ORBIFOLD TECHNIQUES IN DEGENERATION FORMULAS

DAN ABRAMOVICH AND BARBARA FANTECHI

Abstract. We give a new approach for relative and degenerate Gromov–Witten invariants, inspired by that of Jun Li but replacing predeformable maps by transversal maps to a twisted target. The main advantage is a significant simplification in the definition of the obstruction theory. We reprove in our language the degeneration formula, extending it to the orbifold case.

Contents

Introduction 1 1. Roots and twists 9 2. Expanded pairs and degenerations 19 3. Stacks of twisted stable maps and their properness 30 4. Relative and degenerate Gromov-Witten invariants 42 5. Degeneration formula 51 Appendix A. Pairs and nodes 64 Appendix B. Stack constructions 66 Appendix C. Stacks of maps and their obstruction theory 70 References 75 Appendix D. Things to do 78

Introduction

0.1. Gromov–Witten invariants in the smooth case. Gromov– Witten invariants were originally defined for a compact symplectic

Date: September 19, 2010, deg-revisited. Research of D.A. partially supported by NSF grant DMS-0335501 and DMS-

0603284.

Research activities of B.F. partially supported by Italian research grant PRIN 2006 “Geometria delle varietá proiettive”, ESF Research Network MISGAM, Marie Curie RTN ENIGMA, and GNSAGA.

manifold, and in the algebraic language for a smooth projective complex variety. For an extensive bibliography see [C-K]. From an algebraic viewpoint the construction proceeds via the following steps.

Step 1 Definition of a proper moduli stack

\overline{M} := \overline{M}_{g,n}(X, β)

of stable maps to X with fixed discrete invariants β ∈ H_{2}(X, ℤ) and g, n ∈ ℕ, together with an evaluation map ev : \overline{M} → X^{n}.

Step 2 Construction on \overline{M} of a 1-perfect obstruction theory, giving rise to a virtual fundamental class [\overline{M}]^{vir} ∈ A_{d}(\overline{M}), where d is the

expected dimension of \overline{M}.

Step 3 Definition of the invariants by integrating cohomology classes on X^{n} against ev_{∗}((∏ ψ_{i}^{m_{i}})[\overline{M}]^{vir}).

This construction has subsequently been extended to the case of orbifolds, namely smooth Deligne–Mumford stacks, see [CR, AGV]. For Step 1 above this case required care in the definition of stable maps, which were replaced by twisted stable maps in order to preserve properness of the moduli, see [AV]; on the other hand, the obstruction theory in Step 2 stayed essentially the same. The main difference in the formalism of Step 3 is that the evaluation map takes values in \overline{I}(X)^{n}, where \overline{I}(X) is the so called rigidified cyclotomic inertia stack of X.

0.2. Invariants of pairs and degenerate varieties. If X is singular, the moduli stack of stable maps is still proper; however, the natural obstruction theory is not perfect even for very mild singularities, and the construction has to be modified to stand a chance to work.

The issue was addressed for singular varieties W_{0} = X_{1} ⊔_{D} X_{2}, with X − 1, X_{2} and D smooth appearing as fibers in a one-parameter family with smooth total space, by A.M. Li and Y. Ruan [LR]. It was also studied at about the same time by E. Ionel and T. Parker [IP1, IP2], and subsequently worked out in the algebraic language by Jun Li [Li1, Li2]. Here the moduli of stable maps was changed in such a way as to have a perfect obstruction theory while keeping properness. With similar techniques, relative Gromov–Witten invariants were defined for a pair (X, D) with X a smooth projective variety and D a smooth divisor in X. The main tool introduced here is that of expanded degenerations and expanded pairs.

The degeneration formula is a way to express the Gromov–Witten invariants of W_{0} = X_{1} ⊔_{D} X_{2} in terms of the relative invariants of the pairs (X_{i}, D). It is now a key tool in Gromov–Witten theory.

There is also work preceding the cited papers where the ideas involved appear in different guises. The idea of expanded degenerations and its use in enumerative geometry was introduced by Z. Ran [Ra87]. Even earlier Harris and Mumford introduced the related idea of admissible covers [HM], revisited using logarithmic geometry by Mochizuki [Mo]. Related ideas with a different view can be found in [Hi], [AH], [CH], [Ber], [Va]. A simple approach in special but important cases was developed by Gathmann [Ga].

0.3. The twisting method. In this paper we will give an alternative algebraic definition of Gromov–Witten invariants for singular varieties as above, and of relative Gromov–Witten invariants, which extends naturally also to Deligne–Mumford stacks. Our treament follows closely that of Jun Li. However, introducing appropriate auxiliary orbifold structures along the nodes of both source curves and target varieties allows us to give a shorter definition of the the obtruction theory in Step 2 above, and a streamlined proof of the degeneration formula. At the same time we obtain a somewhat more general result, which applies to the orbifold case, see Theorem 0.4.1 below.

In a nutshell, the most difficult point in Jun Li’s approach is to define an obstruction theory on predeformable maps. A predeformable map C → W_{0} from a nodal curve C to a variety W_{0} with codimension-1 nodal singularities locally looks like

\xymatrix{C = Spec \frac{ℂ[u,v]}{(uv)} \ar[r] & Spec \frac{ℂ[x,y,z_{i}]}{(xy)} = W_{0}}
\xymatrix{u^{c} & x \ar@{|->}[l]}
\xymatrix{v^{c} & y, \ar@{|->}[l]}

and z_{i} ↦ f_{i}(u, v) arbitrary. As soon as c, the contact order, is > 1, this predeformability condition is not open on maps but rather locally closed. This means that deformations and obstructions as predeformable maps cannot coincide with deformations and obstructions as maps, so an obstruction theory must be constructed by other means. Jun Li does this by a delicate explicit construction.

Our approach to this is the following: we replace W_{0} by the orbifold W_{0} = [Spec \frac{ℂ[ξ,η,z_{i}]}{(ξη)}/μ_{c}] having W_{0} as its coarse moduli space. Here μ_{c} acts via (ξ, η) ↦ (ζ_{c}ξ, ζ_{c}^{−1}η), and x = ξ^{c}, y = η^{c}. Then the map C → W_{0} locally lifts to

[figure: C; W_{0}; u; ξ; v; η.]

Transversal maps of this type form an open substack of all maps, so an obstruction theory is immediately given by the natural obstruction theory of maps. An identical twisting construction applies in the case of pairs.

This in itself works well when we look at one node of C mapping to a singular locus of W_{0}. When several nodes p_{j} map to the same singular locus, they may have different contact orders c_{j}. If we pick an integer r divisible by all c_{j}, take W_{0} = [Spec \frac{ℂ[ξ,η,z_{i}]}{(ξη)}/μ_{r}], and at each p_{j} put a similar orbifold structure on C with index r/c_{j}, we still obtain a transversal map and therefore a good obstruction theory. In order to keep the moduli stacks separated, we must sellect a way to choose the integer r. We do this using the notion of a twisting choice - a rule that assigns to a collection c = {c_{j}} of contact orders a positive integer r = 𝔯(c) divisible by all the contact orders c_{j}, see Definition 3.4.1.

With this at hand we can define Gromov–Witten invariants. Theorem 4.4.1 shows that our invariants are independent of the twisting choice. Theorem 4.6.1 shows they are defomation invariants.

0.4. The degeneration formula.

Theorem 0.4.1.

⟨∏_{i=1}^{n} τ_{m_{i}}(γ_{i})⟩_{β,g}^{W_{0}}

= ∑_{η∈Ω} \frac{∏_{j∈M} d_{j}}{|M|!} ∑_{\begin{matrix}δ_{j}∈F \\ for j∈M \end{matrix}}(−1)^{ϵ} ⟨ ∏_{i∈N_{1}} τ_{m_{i}}(γ_{i})| ∏_{j∈M} δ_{j}⟩_{Ξ_{1}}^{(X_{1},D)}

· ⟨ ∏_{i∈N_{2}} τ_{m_{i}}(γ_{i})| ∏_{j∈M} δ̃_{j}^{∨}⟩_{Ξ_{2}}^{(X_{2},D)} .

0.4.2. User’s guide - left hand side.

1. W_{0} is a proper Deligne–Mumford stack having projective coarse moduli space W̄_{0}. The rigidified inertia stack of W_{0} is denoted \overline{I}(W_{0}).

2. W_{0} = X_{1} ⊔_{D} X_{2} has first-order smoothable singular locus D separating it in two smooth stacks X_{1}, X_{2}, see Sections A.1 and A.2.

3. g ≥ 0 is an integer and β is a curve class on W_{0}.

4. γ_{1}, …, γ_{n} ∈ H_{orb}^{∗}(W_{0}, ℚ) := H^{∗}(\overline{I}(W_{0}), ℚ) are classes having homogeneous parity, see opening of Section 5. In particular only classes on sectors transversal to D are relevant.

5. m_{1}, …, m_{n} ≥ 0 are integers.

6. Consider a twisting choice 𝔯 (see 3.4.1) and the moduli stack K := K_{Γ}^{𝔯}(W_{0}) of 𝔯-twisted stable maps (see 3.4.5).

7. K carries several universal maps, the coarsest of which is a stable map \underline{C} → W̄_{0} from the coarse contracted curve \underline{C} to the coarse target W̄_{0}. We have n sections s_{i} : K → \underline{C}. We denote ψ_{i} = s_{i}^{∗}c_{1}(ω_{\underline{C}/K}).

8. We have n evaluation maps ev_{i} : K → \overline{I}(W_{0}).

9. Finally we define

⟨∏_{i=1}^{n} τ_{m_{i}}(γ_{i})⟩_{β,g}^{W_{0}} = deg ((∏_{i=1}^{n} ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) ∩ [K]^{vir}) .

0.4.3. Right hand side.

1. F is a homogeneous basis for H^{∗}(\overline{I}(D), ℚ).

2. δ̃^{∨} is the dual of δ ∈ F with respect to the Chen–Ruan pairing, i.e. ∫_{\overline{I}(D)} \frac{1}{r}ι^{∗}δ̃_{j}^{∨} · δ_{i} = ∫_{I(D)} ι^{∗}δ̃_{j}^{∨} · δ_{i} = δ_{i,j}, see 5.2.3.

3. Ω is the set of splittings of the data g, n, β, see 5.1.1 for all details. An element η = (Ξ_{1}, Ξ_{2}) ∈ Ω includes in particular the data below:

4. N_{1}, N_{2} is a decomposition of {1, …, n} in two subsets.

5. Ξ_{1}, Ξ_{2} is a possibly disconnected splitting of the data β, g in two modular graphs having roots labelled by M = {n + 1, …, n + |M|}, see 4.8.1.

6. d_{i} for i ∈ M are assigned intersection multiplicities satisfying condition B of 5.1.1

7. We take K_{1} := K_{Ξ_{1}}^{𝔯}(X_{1}, D) and K_{2} := K_{Ξ_{2}}^{𝔯}(X_{2}, D).

8. For j ∈ M the new evaluation maps are ev_{j} : K_{1} → \overline{I}(D), and similarly for K_{2}.

9. (−1)^{ϵ} is the sign determined formally by the equality

∏_{i=1}^{n} γ_{i} ∏_{j∈M} δ_{j}δ̃_{j}^{∨} = (−1)^{ϵ} ∏_{i∈N_{1}} γ_{i} ∏_{j∈M} δ_{j} ∏_{i∈N_{2}} γ_{i} ∏_{j∈M} δ̃_{j}^{∨}.

10. Finally we define

⟨ ∏_{i∈N_{1}} τ_{m_{i}}(γ_{i})| ∏_{j∈M} δ_{j}⟩_{Ξ_{1}}^{(X_{1},D)}

:= deg (( ∏_{i∈N_{1}} ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) ( ∏_{j∈M} ev_{j}^{∗}δ_{j}) ∩ [K_{1}]^{vir})

and similarly

⟨ ∏_{j∈N_{2}} τ_{m_{j}}(γ_{j})| ∏_{j∈M} δ̃_{j}^{∨}⟩_{Ξ_{1}}^{(X_{2},D)}

:= deg (( ∏_{i∈N_{2}} ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) ( ∏_{j∈M} ev_{j}^{∗}δ̃_{j}^{∨}) ∩ [K_{2}]^{vir}) .

Perhaps the most mysterious part of the formula is the factor ∏_{j∈M} d_{j}. In previous works this arises as a result of delicate deformation theory of admissible or predeformable maps. In this paper it arises as a natural, but still delicate, outcome of the geometry of orbifold maps.

0.5. The logarithmic approach. Jun Li’s study of predeformable maps and their obstruction theory was inspired by log structures: predeformable maps are maps of suitable logarithmic structures on C and W, and in fact predeformable maps with such logarithmic structures are open among maps of logarithmic schemes. However, at the time when he wrote, deformation theory for log schemes was not as advanced as it is now: it applied to log smooth schemes but the moduli spaces are far from log smooth. In addition the passage from a log obstruction theory to a virtual fundamental class had not been addressed. So Jun Li had to resort to ad hoc methods to define his obstruction theory.

The situation is much improved given the later work of Olsson [Ol05]. In order to compare our results with Jun Li’s, one can first replace his obstruction theory by one induced by the natural log structures; this viewpoint is studied by B. Kim [Ki] and M. Gross and B. Siebert (work in preparation).

We are able to compare our obstruction theory with the logarithmic construction. Some further work is needed to understand the relationship of the logarithmic construction with Li’s obstruction theory. Comparing the resulting Gromov–Witten invariants can be done indirectly using deformation invariance, localization, and the work [MP] of Maulik and Pandharipande. We plan to give details for all these comparison issues in [AF].

0.6. Further directions. Taking a slightly broader view, an important reason for revisiting relative Gromov–Witten theory and the degeneration formula and putting them on a firm foundation with either logarithmic geometry or orbifolds, is the need to develop generalizations. We have been working on Gromov–Witten theory relative to a normal crossings divisor, and we are aware of others developing such theories: E. Ionel, B. Parker, J. Davis, Eliashberg–Givental–Hofer, Gross–Siebert. Gross and Siebert have a far reaching program which includes logarithmic stable maps without expanded degenerations

. A related work is currently developed by Q. Chen.

Regarding more immediate questions, the subsequent paper [AF] will address the following questions:

(1) comparison with the logarithmic approach and hopefully with Jun Li’s original obstruction theory,

(2) a few simple generalizations and extensions, e.g. non-rigid targets, and

(3) orbifold relative virtual localization.

0.7. Outline of the paper. In section 1 we review twisted curves and root constructions, as their fine structure is key to our methods.

