphism which is ramiﬁed only over {p, q} ⊆ P1. Then

C∼

=P1and the map is [a, b]7→ [an, bn]for some n.

Proof. First observe that p6=q. Indeed, since the fundamental group of P1without a point is trivial, if p=q

then fmust be an isomorphism.

Then p6=qand consider ωP1(p+q). A local generator for this line bundle around pis dx

x. Locally around

every point piin the preimage of p,fcan be written as x7→ xeifor some integer ei; therefore, the pullback

of dx

xis dxei

xei=eidx

x: we deduce that h∗ωP1(p+q) = ωC(R) where Ris the ramiﬁcation locus of h. But

ωP1(p+q) has degree zero, so also h∗ωP1(p+q) = ωC(R) has degree zero: as Cis a smooth connected curve,

this implies that C∼

=P1,Rconsists of exactly two points, and fis totally ramiﬁed. Then fis of the desired

form. 

4 A. DI LORENZO AND G. INCHIOSTRO

Lemma 2.5. Let P1be the root stack of P1at the points 0 and 1. Assume that there is a representable

morphism f:P1→ BGfor a ﬁnite group G. Then there is an integer e, a map f0:P1→ Bµeand an

embedding ι:Bµe→ BGsuch that f=ι◦f0.

Proof. We use the following fact: given a G-torsor P→Sand a connected component P0⊂P, the classifying

morphism S→ BGfactors as S→ BH→ BG, where His the subgroup of elements of Gﬁxing P0.

Indeed, if P0⊂Pis another connected component, there exists an element g∈Gsuch that gP0⊂P0:

it is immediate to check that this induces an isomorphism P0'P0, hence P'`gH∈G/H P0. In particular,

there is a classifying morphism S→ BHsuch that Spec(k)×BHS'P0. On the other hand, the pullback of

the universal G-torsor Spec(k)→ BGalong BH→ BGis isomorphic to `gH∈G/H Spec(k), from which we

deduce that the pullback of the universal G-torsor along S→ BH→ BGis isomorphic to

gH∈G/H

(Spec(k)) ×BHS'a

gH∈G/H

P0.

We can then conclude that the two classifying maps S→ BGand S→ BH→ BGare isomorphic.

Let C0→ P1be the G-torsor induced by P1→ BG, and let Cbe a connected component of C0. It is

smooth as it is ´etale over P1and it is proper as Spec(k)→ BGis proper. Therefore Cis a curve.

We can take the composition h:C→ P1→P1: this is a cover of P1ramiﬁed at at most two points. But

then from Lemma 2.4, the map hmust be totally ramiﬁed at these two points with ramiﬁcation index e. We

deduce that his given by the two sections Xe

0,(X0−X1)e, the covering C→P1is cyclic or, in other terms,

is a µe-torsor for some integer e. By a descent argument, the subgroup Hof elements in Gthat ﬁx Cmust

be isomorphic to µe.

We can now apply the fact mentioned at the beginning to conclude that the map P1→ BGfactors

through Bµe.

Proposition 2.6. Let Mbe a Deligne-Mumford stack with coarse moduli space M, and let φ:C → M be

a twisted map over Spec(k), with f:C→Mthe induced map on the coarse spaces. Then fis Kontsevich

stable and φis a twisted stable map in the sense of Abramovich-Vistoli.

Proof. Assume by contradiction that it is not, let Dbe an irreducible component of Cwith coarse moduli

space D∼

=P1, and assume that Dis a destabilizing P1. Then f|D:D→Mfactors as D→Spec(k)→M,

where Spec(k)→Mis the inclusion of a closed point p. By the universal property of the ﬁber product,

D → M factors through Spec(k)×MM∼

=BGwhere Gis the stabilizer of the closed point p. By Lemma 2.5,

the map φ|D:D → M factors via D → Bµe→ BG→ M, which is a contradiction. 

2.1.1. Destabilizing components and multidegree. An interesting case is when M= [Spec(A)/Gm] for Spec(A)

an aﬃne Gm-scheme of ﬁnite type over k. Infact Mis an algebraic stack and its good moduli space is pro-

jective if and only if it is a point. In this case, we can consider twisted maps to [Spec(A)/Gm]. Observe

that the composition C → [Spec(A)/Gm]→ BGmcorresponds to a line bundle Lover C. The following

proposition shows that condition (4) of Deﬁnition 2.2 can then be formulated in terms of the degree of Lon

the destabilizing components.

Proposition 2.7. Let (π:C → B, φ :C → [Spec(A)/Gm]) be a pair satisfying the ﬁrst three conditions of

Deﬁnition 2.2. Then it satisﬁes condition (4) (hence it is a twisted map) if and only if for every geometric

point b∈Bthe line bundle Lbon Cbinduced by the composition Cb→[Spec(A)/Gm]→ BGmhas non-zero

degree on every destabilizing component.