In Section 2 we briefly review expanded pairs and expanded degenerations and their twisted versions, and describe their boundary. In addition we introduce a weighted version of the stacks of twisted expanded pairs and expanded degenerations. Finally we treat stable configurations of points on expanded pairs and expanded degenerations.

Section 3 leads to the construction of stacks of 𝔯-twisted stable maps and a proof of their properness. Properness is proven by way of properness of the stack of untwisted predeformable maps. In the orbifold case this does not immediately follow from Jun Li’s result; his proof can however be repeated without much change. We find it instructive to use a different proof, which relies on properness of the stack of stable configurations.

In Section 4 we define Gromov–Witten invariants using 𝔯-twisted stable maps. We prove their deformation invariance and independence of twisting choice.

The degeneration formula is stated and proven in Section 5. We finish by outlining a proof, detailed in [AF], that our invariants must coincide with Li’s.

Results in sections 4 and 5 rely on compatibility results for virtual fundamental classes. We find it useful to systematically use Costello’s result [Co, Theorem 5.0.1] and its generalization in [Ma, Proposition 2, Section 4.3] where a smoothness assumption is removed.

In addition we have three appendices, with necessary material which the knowledgeable reader may only wish to peruse when needed. In Appendix A we review material concerning pairs (X, D), nodal singularities, and transversality. Appendix B is devoted to a number of basic construction with stacks. In Appendix C we review the algebricity of stacks of maps and construction and properties of their obstruction theories.

0.8. Conventions. The following conventions will be in force throughout the paper.

We work over an algebraically closed base field of characteristic 0, denoted by ℂ. We note that the assumption that the field be algebraically closed is mostly for convenience, whereas the characteristic assumption is significantly harder, and for some purposes impossible, to avoid.

All stacks and morphisms are assumed to be locally of finite type over ℂ, unless otherwise specified.

A point in a scheme or algebraic stack (sometimes denoted by p ∈ X) is a ℂ-valued point, unless otherwise specified.

Whenever we say locally we always mean étale locally, unless otherwise specified.

If X is an algebraic stack, by D(X) we denote the derived category of sheaves of O_{X}-modules with coherent cohomology. An object 𝔽 ∈ D(X) is called perfect of perfect amplitude contained in [a, b], or just perfect in [a, b] for brevity, if it is locally isomorphic to a complex of locally free sheaves in degrees a, a + 1, …, b.

An element in a skew commutative graded ring is of homogeneous parity if it is a sum of only even-degree or only odd-degree terms, in which case its parity is even or odd, respectively.

0.9. Notation.

ℂ  fixed algebraically closed field of characteristic 0.
A  The stack [𝔸^{1}/𝔾_{m}].
𝕃_{X/Y}, 𝕃_{f} cotangent complex of a morphism f : X → Y.
k  number of components in an accordion
ℓ  generic splitting divisor
r  twisting index along a divisor
r, 𝔯  twisting sequence on an accordion, twisting choice
n_{X}, e_{i}  number of legs to general point, twisting tuple
n_{D}, f_{j}  number of legs to boundary divisor, twisting tuple
c_{j}, d_{j}  contact order and intersection multiplicity at such a point
d  β · D
m  index of τ in descendant notation τ_{m}(γ)
h, j  number of components of a disconnected graph, their index

Acknowledgements. This work was initiated during a joint visit at the University of Nice. The program was further developed in two visits at the MFO, Oberwolfach, and a program at the de Giorgi Center, Pisa. It was finalized at a program in MSRI, which provided time and immediate interaction with numerous colleagues. We heartily thank Angelo Vistoli for numerous comments as the project evolved, and Martin Olsson who jump-started our understanding of logarithmic structures. Our indebtendness to Jun Li is evident throughout the paper. We also thank Tom Bridgeland, Charles Cadman, Brian Conrad, Bumsig Kim, Joseph Lipman, Amnon Ne’eman, Ravi Vakil and Jonathan Wise for helpful discussions.

1. Roots and twists

1.1. Twisted curves. A prestable twisted curve with n markings is a one dimensional separated connected Deligne–Mumford stack C, with at most nodal singularities, together with a collection of disjoint closed substacks Σ_{1}, …, Σ_{n} of the smooth locus of C such that:

(1) the open locus C^{sm} ∖ ⋃ Σ_{i} in C is a scheme;

(2) each node is a balanced node.

The latter condition means that locally C looks like the model balanced node of index r_{p}

N_{r_{p}} := [ Spec (ℂ[u, v]/(uv)) / μ_{r_{p}}]

where the action of μ_{r_{p}} is balanced, namely (u, v) ↦ (ζ_{r_{p}}u, ζ_{r_{p}}^{−1}v). Similarly, locally near each Σ_{i} the twisted curve C looks like

[Spec ℂ[u] / μ_{e_{i}}]

where μ_{e_{i}} acts via u ↦ ζ_{e_{i}}u. The integers e_{1}, …, e_{n} are the orbifold indices of the markings Σ_{i}.

A family of twisted prestable curves with orbifold indices e_{1}, …, e_{n} is a flat morphism C → S together with closed substacks (Σ_{1}, …, Σ_{n}) of C such that:

(1) Σ_{i} is a gerbe banded by μ_{e_{i}} over S;

(2) each fiber (C_{s}, Σ_{1,s}, …, Σ_{n,s}) is a twisted prestable curve with orbifold indices e_{i}.

1.1.1. Automorphism groups of twisted curves. By [AV, Lemma 4.2.3], see B.1, the 2-groupoid of twisted curves is equivalent to a stack, so we can speak of automorphisms of twisted curves.

Let π : C → C be the morphism from a twisted curve C to its coarse moduli space C. Since the formation of C is functorial, the automorphism group of C acts on C. Consider the group Aut_{C}C of automorphism of C acting trivially on C. As shown in [ACV, Proposition 7.1.1] there is a canonical isomorphism

Aut_{C}C ≃ ∏_{p∈Sing(C)} μ_{r_{p}}.

Notice that nodes contribute but markings do not. These automorphisms are known as ghost automorphisms, as they become “invisible” when looking only at C. The action of μ_{r_{p}} is induced on local coordinates by (ξ, η) ↦ (ζ_{r_{p}}ξ, η), equivalently (ξ, η) ↦ (ξ, ζ_{r_{p}}η). We further discuss these through gluing data in 1.4.5 below.

1.1.2. Deformations of twisted curves. We now consider proper twisted curves. Just like the untwisted case, deformations of twisted curves are unobstructed [AJ, Proposition 2.1.1], [ACV, 3.0.3]. The infinitesimal theory is identical to the untwisted case: first-order infinitesimal automorphisms are in the group Hom(Ω_{C}^{1}(∑ Σ_{i}), O_{C}), first-order deformations in Ext^{1} and obstructions vanish since Ext^{2} = 0.

Again as in the untwisted case, the sheaf Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C}) is a sum of one-dimensional skyscraper sheaves supported at the nodes: Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C}) = ⊕_{p∈Sing(C)}Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C})_{p}. The balanced action condition guarantees that the action of the stabilizers on these skyscraper sheaves is trivial, therefore they are generated by sections. The local-to-global spectral sequence for Ext gives epimorphisms

Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C}) → H^{0} (Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C})_{p})

for all p ∈ Sing(C), corresponding to a divisor Δ_{p} in the versal deformation space - the locus where the node p persists.

1.1.3. Comparing deformations and automorphisms of C and C. Since Aut_{C}C is discrete, it does not affect first-order infinitesimal automorphisms, so the homomorphism

Hom(Ω_{C}^{1}(∑ Σ_{i}), O_{C}) → Hom(Ω_{C}^{1}(∑ Σ̄_{i}), O_{C})

is an isomorphism.

On the other hand the action of Aut_{C}C on H^{0} (Ext^{1}(Ω_{C}^{1}(∑ Σ_{i}), O_{C})) is easily seen to be effective. The deformation spaces of C and C are smooth and have the same dimension. It follows that the deformation space Def_{C,Σ_{i}} of the twisted curve is a branched cover of the deformation space Def_{C,Σ̄_{i}} of the coarse moduli space, with index r_{p} along Δ_{p}. See [A, 3.5], [Ol07a, 1.10]. If we denote by Δp̄ the corresponding divisor in Def_{C,Σ̄_{i}}, then the pullback of the divisor Δp̄ in Def_{C,Σ_{i}} is the divisor r_{p}Δ_{p}. It follows that over the smooth locus of Δ_{p}, the scheme-theoretic inverse image of Δp̄ is locally of the form Δ_{p} × Spec ℂ[ϵ]/(ϵ^{r_{p}}).

1.1.4. Moduli of twisted curves. Families of twisted prestable curves of genus g with n markings form an algebraic stack, denoted 𝔐_{g,n}^{tw}; its connected components are labeled by the indices e_{i} of the markings, and they are all isomorphic to each other ([AGV, Theorem 4.2.1], [Ol07a, Theorem 1.8]). We have an evident embedding 𝔐_{g,n} ↪ 𝔐_{g,n}^{tw}, since a curve is in particular a twisted curve. Taking coarse moduli spaces gives a left inverse morphisms 𝔐_{g,n}^{tw} → 𝔐_{g,n}. Even when fixing e_{i}, the latter morphism is not of finite type and not separated - over the nodal locus of 𝔐_{g,n} there are infinitely many boundary components of 𝔐_{g,n}^{tw} corresponding to different indices at the nodes. Along a boundary component Δ ⊂ 𝔐_{g,n}^{tw} corresponding to a node with index r_{p}, this morphism is branched with index r_{p} as described above. A similar result holds on universal families [A, Section 3.3], given by the usual identification of the universal families in terms of moduli of curves with an additional untwisted marking [?, Corollary 9.1.3].

1.2. Root stacks. We review here the theory developed in [AGV, Appendix B], [C1, Section 2], [M-O, Section 4].

1.2.1. Power morphisms. For a positive integer r, let ν_{r} : 𝔸^{1} → 𝔸^{1} be the morphism given by t ↦ t^{r}; its restriction to 𝔾_{m} is a group endomorphism, hence we get induced endomorphisms which we denote by the same symbol ν_{r} : B𝔾_{m} → B𝔾_{m} and ν_{r} : A = [𝔸^{1}/𝔾_{m}] → A.

1.2.2. Root stack of a line bunle. If X is an algebraic stack, a morphism from X to B𝔾_{m} is a line bundle L on X; we denote by \sqrt[r]{L/X} the fiber product X ×_{L,B𝔾_{m},ν_{r}} B𝔾_{m}. The stack \sqrt[r]{L/X} parametrizes r-th roots of

the pullback of L: objects of \sqrt[r]{L/X} over a base scheme S are triples (f, M, ϕ) where

(1) f : S → X is a morphism, i.e. an object of X(S),

(2) M is an invertible sheaf on S, and

(3) ϕ : M^{r} → f^{∗}L is an isomorphism.

Arrows are given using pullbacks as usual. Note that there is a natural isomorphism \sqrt[r]{L/X} → \sqrt[r]{L^{∨}/X} defined by mapping (f, M, ϕ : M^{⊗r} → f^{∗}L) to (f, M^{∨}, (ϕ^{∨})^{−1}).

1.2.3. Root of a divisor. If D is an effective Cartier divisor on X, it defines a morphism to [𝔸^{1}/𝔾_{m}]; it corresponds to the line bundle O_{X}(D) = I_{D}^{∨} with its canonical section 1_{D}. We denote by X(\sqrt[r]{D}) the fiber product X _{D} ×_{ν_{r}}[𝔸^{1}/𝔾_{m}]. This stack parametrizes symultaneous r-th roots of the pullback of I_{D}^{∨} and of the section: objects over a base scheme S are tuples (f, M, ϕ, s) where

(1) f : S → X is a morphism,

(2) M is an invertible sheaf on S,

(3) ϕ : M^{r} → f^{∗}L an isomorphism, and

(4) s ∈ H^{0}(M) a section,

such that ϕ(s^{r}) = 1_{D}. Again arrows are given using pullbacks.

Notice that in case f^{∗}D is still a Cartier divisor (e.g. if f is dominant), then this structure simply means that f^{∗} is divisible by r: an object of X(\sqrt[r]{D}) with such f is given by M = O(D) and s = 1_{D}, where rD = f^{∗}D. In other words, the object is canonically determined by f.

In particular, the stack X(\sqrt[r]{D}) carries an effective Cartier divisor D such that rD is the inverse image of D. As a stack D ≃ \sqrt[r]{N_{D⊂X}/D}. If (X, D) is a locally smooth pair in the sense of A.1, then so is (X(\sqrt[r]{D}), D); we call it the locally smooth pair obtained from (X, D) by taking an r-th root.

1.2.4. Roots with several divisors. Given finitely many effective Cartier divisors D_{1}, …, D_{k} on X, and given positive integers r_{1}, …, r_{k}, we use the following notation:

X( \sqrt[r_{1}]{D_{1}}, …, \sqrt[r_{k}]{D_{k}}) := X( \sqrt[r_{1}]{D_{1}}) ×_{X} ⋯ ×_{X} X( \sqrt[r_{k}]{D_{k}}).

We will mostly use this notation when the divisors meet transversally.

1.2.5. Comparison of roots. Note that if r = r′ · r′′ then ν_{r} = ν_{r′} ∘ ν_{r′′}. In particular we have canonical morphisms \sqrt[r]{L/X} → \sqrt[r′]{L/X} and X(\sqrt[r]{D}) → X( \sqrt[r′]{D}). In fact this gives a canonical isomporphism

\xymatrix{\sqrt[r]{L/X} \ar[r] & \sqrt[r′′]{M/ ( \sqrt[r′]{L/X})}}

and similarly an isomorphism

\xymatrix{X(\sqrt[r]{D}) \ar[r] & (X( \sqrt[r′]{D})) ( \sqrt[r′′]{D}) .}

1.2.6. Twisted curves as root stacks: the markings. Suppose now C is a twisted curve with markings Σ_{i} with indices e_{i}. Then C is canonically a root stack as follows. Consider the curve C^{u} obtained by gluing the coarse moduli space of the smooth locus C^{sm} with C ∖ (∪Σ_{i}). We have a “partial coarse moduli space” morphism π′ : C → C^{u}. Denote by Σ_{i}^{u} the markings on C^{u}. Then π′^{∗}Σ_{i}^{u} = r_{p}Σ_{i}. This gives a canonical morphism

C → C^{u}( \sqrt[e_{1}]{Σ_{1}}, …, \sqrt[en]{Σ}_{n})

which is easily seen to be an isomorphism [AGV, Theorem 4.2.1].

Generalizing the structure of twisted curves at nodes is a bit more subtle, see 1.4.

1.2.7. Triviality of relative automorphisms. We return to the general setup, and consider Since X(\sqrt[r]{D}) → X is representable over the dense open X ∖ D, the groupoid Aut_{X}(X(\sqrt[r]{D})) is equivalent to a group, see B.1, and we regard it as a group. But since for dominant f : S → X an object (f, M, ϕ, s) is determined by f, the group Aut_{X}(X(\sqrt[r]{D})) is trivial.

1.2.8. Inertia of root stacks when X is a scheme. Inertia stacks are reviewed in B.3.1. Since we are working over ℂ we will identify inertia stacks and cyclotomic inertia stacks. It will be useful

for us to describe the cyclotomic inertia stack of X(\sqrt[r]{D}) and its rigidified version, and similarly for the substack D. The picture is clear when X is a scheme or an algebraic space: since D is a gerbe we have I(D) = ⊔_{i=0}^{r−1}D and \overline{I}(D) = ⊔_{i=o}^{r−1}D_{i}, where D_{i} ≃ \sqrt[g]{N_{D⊂X}/D} and g = gcd(r, i). We similarly have

I(X(\sqrt[r]{D})) = X(\sqrt[r]{D}) ⊔ ∐_{i=1}^{r−1} D,

and

\overline{I}(X(\sqrt[r]{D})) = X(\sqrt[r]{D}) ⊔ ∐_{i=1}^{r−1} D_{i}.

The latter follows from the decompositions into an open substack and closed complement

I(X(\sqrt[r]{D})) = (X ∖ D) ⊔ I(D)

and

\overline{I}(X(\sqrt[r]{D})) = (X ∖ D) ⊔ \overline{I}(D).

1.2.9. Inertia of root stacks when X is an orbifold. In case X itself is an orbifold the picture is almost identical, using the inertia stacks of X and D: we still have decompositions into an open substack and closed complement precisely as above. The coarse moduli space of the stack I(D) consists of r copies of the coarse moduli space of I(D). However the stack structure of the components of I(D) and \overline{I}(D) becomes slightly more involved than in the case when X is representable.

We note that I(B𝔾_{m}) ≃ 𝔾_{m} ×B𝔾_{m}, and the morphism I(B𝔾_{m}) → I(B𝔾_{m}) is simply ν_{r}×ν_{r}. In particular this morphism is a μ_{r}-gerbe over a μ_{r} torsor, coresponding to the gerbe factor B𝔾_{m} → B𝔾_{m} and the torsor factor 𝔾_{m} → 𝔾_{m}. As discussed in B.3.1, forming the inertia is compatible with fiber products. We obtain that I(D) = I(D) ×_{I(B𝔾_{m})} I(B𝔾_{m}) → I(D) is canonically a μ_{r}-gerbe over a μ_{r} torsor.

Since D is assumed Deligne–Mumford, the image of I(D) → 𝔾_{m} is discrete, so the torsor is trivial (though as a group scheme it is a possibly nontrivial extension). In particular every component of I(D) is a μ_{r}-gerbe over the image component of I(D). By definition this is the gerbe associated to the normal bundle of D, namely the pullback of D. We obtain the following formula:

I(D) = μ_{r} × I(D) ×_{D} D.

Of course the group scheme structure of I(D) → D is not a product but the extension of the group shcheme I(D) ×_{D} D → D by μ_{r} corresponding to the normal bundle of D. Explicitly, one can look at local models on X of the form [V/G], where V is smooth and G is the stabilizer of a geometric point. We may assume that D is defined by an eigenfunction x. Denote the character of the action of G on x by χ : G → 𝔾_{m}. Then a local model of X(\sqrt[r]{D}) is given by [Ṽ /G̃], where Ṽ = Spec_{V} O_{V} [u]/(u^{r} − x), and G̃ = G ×_{χ,𝔾_{m},ν_{r}} 𝔾_{m} is the natural

extension of G by μ_{r}.

A similar description follows for rigidified inertia stack. What we will need is the following:

Lemma 1.2.10. Let (x̃, g̃) be an object of a component Z ⊂ I(D). Let (x, g) be the image object of the corresponding component Z ⊂ I(D). Denote by (x̃, g̃) ⟨g̃⟩ and (x, g) ⟨g⟩ the corresponding objects of components \overline{Z} ⊂ \overline{I}(D) and \overline{Z} ⊂ \overline{I}(D). Write |⟨g̃⟩| = \frac{r}{c}|⟨g⟩|. Then the morphism \overline{Z} → \overline{Z} is a gerbe banded by μ_{c}.

Proof. We have that Z → \overline{Z} is a gerbe banded by ⟨g̃⟩ and Z → \overline{Z} is a gerbe banded by ⟨g⟩. Also Z → Z is banded by μ_{r}. Chasing the diagram shows that \overline{Z} → \overline{Z} is indeed a gerbe banded by a cyclic group, and its order is clearly the ratio |⟨g⟩| · r/ |⟨g̃⟩| = c. ♣

1.3. Maps and lifts. We now follow the key observation of Cadman’s work [C1], especially Theorem 3.3.6, relating tangency with orbifold maps. Suppose now (C, Σ) a twisted curve, (X, D) a locally smooth pair and g : C → X a morphism such that g^{∗}D = cΣ. We call c the contact order of C and D at p. (Note that if Σ is twisted with index e, then deg Σ = 1/e and the intersection multiplicity of C and D at P is d = c/e.)

Lemma 1.3.1. Let r, r_{Σ} ≥ 1 be integers, let X′ = X(\sqrt[r]{D}) with divisor D, and similarly C′ = C( \sqrt[r_{Σ}]{Σ}) with divisor Σ̃. We assume c divides r. Then

(1) g : C → X lifts to g̃ : C′ → X′ if and only if r|c · r_{Σ},

(2) when r|c · r_{Σ} such lift is unique up to unique isomorphism,

(3) the lift is representable if and only if r = c · r_{Σ}, and

(4) the lift is transversal if and only if r = c · r_{Σ}.

Proof. (see [C1, Theorem 3.3.6]) For (1), a lift corresponds to a line bundle M on C′ and a section s such that s^{r} = g̃^{∗}1_{D}. But g^{∗}1_{D} = 1_{Σ}^{c} and 1_{Σ} pulls back to 1r_{σ} ; Σ̃ on C′. So g̃^{∗}1_{D} = 1c·r_{Σ} ; Σ̃. It follows that if a lift exists then r|c · r_{σ}. And if rd = c · r_{σ} for some integer d, the pair (O(dΣ̃), 1d ; Σ̃) gives the desired lift C′ → X′.

(2) Uniqueness follows since (O(Σ̃), 1Σ̃)|_{C∖Σ} ≃ (O, 1) which has no nontrivial automorphisms.

(3) It suffices to consider points over Σ. Here automorphisms of an object (L, s = 0, L^{r_{Σ}} ≃ O(Σ̃)) of C′ are given by L →^{ζ} L with ζ a primitive r_{Σ} - root. The action on O(dΣ̃) = L^{d} is via ζ^{d}, which has a nontrivial kernel exactly when d > 1.

(4) We have g̃^{∗}D = dΣ̃, so g̃ is transversal if and only if d = 1. ♣

Remark 1.3.2. A similar result holds without the assumption that c|r, with part (3) modified. See [C1, Theorem 3.3.6].

Given a morphism (C, Σ) → (X, D), the restriction Σ → D is an object of a component Z of \overline{I}(D). When r = cr_{Σ}, the representable and transversal lift (C′, Σ̃) → (X′, D) gives rise to an object Σ̃ → D of a component Z of \overline{I}(D) lying over Z. The following computation will be used in the degeneration formula:

Lemma 1.3.3. The morphism Z → Z is a gerbe banded by μ_{c}.

Proof. The object Σ → D is the rigidification of an object corresponding to (x, g) with |⟨g⟩| = e, and Σ̃ → D is the rigidification of an object corresponding to (x̃, g̃) with |⟨g̃⟩| = e · r_{Σ}. But r_{Σ} = r/c and the result follows from Lemma 1.2.10. ♣

1.4. Twisting along a nodal divisor. We now want to generalize the singularity structure of a twisted curve to the following case. Let W be an algebraic stack, nodal and first order smoothable along a closed substack D, as discussed in A.3. Assume also that W = X_{1} ⊔_{D} X_{2} where X_{1} and X_{2} are closed substacks, which along D are smooth and intersect transversally. The ordering of X_{i} will be kept throughout the discussion. Assume we are given an isomorphism of Ext^{1}(Ω_{W} , O_{W})|_{D} with O_{D}, i.e., an isomorphism α : N_{1} → N_{2}^{∨}, where N_{i} = N_{D/X_{i}}.

Definition 1.4.1. We define the stack obtained by twisting W along D with index r, or adding a balanced node structure of index r along D, denoted W (\sqrt[r]{D}), as follows. Let X′_{i} := X_{i}(\sqrt[r]{D}), and D′_{i} ⊂ X′_{i} the reduced inverse image of D, so that D′_{i} is isomorphic to \sqrt[r]{N_{i}/D}. Let π_{i} : D′_{i} → D be the structure morphism, and let ϕ_{i} : L_{i}^{⊗r} → π_{i}^{∗}N_{i} the universal line r-th root. Let β : D′_{1} → D′_{2} be the morphism defined by the triple (L_{1}^{∨}, π_{1}, α∘(ϕ_{1}^{∨})^{−1}); it is easy to see that β is an isomorphism. We define W (\sqrt[r]{D}) to be the stack obtained by gluing X′_{1} to X′_{2} along the identification of D′_{1} with D′_{2} via β. We let D′ be the closed substack image of either D′_{i}.

Remark 1.4.2. A generalization of this construction and more information about it may be found in [ACFW] and [BV].

1.4.3. Gluing as pushout. In Definition 1.4.1, the term gluing means the existence of a 2-pushout diagram

\xymatrix{D′ \ar[r] \ar[d] & X′_{1} \ar[d] \\ X′_{2} \ar[r] & W (\sqrt[r]{D}).}

See [AGV, Appendix A]. In particular, for every algebraic stack Y, the groupoid of morphisms from W (\sqrt[r]{D}) to Y has as objects the triples (f_{1}, f_{2}, ε) where f_{i} : X′_{i} → Y is a morphism and ε is a 2-morphism between f_{1}|_{D′_{1}} and f_{2}|_{D′_{2}} ∘ β; a 2-morphism from (f_{1}, f_{2}, ε) to (g_{1}, g_{2}, ζ) is a pair of 2-morphisms α_{i} : f_{i} → g_{i} such that the two 2-morphisms from f_{1}|_{D′_{1}} to g_{2}|_{D′_{2}} ∘ β induced respectively by ε and α_{2} and by α_{1} and ζ coincide.

In particular, the structure maps X′_{i} → W and the identity of β define a morphism π : W (\sqrt[r]{D}) → W.

Lemma 1.4.4. (1) W (\sqrt[r]{D}) is an algebraic stack which is nodal

and first-order smoothable along D′;

(2) π is proper, quasifinite and an isomorphism over W ∖ D.

Proof. (1) the only nontrivial part to check is the first-order smoothability. This come from the identification of L_{2} with L_{1}^{∨} via β, and the fact that L_{i} = N_{D′_{i}/X′_{i}}.

(2) Properness is local in the smooth topology of the target, so we may assume that W = 𝔸^{n} × N_{1}; then W (\sqrt[r]{D}) = 𝔸^{n} × N_{r}, and we are done since N_{r} → N_{1} is proper. Quasifiniteness is also smooth local in the target and follows in the same way. The isomorphism over W ∖ D is obvious from the definition, as it is the case D = ∅. ♣

1.4.5. Automorphisms. Unlike root stacks, twisted nodal stacks do have non-trivial relative automorphisms.

Proposition 1.4.6. Let h_{i} be an automorphism of X_{i} restricting to the identity on D, and let h : W → W be the induced automorphism. Each h_{i} acts on N_{i} by multiplication by a nowhere vanishing regular function λ_{i} on D. Then the set of isomorphism classes of automorphisms of W (\sqrt[r]{D}) lifting h (in the sense of Lemma B.1.3) is in natural bijection with the set of regular functions ε on D such that ε^{r} = λ_{1}λ_{2}.

Proof. Write W′ := W (\sqrt[r]{D}) for brevity. We apply 1.4.3 with the stack Y = W′. Let h′ : W′ → W′ be a lifting, and (h′_{1}, h′_{2}, ε) the corresponding triple. Then h′_{1} and h′_{2} are determined up to unique 2-isomorphism by Lemma B.1.3. All that is left is to find a two-isomorphism ε : h′_{1}|_{D′_{1}} → h′_{2} ∘ β lifting the 2-isomorphism λ : h_{1}|_{D_{1}} → h_{2}|_{D_{1}} induced by h. Since h|_{D} acts on L = Ext^{1}(Ω_{W} , O_{W})|_{D} by multiplication with λ_{1}λ_{2}, we have to find an automorphism ε of its r-th root whose rth power is λ_{1}λ_{2}. Every automorphism of a line bundle is a nowhere vanishing function ε, and the lifting condition means ε^{r} = λ_{1}λ_{2}. ♣

Corollary 1.4.7. In particular the group of automorphisms of W (\sqrt[r]{D}) in the sense of B.1 lifting the identity of W is naturally isomorphic to μ_{r}(D); if D is connected, then it is just μ_{r}.

As in the case of curves, elements of this group are called ghost automorphisms.

1.4.8. See [A], Section 3.3: suppose given a one-parameter smoothing of W along D, that is a flat morphism π : W → B with B a smooth curve and an isomorphism of W_{0} := π^{−1}(b_{0}) with W such that W is smooth along D. Then the balanced node structure can be described more directly as follows. Let W′ be the fiber product over W̃ of W(\sqrt[r]{W_{1}}) and W(\sqrt[r]{W_{2}}); by the universal property, the morphism W′ → B induces π̃ : W′ → B(\sqrt[r]{b_{0}}). Then W (\sqrt[r]{D}) is just the fiber of π̃ over a geometric point of B(\sqrt[r]{b_{0}}) over b_{0}.

Note that the one-parameter smoothing induces a trivialization of Ext^{1}(Ω_{W} , O_{W}), unique up to a non-zero scalar (corresponding to a choice of the basis of T_{b_{0}}B).