Proof. Suppose that for some geometric point b∈Bwe have a destabilizing component Db⊂ Cbsuch that

the line bundle L|Dbhas degree zero. Then the coarse moduli space of Dbis isomorphic to P1. As Dbis

DEGENERATIONS OF TWISTED MAPS TO ALGEBRAIC STACKS 5

a Deligne-Mumford stack, there exists an integer esuch that L|⊗e

Dbcomes from a line bundle on the coarse

space Db'P1. As the only line bundle on P1of degree zero is the trivial one, we deduce that L|⊗e

Db' ODb.

This implies that we have a factorization Db→ Bµe→ BGm, hence a map

Db−→ Bµe×BGm[Spec(A)/Gm]'[Spec(A)/µe].

Observe now that the coarse space Db'P1is contracted to a point qin the coarse space of [Spec(A)/µe],

because the latter is aﬃne. This implies that Db→[Spec(A)/µe] factors through some Bµe0, where µe0⊂µe

is the automorphism group of the geometric point whose image in the coarse space Spec(AGm) is q. We showed

that condition (4) in Deﬁnition 2.2 is not satisﬁed.

Viceversa, suppose that condition (4) is not satisﬁed: then there exists a factorization

Db−→ Bµe−→ [Spec(A)/Gm]−→ BGm.

This implies that the line bundle L|Dbcomes from Bµe, i.e. is a torsion line bundle. 

2.2. Why twisted maps with quasi stable coarse space. In this section we report two examples showing

that if one does not allow neither twisted curves nor destabilizing stacky P1s, then there is no hope to ﬁnd

a moduli space satisfying the existence part of the valuative criterion for properness.

2.2.1. Stacky nodes are necessary. This example already appeared in the work of Abramovich and Vistoli.

Consider the two homogeneous polynomials A∈H0(P1,O(4)) and B∈H0(P1,O(6)) such that 4A3+27B2has

12 distinct roots. This data corresponds to a Weierstrass ﬁbration (X, S)→P1with 12 nodal singular ﬁbers,

so it corresponds to a map P1→ M1,1. We can now multiply all the terms in Aof degree greater than 2 and

the terms in Bof degree greater than 3 by a parameter t. This gives two polynomials in H0(P1

k(t),O(4)) and

H0(P1

k(t),O(6)) respectively. The corresponding Weierstrass ﬁbration (Xk(t), Sk(t))→P1

k(t)still corresponds

to a map P1

k(t)→ M1,1which induces a map P1

k(t)→ M1,1→M1,1. We can take the limit of this map to

Spec(k[t](t)). It is easy to see, from how we multiplied the coeﬃcients of Aand Bby t, that if we denote

by C→M1,1the limiting map, the special ﬁber of Cwill have a component Dsuch that D→M1,1has

degree 6. Even if we replace Cwith another scheme C0→Cbirational to C(blowing up the special ﬁber of

C→Spec(R)), the proper transform of Din C0will still map to M1,1with a map of degree 6. This cannot

lift to M1,1: any map from a curve to M1,1has degree divisible by 12.

2.2.2. Destabilizing P1s are necessary. Let Rbe a DVR, and consider two genus 2 curves with two marked

points, {(Ci;pi, qi)}2

i=1, and consider the curve obtained by gluing together p1with p2and q1with q2. Let

C=C×Spec(R), and let ηbe the generic point of Spec(R). Consider the trivial line bundle over the two

components of Cη, glued via the multiplication by a constant on the pis and by the multiplication by the

uniformizer πof Spec(R) on the qi. This corresponds to a map f:Cη→ BGm. One can check that:

(1) the only twisted curve that one can produce as a limit of Cηover Spec(R) is the scheme C, and

(2) the map fdoes not extend to C → BGmas the line bundle does not extend.

In particular, if one does not allow destabilizing P1s, one cannot ﬁll in the limit with just twisted curves.

3. Extending maps from deformations of curves I

Let M= [X/G] be a quotient stack by a linear algebraic group Ghaving a projective good moduli space

M. Let Rbe a DVR over Cwith generic point η, and let π:Cη→ηbe a smooth curve. Suppose to have

a map φη:Cη→ M which induces a map fη:Cη→M. Kontsevich’ theorem tells us that there exists a

unique family of semistable curves C→Spec(R) together with a map f:C→Mthat extends fη. Can we

lift fto a map φ:C→ M that extends φη? A ﬁrst step towards solving this problem would be lifting fat

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DEGENERATIONSOFTWISTEDMAPSTOALGEBRAICSTACKSANDREADILORENZOANDGIOVANNIINCHIOSTROAbstract.Wegiveadenitionoftwistedmaptoaquotientstackwithprojectivegoodmodulispace,andweshowthattheresultingfunctorsatisestheexistencepartofthevaluativecriterionforproperness.1.IntroductionGivenaprojectivevarietyX,aclass...

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DEGENERATIONS OF TWISTED MAPS TO ALGEBRAIC STACKS

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