1.4.9. Twisted curves and twisted nodes. Consider a node ℘ of index r on a twisted curve C, and assume the two branches of C at ℘ belong to two different components of C. Replacing C by an open neighborhood of ℘ we may assume C = C_{1} ⊔_{℘} C_{2}. Let C be the coarse moduli space of C and p the image of ℘. It follows from the definition that C = C(\sqrt[r]{p}).

Even if the node ℘ is not locally separating in the Zariski topology, there is an étale neighborhood where it is, so in fact every twisted node is obtained by this construction locally in the étale topology. See [Ol07a] for a formalism which works in general; we will not use this generality in this paper.

1.4.10. Maps and lifts. Here is the result analogous to 1.3.1 for nodes:

Lemma 1.4.11. Let C = C_{1} ⊔_{Σ} C_{2} be a nodal twisted curve and let W = X_{1} ⊔_{D} X_{2} have first-order smoothable nodal singularities. Let g : C → W be a representable morphism, given via g_{i} : C_{i} → X_{i} ⊂ W and an isomorphism ϕ : (g_{1})|_{Σ} → (g_{2})|_{Σ}. Assume g_{i}^{∗}D = c · Σ as Cartier divisor on C_{i} (so the two contact orders agree). Let r, r_{Σ} ≥ 1 be an integers, and consider the twisted structures W′ = Y (\sqrt[r]{D}) and C′ = C( \sqrt[r_{Σ}]{D}). We again assume c divides r. Then

(1) g : C → W lifts to g̃ : C′ → X′ if and only if r|c · r_{Σ},

(2) such lift is representable if and only if r = c · r_{Σ}, and

(3) such lift is transversal if and only if r = c · r_{Σ}.

Proof. The necessity in (1) as well as (2) and (3) hold since the Lemma 1.3.1 applies to g_{i}. To show that r|c · r_{Σ} is sufficient in (1), consider the lift g̃_{i} of g_{i} corresponding to the line bundle O(dΣ̃) on (C_{i})′ with section s_{i} = 1d ; Σ̃, where rd = cr_{Σ} as in Lemma 1.3.1; it clearly satisfies s_{i}^{r} = g̃_{i}^{∗}1_{E}.

The isomorphism β of 1.4.1 sends O_{(C_{1})′}(Σ̃)|Σ̃ to O_{(C_{2})′}(Σ̃)^{∨}|Σ̃. In particular it sends O_{(C_{1})′}(dΣ̃)|Σ̃ to O_{(C_{1})′}(dΣ̃)|Σ̃, giving the gluing data for a lift g̃ : C′ → X′.

♣

Remark 1.4.12. In this case we do not have uniqueness since we may compose by ghost automorphisms.

2. Expanded pairs and degenerations

2.1. Expanded pairs. Convention 2.1.1. In this subsection we fix a locally smooth pair (X, D), i.e. X is an algebraic stack smooth along a smooth divisor D. Write N for N_{D/X}. Let ℙ := ℙ_{D}(N ⊕ O_{D}), with its sections D^{−} and D^{+} having normal bundle N^{∨} and N respectively. Let ℙ_{i} (for i ≥ 1) be copies of ℙ, and let D_{i}^{−} and D_{i}^{+} be the corresponding sections. For a positive integer r, let D_{r} := \sqrt[r]{N/D}, see 1.2. Definition 2.1.2. Let k ≥ 0 be an integer. Define X[k] to be the algebraic stack obtained by gluing X to ℙ_{1} via the natural isomorphism of D with D_{1}^{−}, and for every i ∈ {1, …, k − 1} gluing ℙ_{i} with ℙ_{i+1} via the natural isomorphism of D_{i}^{+} with D_{i+1}^{−}.

Note that X[k] has nodal first-order smoothable singularities along its singular locus, which is the disjoint union of D_{0}, …, D_{k−1} (where D_{i} is the image of either D_{i+1}^{−} or D_{i}^{+}). Write D_{k} for the image of D_{k}^{+}; note that X[k] is smooth along D_{k} and that ι^{∗}N_{D_{k}/X[k]} = N, if ι : D → D_{k} ⊂ X[k] is the natural isomorphism.

We call the sequence of morphisms D → X[k] → X an untwisted expanded pair. (It is also sometimes called an untwisted half accordion.) of length k. Definition 2.1.3. Let r := (r_{0}, …, r_{k}) be a sequence of positive integers. Define X[k](r) → X[k] to be the morphism obtained by twisting X[k] along D_{i} with index r_{i}; note that we can apply the construction in section 1.4 since the relevant normal bundles are naturally dual to each other. We call the sequence of morphisms D_{r_{k}} →^{ι} X[k](r) → X the r-twisted expanded pair of length k over (X, D), or just a twisted expansion of (X,D). Its twisting index

is defined to be the integer r_{k}.

Note that an untwisted expansions of length k is the same as a 1twisted expansion of length k, where 1 = (1, …, 1). Moreover, there is a natural morphism X[k](r) → X[k] which, if X is a scheme or algebraic space, is just the morphism to the coarse moduli space.

2.1.4. Stacks of expanded pairs. Forming stacks of expanded pairs is slightly more subtle than it might seem, since in general not all deformations are good. The stack T^{u} of untwisted expanded pairs was defined by Jun Li, see [Li1, Proposition 4.5] where the notation ℨ^{rel} is used for its universal family, and [GV, Section 2.8] where the notation T for the stack we denote here by T^{u} is introduced. In this paper we also use the stack T^{tw} of twisted espanded pairs. The stacks T^{u} and T^{tw} are studied in [ACFW], where various alternative definitions and properties are described. Here we give a definition which is quick but somewhat unsatisfying, specific for the case where 𝔾_{m}(D) = ℂ^{∗}.

Definition 2.1.5. Assume 𝔾_{m}(D) = ℂ^{∗}. Fix a positive integer r. The stack T_{r}^{tw} of twisted expanded pairs of index r of (X, D) is defined as follows.

An object over a reduced scheme S is a sequence of proper morphisms D → X → X × S such that

(1) D → X is a closed embedding;

(2) the composite morphism X → S is flat;

(3) for every point s ∈ S the fiber D_{s} → X_{s} → X is isomorphic to a half accordion of index r over (X, D).

A morphism of twisted expanded pairs of index r over a morphism ϕ : S → S′ is an equivalence class of morphisms ψ : X → X′ such that the following diagram is 2-cartesian:

\xymatrix{D \ar[r] \ar[d] & D′ \ar[d] \\ X \ar[r]^{ψ} \ar[d] & X′ \ar[d] \\ X × S \ar[r]^{Id ×ϕ} & X × S′.}

Note that the implicit 2-arrows making such a diagram cartesian are necessarily unique by Lemma B.1.3. Two such morphisms ψ, ψ̄ are equivalent if there exists a (necessarily unique) 2-morphism λ : ψ ⇒ ψ̄ satisfying the appropriate compatibility with the given 2-arrows.

We define the stack of twisted expanded pairs of (X, D) to be T^{tw} = ⊔_{r}T_{r}^{tw}.

The stack T^{u} of untwisted expanded pairs is defined as the substack of T^{tw} where in (3) above we insist that D_{s} → X_{s} → X is isomorphic to an untwisted half accordion over (X, D). In this case D = D × S. Note however that for twisted expansions D → S is in general not a product - it only becomes a product over a μ_{r}-gerbe over S.

We denote the universal families of expanded pairs X_{univ}^{u} → T^{u} and X_{univ}^{tw} → T^{tw}.

Proposition 2.1.6. (1) The stacks T_{r}^{tw} and T^{u} are smooth con-

nected algebraic stacks of dimension 0.

(2) There is a natural open embedding T^{u} ↪ T^{tw} and a natural right inverse T^{tw} → T^{u}.

(3) The stacks T^{tw} and T^{u} do not depend on the choice of the pair (X, D).

(4) If furthermore P ic(D) is representable and discrete, the objects and arrows of T^{tw}(S) agree with those given as in definition 2.1.5 also for non-reduced schemes.

A proof is given in [ACFW].

Part (3) of the proposition was observed in [GV]. Part (4) applies for X a curve and D a point, and Graber and Vakil identify T^{tw} with a familiar stack of pointed semistabe curves: the substack of the stack 𝔐_{0,3} of 3-pointed presetable curves of genus 0, where the pointed curve curve is semistable and the points marked 1, 2 are on the same component. This enables us to read off properties of T^{tw}, T^{u} from those of stacks of twisted curves.

2.1.7. Defining the stack of expansions in general. As in part (4) of the proposition, Definition 2.1.5 is valid also for non-reduced base schemes S if we assume in addition that P ic(D) is representable and H^{1}(D, O_{D}) = 0, i.e. P ic(D) discrete. But in general a half accordion admits deformations which are no longer half accordions. Since the stacks T^{u} and T^{tw} are smooth they are determined by their restriction to reduced schemes.

A definition of the stacks T^{tw}, T^{u} without any restrictions on D, as well as a proof of Proposition 2.1.6, can be found

in [ACFW]. Three approaches for doing this are given (along with several variations and relationships between them).

The first approach is the original approach of Jun Li, which is done in terms of an explicit limit presentation of the stacks and their universal families. With this approach one can describe the structure of T^{tw}, T^{u} and their universal families explicitly. The modular properties of the stack, which is used in the proof of the degeneration formula, require a further argument.

Another approach relies on the fact that the description above holds for all base schems when X is a curve. The same holds for the “one dimensional geometry” (A, B𝔾_{m}). One then shows that for such X the stacks T^{tw} and T^{u} are independent of X as in Proposition 2.1.6 (3), and in fact for every X the universal family can be constructed as a pull-back via the map X → A associated to D.

The third approach relies on the fact that half accordions carry a canonical log structure, and deformations of half accordions preserving the log structure are automatically half accordions, so the issue of bad deformations does not arise.

2.1.8. Charts for the stacks. Here we give a presentation of the base stacks, as this is used in this paper. For the universal families see [ACFW]:

(1) Jun Li’s stack T^{u} is the limit of the stacks U_{k} = [𝔸^{k}/𝔾_{m}^{k}] ≃ A^{k}, with k a positive integer, carrying a universal family of untwisted expansions with an order-preserving injective labeling of the splitting divisors by elements of {1, …, k}. The locus Δ_{ℓ} where the splitting divisor labeled by ℓ persists corresponds to x_{ℓ} = 0 in 𝔸^{k}. There are natural injective maps U_{k} → U_{k′} for each strictly increasing map {1, …, k} → {1, …, k′}, and the limit is taken in the categorical sense. This presentation can be read off directly from Graber-Vakil’s realization of T^{u} in terms of semistable curves.

(2) The stack T_{1}^{tw} has a presentation similar to that of T^{u}: it is the limit of U_{k,r} = [𝔸^{k}/𝔾_{m}^{k}] ≃ A^{k}, where k is a positive integer and r = (r_{1}, …, r_{k}) is a vector of positive integers (indicating twisting indices). But now U_{k,r} carries a universal family of twisted expansions of (X, D) with splitting divisors labelled by elements of {1, …, k}, and the splitting divisor corresponding to ℓ is twisted with index r_{ℓ}. We now take the same maps U_{k,r} → U_{k′,r′}, but only corresponding to those strictly increasing maps ϕ : {1, …, k} → {1, …, k′} respecing the twisting, namely r′_{ϕ(ℓ)} = r_{ℓ}. Again the limit is taken in the categorical sense. This presentation comes by presenting T_{1}^{tw} as a stack of twisted semistable curves.

(3) The stacks T_{r}^{tw} are all isomorphic to T_{1}^{tw}: the universal family is obtained by replacing the universal family by its r-th root X (\sqrt[r]{D_{k}}).

(4) The embedding T^{u} → T^{tw} is given by mapping U_{k} isomorphically to U_{k,1}, where 1 = (1, …, 1).

(5) The map T^{tw} → T^{u} is given by maps U_{k,r} → U_{k}, induced by the map 𝔸^{k} → 𝔸^{k} given by (z_{1}, …, z_{k}) ↦ (z_{1}^{r_{1}}, …, z_{k}^{r_{k}}). This in particular gives that

U_{k,r} = U_{k}( \sqrt[r_{1}]{Δ_{1}}, …, \sqrt[r_{k}]{Δ_{k}}).

See 1.4.8.

(6) This immediately gives a relationship of boundary divisors. Consider Δ ⊂ T^{u} the boundary divisor. Its inverse image in T^{tw} is the divisor ∑_{r} rΔ_{r}, where Δ_{r} is the divisor corresponding to expansions having a splitting divisor with twisting index r. See 1.1.3, 1.1.4.

2.2. Twisted expanded degenerations. We now discuss stacks analogous to the Artin stack underlying Jun Li’s family 𝔚 of expanded degenerations.

Convention 2.2.1. In this subsection we fix π : W → B, a flat morphism such that B is a smooth curve, W is a smooth stack, and b_{0} ∈ B is the unique critical value of π; we set W_{0} := π^{−1}(b_{0}) an assume W_{0} = X_{1} ⊔_{D} X_{2} is the union of two smooth closed substacks X_{1} and X_{2} intersecting transversally along D, a smooth divisor in each X_{i}. This implies that W_{0} is nodal and first-order smoothable along its singular locus D.

Definition 2.2.2. (1) Let k ≥ 0 be an integer. Define W_{0}[k] to be the Deligne–Mumford stack obtained by gluing the k-th untwisted half accordion X_{1}[k] over (X_{1}, D) to X_{2} via the identification of D_{k} ⊂ X_{1}[k] with D ⊂ X_{2}; denote the image of D_{i} ⊂ X_{1}[k] again by D_{i}.

(2) Let r := (r_{0}, …, r_{k}) be a sequence of positive integers. Define W_{0}[k](r) → W_{0}[k] to be the morphism obtained by twisting W_{0}[k] along D_{i} with index r_{i}.

(3) We call the morphism W_{0}[k](r) → W_{0} the (k, r) twisted accordion or over W_{0}, or just an expansion of W_{0}. We call it an untwisted accordion if r = (1, …, 1) or equivalently if it has the form W_{0}[k] → W_{0}. An expanded degeneration of W is either a general fiber or a twisted or untwisted accordion.

Definition 2.2.3. Assume 𝔾_{m}(D) = ℂ^{∗}. The stack 𝔗_{B,b_{0}}^{tw} of expanded degenerations of π is defined as a stack over B as follows.

An object over a reduced B–scheme S is a locally projective morphism W → S ×_{B} W of algebraic stacks satisfying the following conditions:

(1) the composite morphism W → S is flat;

(2) for every point s ∈ S the fiber W_{s} → W_{p(s)} is an isomorphism if p(s) ≠ b_{0}, and is an accordion over W_{0} if p(s) = b_{0}.

A morphism from W → S to W′ → S′ over a morphism of B–schemes ϕ : S → S′ is an equivalence class of morphisms ψ : W → W′ such that the following diagram is 2-cartesian:

\xymatrix{W \ar[r]^{ψ} \ar[d] & W′ \ar[d] \\ S ×_{B} W \ar[r]_{ϕ} & S′ ×_{B} W.}

Note that the implicit 2-arrow is uniquely determined by ψ by Lemma B.1.3, since there exists an open scheme-theoretically dense substack of W such that W → S ×_{B} W is representable. Two morphisms ψ and ψ̄ are equivalent if there exists an isomorphism λ : ψ → ψ̄ compatible with the given 2-arrows.

The stack 𝔗_{B,b_{0}}^{u} of untwisted expanded degenerations is defined by insisting that the fiber W_{s} → W_{p(s)} is an untwisted accordion over W_{0} if p(s) = b_{0}.

We denote the universal families of expanded degenerations W_{univ}^{u} → 𝔗^{u} and W_{univ}^{tw} → 𝔗^{tw}.

Here is the analogue of Proposition 2.1.6:

Proposition 2.2.4. (1) The stacks 𝔗_{B,b_{0}}^{tw} and 𝔗_{B,b_{0}}^{u} only depend on

the base (B, b_{0}) but not on the family W.

(2) The stacks 𝔗_{B,b_{0}}^{tw} and 𝔗_{B,b_{0}}^{u} are smooth, algebraic connected stacks of dimension 1;

(3) There is a natural open embedding 𝔗_{B,b_{0}}^{u} ↪ 𝔗_{B,b_{0}}^{tw} and a natural right inverse 𝔗_{B,b_{0}}^{tw} → 𝔗_{B,b_{0}}^{u};

(4) If P ic(D) is representable and discrete, the objects of 𝔗_{B,b_{0}}^{tw}(S), 𝔗_{B,b_{0}}^{u}(S) are described as above even for nonreduced S.

2.2.5. Definition in general and charts. The analogous discussion 2.1.7 holds for 𝔗_{B,b_{0}}^{tw} and 𝔗_{B,b_{0}}^{u} as well: the definition in terms of reduced base schemes is sufficient since the stacks are smooth; but families over nonreduced bas require care because not all deformations of an expansion are expansions. The general base case, with no assumptions on D, is treated in [ACFW]. The logarithmic approach and the approach through one-dimensional geometry are almost identical to the case of pairs.

Again we present the base stacks using Li’s limit approach. The following charts are read off from a description of 𝔗_{B,b_{0}}^{tw} in terms of semistable curves.

(1) For k a positive integer consider the morphism A^{k} → A given by the product map, and the morphism B → A given by the divisor b_{0}. This time let U_{k} = A^{k} ×_{A} B. Jun Li’s stack 𝔗_{B,b_{0}}^{u} is the limit of the stacks U_{k}. Again this carries a universal family of untwisted expansions with an order-preserving labeling of the splitting divisors by elements of {1, …, k}. And again the locus Δ_{ℓ} where the ℓ-th splitting divisor persists corresponds to x_{ℓ} = 0 in 𝔸^{k} → A^{k}. There are again natural injective maps U_{k} → U_{k′} for each strictly increasing map {1, …, k} → {1, …, k′}, and the limit is taken in the categorical sense.

(2) The stack 𝔗_{B,b_{0}}^{tw} has a presentation similar to that of 𝔗_{B,b_{0}}^{u}: it is the limit of U_{k,r} defined as follows. Fix k a positive integer and r = (r_{1}, …, r_{k}) a vector of positive integers (indicating twisting indices). Consider the map A^{k} → A induced by (x_{1}, …, x_{k}) ↦ ∏ x_{i}^{r_{i}}. We define U_{k,r} = A^{k} ×_{A} B using this map. Now U_{k,r} carries a universal family of twisted expansions with splitting divisors labelled by elements of {1, …, k}, and the splitting divisor corresponding to ℓ is twisted with index r_{ℓ}. We now take the maps U_{k,r} → U_{k′,r′}, but for only those strictly increasing map ϕ : {1, …, k} → {1, …, k′} respecing the twisting, namely r′_{ϕ(ℓ)} = r_{ℓ}. Again the limit is taken in the categorical sense.

(3) The embedding 𝔗_{B,b_{0}}^{u} → 𝔗_{B,b_{0}}^{tw} is given by mapping U_{k} isomorphically to U_{k,1}, where 1 = (1, …, 1).

(4) The map 𝔗_{B,b_{0}}^{tw} → 𝔗_{B,b_{0}}^{u} is given by maps U_{k,r} → U_{k}, induced by the map 𝔸^{k} → 𝔸^{k} given by (z_{1}, …, z_{k}) ↦ (z_{1}^{r_{1}}, …, z_{k}^{r_{k}}). Again this gives that

U_{k,r} = U_{k}( \sqrt[r_{1}]{Δ_{1}}, …, \sqrt[r_{k}]{Δ_{k}}).

(5) This gives the same relationship of boundary divisors: denote the boundary divisor, namely the inverse image of b_{0}, by 𝔗_{0}^{u} ⊂ 𝔗_{B,b_{0}}^{u}. It is evidently a reduced normal crossings divisor. Its inverse image in 𝔗^{tw} is the divisor 𝔗_{0}^{tw} = ∑_{r} r𝔗_{0}^{r}, where 𝔗_{0}^{r} is the divisor corresponding to expansions having a splitting divisor with twisting index r. Its reduction ∑_{r} 𝔗_{0}^{r} is a reduced normal crossings divisor.

(6) An observation about the universal family are useful. Let W_{k,r} → U_{k,r} be the universal family. By definition of Δ_{ℓ} ∈ U_{k}, we have a splitting of the restriction W_{k,1} ×_{U_{k}} Δ_{ℓ} as W_{1}^{ℓ} ⊔_{D^{ℓ}} W_{2}^{ℓ}. Then

W_{k,r} = W_{k,1}( \sqrt[r_{1}]{W_{1}^{1}}, \sqrt[r_{1}]{W_{2}^{1}}, …, \sqrt[r_{k}]{W_{1}^{k}}, \sqrt[r_{k}]{W_{2}^{k}}).

This is obtained by iterating 1.4.8.

2.3. The singular case in the absence of smoothing. We have defined 𝔗_{0}^{u} = 𝔗_{B,b_{0}}^{u} ×_{B} {b_{0}} and 𝔗_{0}^{tw} = 𝔗_{B,b_{0}}^{tw} ×_{B} {b_{0}}. When π :

W → B is as in 2.2 above, these stacks parametrize untwisted and twisted expansions of the singular fiber W_{0}. But more is true: consider a first-order smoothable W_{0} = X_{1} ⊔_{D} X_{2}. We define untwisted and twisted accordions on W_{0} as before, and they obviously coincide with the previous definition if W_{0} is provided with a smoothing over some curve B. In [ACFW] we define families of expansions of W_{0}. Then we have the following.

Proposition 2.3.1. (1) The stacks 𝔗_{0}^{tw} and 𝔗_{0}^{u} are the stacks of

twisted and untwisted expansions of W_{0}.

(2) These stacks 𝔗_{0}^{tw} and 𝔗_{0}^{u} are independent of W_{0}.

In particular, if W_{0} is a fiber in a one-parameter family π : W → B, the universal families over 𝔗_{0}^{tw} and 𝔗_{0}^{u} are independent of the smoothing.

Note that the stacks 𝔗_{0}^{tw} and 𝔗_{0}^{u} are singular, and 𝔗_{0}^{tw} is nonreduced. So a direct definition analogous to 2.1.5 and 2.2.3 in terms of families over reduced schemes is insufficient.

2.4. Split expansions. As the stacks 𝔗_{0}^{tw} and 𝔗_{0}^{u} are normal crossings, their normalizations are of interest.

We define the stack 𝔗_{0}^{r,spl} to be the stack of twisted accordions with a choice of a splitting divisor D of twisting index r. We similarly 𝔗_{0}^{u,spl} in the untwisted case. We have a natural map 𝔗_{0}^{r,spl} → 𝔗_{0}^{r} ⊂ 𝔗_{0}^{tw} and 𝔗_{0}^{u,spl} → 𝔗_{0}^{u}. On the level of charts these are defined as follows: recall the chart U_{k,r} of 𝔗_{0}^{tw}. For an index ℓ such that r_{ℓ} = r consider the locus Δ_{k,r,ℓ} ⊂ U_{k,r} defined by x_{ℓ} = 0. Then 𝔗_{0}^{r,spl} is the limit of the stacks Δ_{k,r,ℓ}, where the maps Δ_{k,r,ℓ} → Δ_{k′,r′,ℓ′} are taken only with respect to those strictly increasing maps ϕ : {1, …, k} → {1, …, k′} respecting r, r′ as before such that in addition ϕ(ℓ) = ℓ′. The universal splitting divisor over Δ_{k,r,ℓ} is denoted D^{ℓ}.

Consider now the stack 𝔔 = 𝔗_{0}^{u,spl} ×_{𝔗_{0}^{u}} 𝔗_{0}^{tw}. It decomposes as a disjoint union 𝔔 = ⊔_{r}𝔔_{r}, where over he reduction of 𝔔_{r} the splitting divisor in the universal family is twisted with index r. On the level of charts 𝔔_{r} is given by rΔ_{k,r,ℓ} ⊂ U_{k,r} defined by x_{ℓ}^{r} = 0. It follows that

Lemma 2.4.1. The morphism 𝔗_{0}^{r,spl} → 𝔔_{r} is of degree 1/r.

Consider the universal family W_{univ}^{r,spl} over 𝔗_{0}^{r,spl}. Taking the partial normalization along the splitting divisor D gives two families D → X_{1} and D → X_{2} of expanded pairs, which define a morphism 𝔗_{0}^{r,spl} → T_{r}^{tw} × T_{r}^{tw}.

Lemma 2.4.2. The natural morphism

𝔗_{0}^{r,spl} → T_{r}^{tw} × T_{r}^{tw}

corresponding to the two components of the partial normalization of the universal family W_{univ}^{r,spl} is a gerbe banded by μ_{r}; in particular it has degree 1/r.

Proof. This is identical to [AGV, Section 5]

. The gluing datum corresponds to an isomorphism of the residual gerbes reversing the band. Such isomorphisms form a torsor under the group-stack Bμ_{r}.

♣

2.5. Weighted and 𝔯-twisted expansions. In Theorem 4.4.1 we will need a slightly refined version of the stacks T^{tw} of expanded pairs and 𝔗^{tw} of expanded degenerations, namely the stacks T^{𝔯} and 𝔗^{𝔯} of 𝔯twisted Δ-wighted expansions.

2.5.1. Δ-weighted expansions. As a first step we consider a set Δ and stacks T^{Δ} and 𝔗^{Δ} in which the splitting divisors of the universal object are weighted, in a sense analogous to Costello [Co, Section 2], by elements of Δ. In our application of Theorem 4.4.1 the set Δ is the set of finite multi-sets c = {c_{1}, …, c_{n}} of positive integers, which will indicate contact orders of branches of a curve along a splitting divisor. (Recall that a multi-set is an unordered sequence, with elements possibly appearing several times). Unlike Costello’s situation we will not need a monoid structure on the weight set.

The construction and algebricity of these weighted stacks can be obtained by a slight modification of their various constructions for the stacks without weights. An approach as in [ACFW] close to Jun Li’s original construction described in 2.1.7 and 2.2.4 is as follows: the stack 𝔗^{tw} is obtained as a direct limit of stacks of the form U_{k,r}, where r denotes the twisting sequence of the k splitting divisors. The stack 𝔗^{Δ} is similarly defined as a direct limit of labelled copies of the same:

U_{k,r,𝔠} ≃ U_{k,r}.

Here 𝔠 = (c^{1}, …, c^{k}) and c^{ℓ} ∈ Δ denote the weight on the ℓ-th splitting divisor; the arrows U_{k,r,𝔠} → U_{k′,r′,𝔠′} are included in the limiting category only when r′ and 𝔠′ are both compatible with the underlying increasing map ϕ : {1, … k} → {1, … k′}, namely r′_{ϕ(ℓ)} = r_{ℓ} and c′^{ϕ(ℓ)} = c^{ℓ}. The construction for T^{Δ} is similar.

There are evident forgetful maps 𝔗^{Δ} → 𝔗^{tw} and T^{Δ} → T^{tw}. By construction these maps are étale and representable. The universal families on 𝔗^{Δ} and T^{Δ} are given by pullback along the respective forgetful map. As before we define 𝔗_{0}^{Δ} to be the inverse image of b_{0}. The relationship of families in 2.2.5 (6) immediately applies to charts of the stacks 𝔗^{Δ}, T^{Δ}.

2.5.2. 𝔯-twisted expansions. Next we consider a function 𝔯 : Δ → ℕ with positive integer values. An object of 𝔗^{Δ} or T^{Δ} is said to be 𝔯-twisted if the ℓ-th splitting divisor is twisted with index 𝔯(c^{ℓ}). We obtain open substacks T^{𝔯} ⊂ T^{Δ} and 𝔗^{𝔯} ⊂ 𝔗^{Δ} of 𝔯-twisted, Δ-weighted expansions. Indeed these are simply obtained by only taking the charts U_{k,r,𝔠} for which r_{ℓ} = 𝔯(c^{ℓ}) in the notation above. Again 𝔗_{0}^{𝔯} denotes the inverse image of b_{0}

We define a partial ordering ≺ on functions 𝔯 : Δ → ℕ by divisibility: 𝔯 ≺ 𝔯′ if an only if 𝔯(c)|𝔯′(c) for all c ∈ Δ.

Lemma 2.5.3. Assume that we are given functions 𝔯 and 𝔯′ : Δ → ℕ such that 𝔯 ≺ 𝔯′. Then are natural partial untwisting morphisms 𝔗_{B,b_{0}}^{𝔯′} → 𝔗_{B,b_{0}}^{𝔯}, 𝔗_{0}^{𝔯′} → 𝔗_{0}^{𝔯} and T^{𝔯′} → T^{𝔯}. These morphisms lift canonically to universal families, i.e.: for every algebraic stack S and every family of expansions W′ → S corresponding to a morphism f′ : S → 𝔗^{𝔯′}, let W → S be the family induced by the composite morphism f : S → 𝔗^{𝔯}; then there is a natural morphism p : W′ → W lifting 𝔗_{B,b_{0}}^{𝔯′} → 𝔗_{B,b_{0}}^{𝔯}.

Proof. This can be described on the level of charts defined above. We have a canonical morphism from U_{k,r′,𝔠} = U_{k}( \sqrt[r′_{1}]{Δ_{1}}, …, \sqrt[r′_{k}]{Δ_{k}}) to U_{k,r,𝔠} = U_{k}( \sqrt[r_{1}]{Δ_{1}}, …, \sqrt[r_{k}]{Δ_{k}}) and similarly on the universal families, see 1.2.5. ♣

The following proposition is a manifestation of the well-known fact that given a twisted curve, there is essentially a unique way to increase its indices by any given amount.

Proposition 2.5.4. The morphisms 𝔗_{B,b_{0}}^{𝔯′} → 𝔗_{B,b_{0}}^{𝔯}, 𝔗_{0}^{𝔯′} → 𝔗_{0}^{𝔯} and T^{𝔯′} → T^{𝔯} are proper, quasi-finite, flat and surjective. Moreover this map has pure degree 1 in the sense of [Co, Section 5].

Proof. This can be tested locally on the target, and the statement for 𝔗_{0}^{𝔯′} → 𝔗_{0}^{𝔯} follows from that for 𝔗_{B,b_{0}}^{𝔯′} → 𝔗_{B,b_{0}}^{𝔯}. It suffices to consider a chart U_{k,r,𝔠} in 𝔗_{B,b_{0}}^{𝔯} or T^{𝔯}, and note that there is a canonical chart U_{k,r′,𝔠} in 𝔗_{B,b_{0}}^{𝔯′} or T^{𝔯′} and a proper quasi-finite birational surjective map U_{k,r′,𝔠} → U_{k,r,𝔠}. This is because U_{k,r′,𝔠} is the root stack of U_{k,r,𝔠} with indices given by r′_{ℓ}/r_{ℓ} along the divisor corresponding to the ℓ-th node. Since these charts are smooth, the map is flat. ♣ 2.6. Stable expanded configurations. We now consider stable maps of collections of points to expanded pairs and expanded degenerations. If X is a smooth stack, we can view X^{n} as a (smooth) parameter space for maps from n ordered points to X. We want to generalize this to the case of a pair (X, D) and to that of the fibers of a one-parameter degeneration π : W → B, where we insist that none of the n points maps to either a singular point or the boundary divisor.

In the case of a pair (X, D) this was done by Kim and Sato in [KS]. The case of a degeneration is similar, but we approach it from a slightly different angle, which can also be adapted to the case of pairs. Further details on both cases are available in [ACFW]. To avoid notation conflicts with [KS], we will consider configurations of points labeled by {1, …, n}. The unlabeled case is obtained as usual by taking the quotient by the symmetric group S_{n}.

Convention 2.6.1. We fix a morphism W → B and a point b_{0} ∈ B with the same assumptions as in Convention 2.2.1.

Definition 2.6.2. A stable expanded configuration (W → W, σ_{i}) of degree n on W → B consists of

(1) an object W → W_{b} of 𝔗_{B,b_{0}}^{u}(ℂ), namely either a general fiber or an untwisted accordion, and

(2) an ordered collection of n smooth points σ_{i} ∈ W^{sm}, such that, in case W is an accordion, for each component ℙ_{j} there exists an i_{j} such that σ_{i_{j}} ∈ ℙ_{j}.

An isomorphism ρ : (W → W, σ_{i}) → (W′ → W′, σ′_{i}) is an isomorphism ρ : W → W′ over W such that ρ(σ_{i}) = σ′_{i}.

Definition 2.6.3. A family of stable expanded configurations of degree n over a B–scheme S is given by

(1) an object W_{S} → W × S of 𝔗_{B,b_{0}}(S);

(2) n sections σ_{i} : S → W_{s}; such that for every s ∈ S the fiber over s is a stable expanded configuration of degree n.

Morphisms of families of stable expanded configurations are defined using cartesian diagrams as usual; we denote the the resulting category W_{π}^{[n]}. It is clearly a stack. Composing σ_{i} with W → W, we obtain an object of the fibered power W_{π}^{n} of W over B, giving a morphism W_{π}^{[n]} → W_{π}^{n}.

The notation should not be confused with the notation for the degreen Hilbert scheme of a surface.

Proposition 2.6.4. The stack W_{π}^{[n]} is smooth and algebraic, and the morphism W_{π}^{[n]} → W_{π}^{n} is projective. In particular, if W is a scheme, or a Deligne–Mumford stack, so is W_{π}^{[n]}.

The proof of this proposition, as well as the analogous result of [KS], can be found

in [ACFW]. As before, the construction also applies for a first-order smoothable variety W_{0} without a deformation.

3. Stacks of twisted stable maps and their properness

3.1. Conventions on stacks of maps. We will extend Jun Li’s construction [Li1, Section 3] to the case where we degenerate an orbifold instead of a smooth variety, and we introduce the twisted version of that stack which allows us to replace predeformable maps by transversal ones. We do the same for pairs as well. In this sections, the ambient space for both degenerations and pairs will be denoted by the letter W.

We will use data as set in one of the following two conventions:

Convention 3.1.1 (Data for a degeneration). Consider a morphism π : W → B and W_{0} = X_{1} ⊔_{D} X_{2} as in Section 2.2. We also fix Γ = (β, g, N, e) where

(1) β is a curve class in the fiber of W → B;

(2) g ≥ 0 is an integer;

(3) N is a finite ordered set, possibly empty, which we may take to be {1, …, n}.

(4) e = (e_{i})_{i∈N} is a tuple of positive integers such that I_{e_{i}}(W_{b}) ≠ ∅ for all b ∈ B, i ∈ N; note that if W is a variety, one must have e_{i} = 1 for all i.

Convention 3.1.2 (Data for a pair). Fix

a smooth pair (W, D) with W a Deligne–Mumford stack. We also fix Γ = {β, g, N, M, e, f, c} where

(1) β is a curve class on X;

(2) g ≥ 0 is an integer;

(3) N, M are disjoint finite ordered sets, which we may take to be {1, …, n} and {n + 1, …, n + |M|}.

(4) e = (e_{i})_{i∈N} as above, and f = (f_{j})_{j∈M} is similarly a tuple of positive integers such that I_{f_{j}}(D) ≠ ∅ for all j ∈ M.

(5) c = (c_{j})_{j∈M} are positive integers such that ∑_{j∈M} c_{j}/f_{j} = (β ·

D)_{X}. We denote by d = (d_{j})_{j∈M} the tuple formed by d_{j} = c_{j}/f_{j}.

Remark 3.1.3. Following Jun Li, we will later find it useful to think in either case of the data Γ as a weighted modular graph. In the degenerate case it has one vertex marked with (g, β), no edges or loops, legs corresponding to the set N and weighted by e_{i}, and no roots. In the pair case it has again one vertex, no edges or loops, legs corresponding to the set N and weighted by e_{i}, and roots corresponding to the set M and weighted by (f_{j}, c_{j}). See definition 4.8.1. Indeed, the degeneration formula requires working with disconnected graphs. To avoid heavy notation here we postpone the disconnected case to section 4.8.

Consider the universal families W_{univ}^{tw} → 𝔗_{B,b_{0}}^{tw} and W_{univ}^{u} → 𝔗_{B,b_{0}}^{u} of twisted and untwisted expanded degenerations. They both satisfy the assumptions in Convention C.1.1 with D = ∅. Similarly consider the universal families of pairs (W_{univ}^{tw}, D_{univ}^{tw}) → T^{tw} and (W_{univ}^{u}, D_{univ}^{u}) → T^{u}. These satisfy the assumptions in C.1.1 with D = D_{univ}.

Convention 3.1.4 (Shorthand for stacks of maps). The notation 𝔎^{tw}, 𝔎^{u}, and K(W) will be used for either one of the following three cases

(1) (Degeneration case) W → B is a degeneration as in 3.1.1, T^{tw} = 𝔗_{B,b_{0}}^{tw}, T^{u} = 𝔗_{B,b_{0}}^{u}.

𝔎^{tw} := 𝔎_{Γ}(W_{univ}^{tw}/𝔗_{B,b_{0}}^{tw}), 𝔎^{u} := 𝔎_{Γ}(W_{univ}^{u}/𝔗_{B,b_{0}}^{u})),

K(W) := 𝔎_{Γ}(W/B).

(2) (Singular case) W = W_{0} is first-order smoothable, B = Spec ℂ, T^{tw} = 𝔗_{0}^{tw}, T^{u} = 𝔗_{0}^{u}.

𝔎^{tw} := 𝔎_{Γ}(W_{0 univ}^{tw}/𝔗_{0}^{tw}), 𝔎^{u} := 𝔎_{Γ}(W_{0,univ}^{u}/𝔗_{0}^{u})),

K(W) := 𝔎_{Γ}(W_{0}).

(3) (Relative case) (W, D) a pair, T^{tw} = T^{tw}, T^{u} = T^{u}.

𝔎^{tw} := 𝔎_{Γ}((W_{univ}^{tw}, D_{univ}^{tw})/T^{tw}), 𝔎^{u} := 𝔎_{Γ}((W_{univ}^{u}, D_{univ}^{u})/T^{u})),

K(W) := 𝔎_{Γ}(W, D).

We follow the notation K, 𝔎_{nd} and K_{nd} of C.1.6, adding a superscript tw or u to denote the corresponding substacks of 𝔎^{tw} or 𝔎^{u}. We will suppress the superscript tw or u when statements hold for wither one.

In either of the three cases we will write f : (C, Σ) → W → W for a stable map belonging to 𝔎 (i.e. 𝔎^{tw} or 𝔎^{u}). We will indicate the divisor D only when necessary.

Following [Li1], we can characterize points of 𝔎 belonging to the Deligne–Mumford locus K using semistable components:

Definition 3.1.5. Let f : (C, Σ) → W → W be a stable map corresponding to a point in 𝔎^{tw}. Let E ⊂ C be an irreducible component.

We say that E is a semistable component if it is a standard cyclic cover of a line occuring as a fiber in some ℙ, explicitly if

(1) E is smooth, irreducible, of genus zero, contains no marked points;

(2) E maps to a fiber F of W → W;

(3) f|_{E} : E → F is branched at most over the intersection of F with the singular locus of W;

(4) E ∩ C^{sing} maps to W^{sing}.

Lemma 3.1.6. A point of 𝔎 given by a stable map f : (C, Σ) → W → W is in K if and only if there is no irreducible component ℙ_{i} such that every component of C whose image meets ℙ_{i} ∖ (D_{i}^{+} ∪ D_{i}^{−}) is a semistable component.

Proof. The point is in K if and only if no positive dimensional subgroup of the group of automorphisms of the accordion W lifts to an automorphism of C. Up to the finitely many ghost automorphisms, such automorphism are given by a copy of 𝔾_{m} for every component ℙ_{i}. The Deligne–Mumford condition is equivalent to ensuring that for each i there is at least one component X_{i} mapping to ℙ_{i} to which the 𝔾_{m} action doesn’t lift. It is easy to see that the only components whose image meets ℙ_{i} ∖ (D_{i}^{+} ∪ D_{i}^{−}) to which the action lifts are exactly the semistable components. ♣

3.2. Transversal maps and predeformable maps. Recall the natural morphisms between 𝔗_{B,b_{0}}^{tw} and 𝔗_{B,b_{0}}^{u} in Proposition 2.2.4 (3), and the corresponding ones for the singular and relative cases. By the functoriality of [AV, Corollary 9.1.2], these induce an open embedding 𝔎^{u} → 𝔎^{tw} and an untwisting morphism 𝔎^{tw} → 𝔎^{u} which is its left inverse. These are compatible with the morphisms 𝔎^{u} → K(W), 𝔎^{tw} → K(W).

Definition 3.2.1. Let f : (C, Σ) → W → W be a stable map corresponding to a point in 𝔎^{tw}. We say that f is transversal if it is transversal to both the singular locus and the boundary divisor in the sense of A.2.

Note that the condition is vacuous when W is nonsingular and D empty.

Remark 3.2.2. If f is a transversal stable map, then it is nondegenerate.

By Lemma A.2.2 the transversal condition is open. This allows us to formulate the following:

Definition 3.2.3. We define 𝔎^{tr} ⊂ 𝔎^{tw} to be the open substack of transversal maps; we write K^{tr} := K^{tw} ∩ 𝔎^{tr}.

The main objects we will use are transversal maps. However we find it appropriate to relate them to predeformable maps, a notion used by Jun Li and other previous authors. Accordingly, our arguments will go by way of predeformable maps, even though this detour can be avoided.

Definition 3.2.4. We say that a non-degenerate map C → W is predeformable if it is in the set-theoretic image of 𝔎^{tr} in 𝔎^{u}.

3.2.5. Contact orders and Li’s predeformability. In [Li1, Definition 2.5], a nondegenerate morphism f : C → W over an algebraically closed field is defined to be predeformable if at every point p ∈ C mapping to a singular component D^{ℓ} of W, locally the map is described as

(Spec k[u, v]/(uv))^{sh} ⟶ (Spec k[x, y, z_{1}, …, z_{m}]/(xy))^{sh}

where

x ↦ u^{c_{p}}, y ↦ v^{c_{p}}.

for some positive integer c_{p}, called the contact order at p. Note that we put no condition at points mapping to the boundary divisor, if nonempty; however for points p mapping to the boundary divisor we define a contact order as usual: locally the map is described as

(Spec k[u])^{sh} ⟶ (Spec k[x, z_{1}, …, z_{m}])^{sh}

where x ↦ u^{c_{p}}, and the contact order is again c_{p}.

We now show that the two notions of predeformability coincide in a precise way. The key is Lemmas 1.3.1 and 1.4.11.

Lemma 3.2.6. (1) Let C → W be a predeformable map in the sense of Definition 3.2.4. Then C → W is predeformable in the sense of [Li1, Definition 2.5].

(2) Let f : C → W be predeformable in the sense of [Li1, Definition 2.5]. For any component D of the singular locus of W fix a positive integer r_{D} divisible by the contact order c_{p} for every node of C mapping to the given singular component D. Let W be the root stack of W with index r_{D} over each component D of the singular locus. Then there is a transversal map f̃: C → W mapping to f : C → W; in particular f is predeformable in the sense of Definition 3.2.4.

(3) Any twisted stable map f̃ : C → W, with W as in (2) above, and lifting f : C → W is transversal.

Proof. (1) Let C → W be a transversal map which lifts C → W. Consider a point p ∈ C, and a lifting p̃ in C. In local coordinates, if p is a node the map C → W is given by x ↦ u, y ↦ v and arbitrary z_{i} ↦ f_{i}(u, v). The coordinates on C are ū = u^{r_{p}} and v̄ = v^{r_{p}}. The coordinates on W are x̄ = x^{r_{D}}, ȳ = y^{r_{D}}, and z_{i} unchanged. Then x̄(u^{r_{p}}) = u^{r_{D}} and similarly for ȳ. We thus have r_{D} = c_{p} · r_{p} for some c_{p} and may take ū = x̄^{c_{p}}, v̄ = ȳ^{c_{p}} as needed. When p maps to boundary divisor the calculation is similar.

(2) Applying Lemmas 1.3.1 and 1.4.11 a transversal representable lift exists with C having twisting index r_{p} = r_{D}/c_{p}, locally at each p mapping to a component D. Since such a lift is unique away from the union of the D, these local lifts glue to a lift C → W.

(3) This follows from part (4) of Lemma 1.3.1 and part (3) of

Lemma 1.4.11.

♣

We now consider the scheme structure for stacks of transversal and predeformable maps. Since there are infinitely many choices for the twisting index r_{D} in Lemma 3.2.6, we need the following for constructibility:

Lemma 3.2.7. (1) (Base change property.) Let C → W → S be a family of transversal maps over S, and let W̃ → W be a family of root stacks. Let C̃ = W̃ ×_{W} C. Then C̃ → S is a twisted nodal curve and C̃ → W̃ → S is a family of transversal maps.

(2) (Descent property.) Assume C̃ → W̃ → S is a family of transversal maps with underlying untwisted maps C → W → S. Assume that there is an integer d, and D ⊂ W, either a component of the singular locus or the boundary divisor, such that W̃ is twisted at D with twisting index r_{D} such that for any point p of C mapping to D^{ℓ}, we have d · c_{p}|r_{D}. Write r = \frac{r_{D}}{d} and let W → W be isomorphic to W (\sqrt[r]{D}) near D and to W̃ elsewhere. Then the representable map C → W → S obtained by stabilizing C̃ → W → S is transversal.

Proof. The transversality property is tested on fibers. The base change property is now a local computation, e.g. in case of a node

(Spec k[u, v]/(uv))^{sh} ⟶ (Spec k[x, y, z_{1}, …, z_{m}]/(xy))^{sh} where x ↦ u, y ↦ v and z_{}↦ f_{i}(u, v), the map C̃ → W̃ is given by

(Spec k[ũ, ṽ]/(ũṽ))^{sh} ⟶ (Spec k[x̃, ỹ, z_{1}, …, z_{m}]/(x̃ỹ))^{sh}

where x̃ ↦ ũ, ỹ ↦ ṽ and z_{}↦ f_{i}(u, v) as before.

For the descent property, note that C is obtained as the relative coarse moduli space of C̃ → W. Its formation commutes with base change so we can restrict to fibers again. The integer r is divisible by the contact orders, so Lemmas 1.3.1 and 1.4.11 apply. In particular, since the map is representable we have r = f_{p}c_{p}, and therefore it is transversal. ♣

Lemma 3.2.8. The collection of predeformable maps is closed in 𝔎_{nd}^{u} ⊂ 𝔎^{u}, the open locus of nondegenerate maps.

Proof. This is a local statement in the étale topology; hence we can use the proof given by Jun Li, see [Li1, Lemma 2.7].

♣

Given a deformation of a nondegenerate map over a base of finite type, only finitely many contact orders occur. Lemma 3.2.7 implies that the following gives a well defined closed substack:

Definition 3.2.9. We define the stack of predeformable maps 𝔎_{pd}^{tw} to be the stack-theoretic image of 𝔎^{tr} in 𝔎^{u}. We define 𝔎_{pd}^{u}, K_{pd}^{u}, K_{pd}^{tw} as the intersections of the appropriate sunbstacks of 𝔎^{tw} with 𝔎_{pd}^{tw}.

Remark 3.2.10. A more precise form of the statement in Lemma 3.2.6, which we do not use in this paper, is the following: if W → B is a projective morphism of schemes, then the stack K_{pd}^{u} is naturally isomorphic to 𝔐(W, Γ) in [Li1].

Set theoretically this is shown in Lemma 3.2.6. The subtle scheme structure in J. Li’s stack relies on [Li1, Lemmas 2.3], which describes the scheme structure of a predeformable map over a base scheme, and [Li1, Lemmas 2.4] which shows that the scheme structure behaves well under base change and can be glued. One can lift these to the root stacks described in Lemma 3.2.6 and show that indeed a family of predeformable maps underlies a family of transversal maps.

Remark 3.2.11. As remarked by Jun Li, 𝔎_{pd}^{u} is locally closed, but in general not open, in 𝔎^{u}; this makes it hard to write down a perfect obstruction theory for it, see [Li2, 1.2-1.3 p. 213-129 and Appendix, p. 284-288].

3.3. Properness of the stack of predeformable maps. Our goal is to prove properness for certain stacks of transversal maps, which can be done directly. However we find it appropriate to relate this to previous work and go through properness of predeformable maps.

We will need the following lemma, which is a slightly stronger statement of part of Lemma 3.2.8.

Lemma 3.3.1. Let (f : C → W, Σ) ∈ 𝔎^{u} , and let X ⊂ C be an irreducible nondegenerate component. Assume there is a point z ∈ X ∩ C^{sm} such that f(z) ∈ W^{sing}. Then (f : C → W, p_{i}) ∉ \overline{𝔎_{pd}^{u}} ⊂ 𝔎^{u}.

Proof. We argue by contradiction and assume there exist

• a spectrum Δ of a discrete valuation ring R with generic point η and special point s;

• a family (f_{Δ} : C_{Δ} → W_{Δ}, Σ_{Δ}) in 𝔎^{u} such that (f_{η} : C_{η} → W_{η}, Σ_{η}) ∈ 𝔎_{pd}^{u} and (f_{s} : C_{s} → W_{s}, Σ_{s}) = (f : C → W, Σ).

Let z be a point on X which is smooth on C but maps to W^{sing}. The problem is local at z so we may assume

W_{Δ}^{sh} = (Spec R[u, v, w_{1}, …, w_{m}]/(uv − a))^{sh}

for some a ∈ R,

C_{Δ}^{sh} = (Spec R[x])^{sh} ,

and on the central fiber

f^{∗}u = 0, f^{∗}v = x^{r}.

Here sh means strict henselization and may be thought of as a choice of local analytic coordinates.

Consider the homomorphism f_{Δ}^{∗} : R[u, v, w_{i}]/(uv−a) → R[x] and let u_{Δ}, v_{Δ} be the images of u and v; they must satisfy u_{Δ}v_{Δ} = a, and be equal to (0, x^{r}) modulo m_{s}. In particular v_{Δ}, viewed as a polynomial in x, has positive degree; therefore the only possibility that its product with u_{Δ} has degree zero is that u_{Δ} = a = 0, which means that f_{η} is not predeformable. ♣

Notation 3.3.2. When we use the valuative criterion, we always take Δ = Spec R for R a discrete valuation ring, with general point η and closed point s. Given a commutative diagram

\xymatrix{η \ar[r]^{f_{η}} \ar[d] & X \ar[d]^{p} \\ Δ \ar[r]_{g} & Y}

we will view liftings of g to X as a groupoid whose objects are the two-commutative diagrams

\xymatrix{η \ar[r]^{f_{η}} \ar[d] & X \ar[d]^{p} \\ Δ \ar@{-->}[ur]^{f} \ar[r]_{g} & Y}

and whose morphisms f ⇒ f′ are those sending the two-commutative diagram of f to that of f′. We will sometimes write “there is a unique lifting with certain properties” to mean “there exists a lifting with certain properties, and it is unique up to unique isomorphism”.

Lemma 3.3.3. The commutative diagram

\xymatrix{𝔎^{u} \ar[r] \ar[d] & T^{u} \ar[d] \\ K(W) \ar[r] & B}

induces a proper morphism 𝔎^{u} → K(T) ×_{B} T^{u}.

Proof. The fibered product is 𝔎_{Γ}(W′/T^{u}) where W′ := W ×_{B} T^{u}. The structure map W_{univ}^{u} → W′ induces the morphism from 𝔎^{u} to the fiber product, and it is proper by C.1.8. ♣

Theorem 3.3.4. The natural morphism K_{pd}^{u} → K(W) is proper.

The proof of this theorem will be given at the end of this section 3.3. It is closely related to the corresponding proof in [Li1, Lemmas 3.8, 3.9]; the reader who is familiar with both Jun Li’s proof and the definition of twisted stable maps may well be able to directly modify Jun Li’s proof to cover the orbifold case treated here.

Notation 3.3.5. To prove properness of the morphism K_{pd}^{u} → K(W) we use the valuative criterion. We will fix from now on a commutative diagram

\xymatrix{η \ar[r] \ar[d] & K_{pd}^{u} \ar[d] \ar[r] & T^{u} \ar[d] \\ Δ \ar[r] & K(W) \ar[r] & B,}

and our aim will be showing that, after a base change Δ̃ → Δ, there is a unique lifting of Δ → K(W) to Δ̃ → K_{pd}^{u}. We will denote by (C̄, Σ̄, f̄) the family of stable maps over Δ corresponding to Δ → K(π) and by (C_{η}, Σ_{η}, f_{η} : C_{η} → W) the family corresponding to η → K_{pd}^{u}.

Remark 3.3.6. From Lemma 3.3.3 it follows that there exists an equivalence, compatible with base change, between liftings of Δ → B to T^{u} and liftings (up to base change) of Δ → K(W) to 𝔎^{u}. Given a lifting a : Δ̃ → T^{u}, we will denote by (C^{a}, Σ^{a}, f^{a} : C^{a} → W) the family of stable maps corresponding to the lifting Δ̃ → 𝔎^{u} induced by a.

The key to producing a lifting of Δ → B to T^{u} is to use an auxiliary choice of a stable configuration of points as a guide.

Proposition 3.3.7. After a base change Δ̃ → Δ, there exist a positive integer N and closed subschemes (p̄_{1}, …, p̄_{N}) =: P̄ ⊂ C̄ such that

(1) the induced morphisms p̄_{i} → Δ̃ are isomorphisms;

(2) each p̄_{i} is contained in the smooth, nontwisted and not marked locus of both C̄_{η} and C̄_{s};

(3) P̄_{η} lifts uniquely to P_{η} ⊂ C_{η};

(4) every irreducible component of C̄_{s} intersects P̄_{s}.

Proof. First assume that the morphism \underline{W} → B, where \underline{W} is the coarse moduli space of W, is quasiprojective. Consider the line bundle

L := (ω_{C̄/Δ}(Σ) ⊗ \underline{f}^{∗}(H^{⊗3}))^{⊗3},

where H is an ample line bundle on the fibers of W → B. The line bundle L is very ample for C over Δ; we can choose the p̄_{i} to be the zero locus of a general section of L, suitably ordered by taking a base change. In particular, in this case the degree of L, and hence the number N of added marked points, is given a priori in terms of the numerical data.

For the general case, we remark that we can replace in the previous argument W → B with its closed substack which is the scheme theoretic image of \underline{C} → W_{Δ}. Its coarse moduli space is automatically quasiprojective because it is a family of curves over a discrete valuation ring

. ♣

We now replace Δ by Δ̃, so we may assume sections as in the proposition exist over Δ.

Proposition 3.3.8. (1) The morphism η → W^{N} induced by f_{η} and P_{η} ⊂ C_{η} defines a stable configuration, and hence a morphism η → W_{π}^{[N]} which lifts the morphism Δ → W^{N} given by f̄ and P̄;

(2) There exists a unique lifting of Δ → B to a_{0} : Δ → T^{u} such that the induced map s → W^{[N]} is a stable configuration.

Proof. (1) If X ⊂ C_{η} is an irreducible component then X contracts to a point in C̄_{η} if and only if it contains no p_{i}, if and only if it is semistable. The result then follows from Lemma 3.1.6.

(2) This follows from the fact W_{π}^{[N]} → W_{B}^{N} is proper and representable. ♣

Proposition 3.3.9. The point (C_{s}, Σ_{s}, f_{s}) ∈ 𝔎^{u} induced by the lifting a_{0} is in K_{nd}^{u}.

Proof. Step 1: we prove that any connected component of the contracted locus of C_{s} → C̄_{s} is a chain of semistable components. We argue by contradiction. Let X be a leaf (i.e., an end component) which is not part of a semistable chain of the contracted locus. So X contains exactly one node of C_{s} and at most one point of P. Since (C_{s} → W, Σ_{s} + P) ∈ 𝔎^{u}, and since the image of X in W maps to a point in W, it follows that f_{s}|_{X} is a non constant map to a fiber of W → W and X ∩ f_{s}^{−1}W^{sing} is discrete with cardinality ≥ 2. This contradicts Lemma 3.3.1.

Step 2: Any semistable chain lies over a singular point of C̄. ♣

Proposition 3.3.10. Let a : Δ → T^{u} be any lifting of Δ → B. If the induced point (C_{s}^{a}, Σ_{s}^{a}, f_{s}^{a}) ∈ 𝔎^{u} is in K_{nd}^{u}, then a is isomorphic to a_{0}.

Proof. Same argument as in Proposition 3.3.8(1).

♣

Proof of Theorem 3.3.4. We use the valuative criterion of properness, hence we assume that we are in the set-up provided by Notation 3.3.5. In view of Corollary 3.3.6, it is enough to show that there exists a unique lifting a of Δ to T^{u} such that the induced point (C^{a}, Σ^{a}, f^{a}) is contained in K_{nd}^{u} (since K_{pd}^{u} is closed in K_{nd}^{u}). Hence it is enough to prove that a lifting a : Δ → T^{u} induces a map to K_{pd}^{u} if and only if it is isomorphic to a_{0}, where a_{0} is defined in Part (2) of Proposition 3.3.8. The “if” part is then Proposition 3.3.9 and the “only if” part is Proposition 3.3.10. ♣

Corollary 3.3.11. Under the assumptions for this section, assume moreover that W → B is proper and has projective coarse moduli space. Then K_{pd}^{u} is proper over B.

Proof. This follows since by [AV, Theorem 1.4.1] the stack K(W) is proper. ♣ 3.4. 𝔯–twisted stable maps and their properness.

Definition 3.4.1. A twisting choice is a map 𝔯 which associates to every finite multi-set of positive integers c = {c_{1}, …, c_{n}} a positive integer 𝔯(c) such that c_{j}|𝔯(c) for all j.

Definition 3.4.2. We define a partial order on twisting choices by saying that 𝔯 ≺ 𝔯′ if 𝔯(c) divides 𝔯′(c) for every c.

Note that there is a unique minimal twisting choice, namely

𝔯_{min}(c) = lcm(c_{1}, …, c_{k}).

Similarly, given any two twisting choices we obtain a third larger than both by laking their least common multiple.

Remark 3.4.3. We find it important to allow choosing a twisting choice, for two reasons. First, a non-minimal choice is used in [ACW]). Second, keeping track of the fact that the invariants we define do not depend on the twisting choice helps make sure that we are doing things right.

Definition 3.4.4. Let 𝔯 be a twisting choice. A map f : (C, Σ) → W is called an 𝔯-twisted stable map if it is in K^{tr} and the following holds. Let f^{u} : C^{u} → W^{u} be the image of f in K_{pd}^{u}. Consider any splitting divisor D_{ℓ} of W, and let c^{ℓ} be the multiset of contact orders of the nodes in C^{u} mapping to D_{ℓ}^{u}. Then the twisting index of D_{ℓ} is 𝔯(c^{ℓ}).

Finally we arrive at the main moduli stacks of this paper:

Definition 3.4.5. If 𝔯 is a twisting choice, we define the substack K^{𝔯} of K^{tr} to be the full substack of families whose points are 𝔯-twisted stable maps. In the following sections we will use case-specific notations for K^{𝔯}:

(1) (Degeneration case) K_{Γ}^{𝔯}(W/B),

(2) (Singular case) K_{Γ}^{𝔯}(W_{0}),

(3) (Pair case) K_{Γ}^{𝔯}(X, D).

Lemma 3.4.6. The stack K^{𝔯} is open in K^{tr}.

Proof. It is enough to prove that it is stable under generalization. Assume we have a family of twisted transversal stable maps over Spec R, with R a discrete valuation ring, and that the fiber over the special point is 𝔯-twisted. Let D_{ℓ} be a splitting divisor on the general fiber; then it induces a splitting divisor on the special fiber, and since twisting indices are locally constant along deformations so long as the node (or the splitting divisor) doesn’t get smoothed out, all the contact orders and the twisting of the splitting divisor are the same at the special point and at the general point of Spec R. ♣

Theorem 3.4.7. Let 𝔯 be a twisting choice. Then the canonical morphism K^{𝔯} → K_{pd}^{u} is proper.

Proof. First we claim that the morphism is of finite type locally over K_{pd}^{u}

. Choose an open covering of K_{pd}^{u} where in each chart only finitely many contact orders apear. Then objects in K^{𝔯} involve maps to targets with bounded twisting, and the stack of these targets is of finite type over the base. Since K^{𝔯} is proper over that stack the claim follows.

It now suffices to use the valuative criterion of properness. We keep the conventions established in Notation 3.3.2, and assume we are given a commutative diagram

\xymatrix{η \ar[r] \ar[d] & K^{𝔯} \ar[d] \ar[r] & T^{tw} \ar[d] \\ Δ \ar[r] & K_{pd}^{u} \ar[r] & T^{u}}

and we want to find a unique lifting Δ → K^{𝔯}. We denote the induced families of stable maps as (C, Σ, f : C → W) and (C̃_{η}, Σ̃_{η}, f̃_{η} : C_{η} → W_{η}). The stack of twisted stable maps into W_{univ}^{tw} is proper over T^{tw}. Therefore it is enough to show that there is a unique lifting Δ → T^{tw} such that the induced family’s central fiber (C̃_{s}, Σ̃_{s}, f̃_{s}) is in K^{𝔯}.

Consider the set Δ of multisets of positive integers. We have a canonical lifting of K_{pd}^{u} → T^{u} to a morphism K_{pd}^{u} → T^{Δ} to the stack of Δ-weighted expansions 2.5, where we weigh the expansions by the contact order at each splitting divisor. Similarly we have a canonical lifting of K^{𝔯} → T^{u} to a morphism K^{𝔯} → T^{𝔯} to the stack of 𝔯-twisted Δ-weighted expansions, since the expansions in K^{𝔯} are by definition 𝔯-twisted Δ-weighted expansions. We obtain he following refinement of the previous commutative diagram:

\xymatrix{η \ar[r] \ar[d] & K^{𝔯} \ar[d] \ar[r] & T^{𝔯} \ar[d] \\ Δ \ar[r] & K_{pd}^{u} \ar[r] & T^{Δ}.}

Note that T^{Δ} = T^{1} is the stack of 1-twisted (namely untwisted) Δweighted expansions, so by Proposition 2.5.4 the morphism T^{𝔯} → T^{Δ} is proper. Therefore, after a suitable base change we obtain a unique lifting Δ → T^{𝔯}. By Lemma 3.2.6 (3), any lift of {s} → K_{pd}^{u} ×_{T^{u}} T^{𝔯} is in K^{𝔯}, in particular the limit (C̃_{s}, Σ̃_{s}, f̃_{s}) is in K^{𝔯}, as needed. ♣

Corollary 3.4.8. If W/B is proper with projective coarse moduli space, then the stack K^{𝔯} is proper for every twisting choice 𝔯.

Proof. This follows from Corollary 3.3.11. ♣

Remark 3.4.9. One could put the relative and degeneration cases together and define invariants for a degeneration with a divisor smooth over B disjoint from W^{sing}. The degeneration formula still holds, see [AF]

4. Relative and degenerate Gromov-Witten invariants

4.1. Curve classes and orbifold cohomology on nodal DM stacks. Intersection conditions for curve classes; transversal part of orbifold cohomology; issue of monodromy, etc.

4.2. Gromov–Witten invariants for nodal DM stacks. In this Section 4.2 we fix a nodal, first order smoothable proper DM stack W_{0} = X_{1} ⊔_{D} X_{1} with a projective coarse moduli scheme. We also fix data Γ = (β, g, N, e) as in 3.1.1.

We denote by K_{Γ}^{𝔯}(W_{0}) stack of transversal, 𝔯-twisted stable maps to expansions of W_{0}. This stack is proper by Theorem 3.4.7. The structure morphism to 𝔗_{0}^{tw} has a relative perfect obstruction theory, by a standard construction described in Section C.2. Since 𝔗_{0}^{tw} has pure dimension zero, it has a natural fundamental class,

and we can therefore define an induced virtual fundamental class [K_{Γ}^{𝔯}(W_{0})]^{vir}.

Definition 4.2.1. The stack K_{Γ}^{𝔯}(W_{0}) carries a universal family of twisted stable maps to expansions of W_{0} denoted as follows:

\xymatrix{Σ_{i} \ar@{^{(}->}[r] \ar[dr] & C^{tw} \ar[r]^{f^{tw}} \ar[d]^{p^{tw}} & W_{0,univ}^{tw} \\  & K_{Γ}^{𝔯}(W_{0}).}

We denote the underlying family of predeformable maps as follows:

\xymatrix{Σ_{i} \ar@{^{(}->}[r] \ar[dr] & C^{u} \ar[r]^{f^{u}} \ar[d]^{p^{u}} & W_{0,univ}^{u} \\  & K_{Γ}^{𝔯}(W_{0}).}

Composing with the morphism W_{0,univ}^{u} → W_{0} and stabilizing, we obtain a diagram

\xymatrix{Σ_{i} \ar@{^{(}->}[r] \ar[dr] & C \ar[r] \ar[d] & W_{0} \\  & K_{Γ}^{𝔯}(W_{0}).}

The markings Σ_{i} remain the same since the curves C^{tw}, C^{u} and C are isomorphic in a neighborhood of the Σ_{i}.

For i ∈ N the morphism Σ_{i} → W_{0} induces an evaluation morphism ev_{i} : K_{Γ}^{𝔯}(W_{0}) → \overline{I}_{e_{i}}(W_{0}) ⊂ \overline{I}(W_{0}), where \overline{I}(W_{0}) is the rigidified inertia stack of W_{0}. See [AGV, 4.4].

Finally, the family of coarse curves \underline{C} → K_{Γ}^{𝔯}(W_{0}) has sections s_{i} : K_{Γ}^{𝔯}(W_{0}) → \underline{C} induced by Σ_{i}. Following [AGV, 8.3] we denote

ψ_{i} = c_{1} (s_{i}^{∗}ω_{\underline{C}/K_{Γ}^{𝔯}(W_{0})}) .

Definition 4.2.2. Assume for i ∈ N we are given positive integers m_{i} and cohomology classes

γ_{i} ∈ H^{∗}(\overline{I}(W_{0})). We define the Gromov–Witten invariant

⟨∏_{i∈N} τ_{m_{i}}(γ_{i})⟩_{Γ}^{W_{0}} := deg (∏_{i∈N}(ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) ∩ [K_{Γ}^{𝔯}(W_{0})]^{vir}) .

Remark 4.2.3. Recall that in 𝔯-twisted stable maps, the image of a marked point is never in a splitting divisor. This implies that the evaluation maps only land in sectors on \overline{I}(W_{0}) transversal to D. In particular Gromov–Witten invariants involving a class γ_{i} from a sector supported in \overline{I}(D) vanish.

4.3. Relative Gromov–Witten invariants. Here we fix a proper DM smooth pair (X, D) with a projective coarse moduli scheme. We also fix data Γ = (β, g, N, M, e, f, c) as in 3.1.2.

We denote by K_{Γ}^{𝔯}(X, D) stack of transversal, 𝔯-twisted stable maps to expansions of (X, D). Again this stack is proper by Theorem 3.4.7, and the structure morphism to T^{tw} has a relative perfect obstruction theory described in Section C.2, with induced virtual fundamental class [K_{Γ}^{𝔯}(X, D)]^{vir}.

Evaluation maps ev_{i} : K_{Γ}^{𝔯}(X, D) → I(X) and classes ψ_{i} for i ∈ N are defined as in 4.2.1. Note however that for j ∈ M the markings Σ_{j}^{𝔯} ⊂ C^{𝔯} and Σ_{j} ⊂ C^{u} are not isomorphic: Σ_{j}^{𝔯} → Σ_{j}^{u} is a gerbe banded by μ_{r/c_{j}}, where r = 𝔯({c_{j}}_{j∈M}) is the twisting of the divisor.

An additional subtlety is the fact that the stack D → T^{tw} is not a product, and we are only interested in the relative part of inertia. Since the root markings Σ_{j} map to D and Σ_{j}^{𝔯} map to D we can define the following:

Definition 4.3.1. We denote by ev_{j} : K_{Γ}^{𝔯}(X, D) → \overline{I}_{f_{j}}(D) ⊂ \overline{I}(D) the evaluation map induced by Σ_{j} → D. We denote by ev_{j}^{𝔯} : K_{Γ}^{𝔯}(X, D) → \overline{I}(D/T^{tw}) the evaluation map induced by Σ_{j}^{𝔯} → D. Note that these land in \overline{I}_{f_{j}r/c_{j}}(D/T^{tw}) ⊂ \overline{I}(D/T^{tw}).

The maps ev_{j}^{𝔯} will play a role in the proof of the degeneration formula. Gromov–Witten invarians involve only ev_{j}:

Definition 4.3.2. Let m_{i}, i ∈ N be nonnegative integers; γ_{i} ∈ H^{∗}(X), i ∈ N and γ_{j} ∈ H^{∗}(\overline{I}(D)), j ∈ M. We define relative Gromov–Witten invariants with gravitational descendants by the formula

⟨∏_{i∈N} τ_{m_{i}}(γ_{i})| ∏_{j∈M} γ_{j}⟩_{Γ}^{(X,D)}

:= deg ((∏_{i∈N} ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) · ( ∏_{j∈M} ev_{j}^{∗}γ_{j}) ∩ [K_{Γ}^{𝔯}(X, D)]^{vir}) .

Remark 4.3.3. Note that unless the condition

(1)  ∑_{i∈M} d_{i} = β · D

is satisfied, the moduli stack K_{Γ}^{𝔯}(X, D) is empty and hence the invariant is zero.

4.4. Independence of twisting choice.

Theorem 4.4.1. The Gromov–Witten invariants defined above are independent of the twisting choice 𝔯.

The proof requires some preparation. Let Δ be the set of multisets c = {c_{1}, …, c_{k}} of positive integers, and recall the stack 𝔗_{0}^{𝔯} ⊂ 𝔗_{0}^{Δ} of 𝔯-twisted, Δ-weighted accordions, defined in section 2.5. There is a natural morphism K_{Γ}^{𝔯}(W_{0}) → 𝔗_{0}^{Δ} mapping each stable map to the labeling of each divisor by the multiset of contact orders of the associated predeformable map. Its image is clearly contained in the open substack 𝔗_{0}^{𝔯}. Similarly we have K_{Γ}^{𝔯}(X, D) → T^{𝔯}.

Proposition 4.4.2. Assume that 𝔯 and 𝔯′ are twisting choices with 𝔯 ≺ 𝔯′. Write for brevity K → T for K_{Γ}^{𝔯}(W_{0}) → 𝔗_{0}^{𝔯} (respectively, K_{Γ}^{𝔯}((X, D)) → T^{𝔯}) and K′ → T′ for K_{Γ}^{𝔯′}(W_{0}) → 𝔗_{0}^{𝔯′}, (respectively, K_{Γ}^{𝔯′}((X, D)) → T^{𝔯′}); similarly for the maps ev_{i}, ev′_{i} and classes ψ_{i}, ψ′_{i}.

(1) There is a natural morphism K′ → K.

(2) This morphism induces a 2–cartesian diagram

\xymatrix{K′ \ar[r]^{ϕ} \ar[d] & K \ar[d] \\ T^{𝔯′} \ar[r] & T^{𝔯},}

where the lower arrow is given by Lemma 2.5.3.

(3) There is a natural isomorphism ϕ^{∗}𝔼_{K/T^{𝔯}} → 𝔼_{K′/T^{𝔯′}}.

(4) We have ψ′_{i} = ϕ^{∗}ψ_{i}, and there is a natural equivalence between ev′_{i} and ev_{i} ∘ ϕ for i ∈ N ⊔ M.

Proof. (1) Let (C′, Σ′) → W′ → W be a family of 𝔯′-twisted maps over a base scheme S. By Lemma 2.5.3 there are morphisms T^{𝔯′} → T^{𝔯} and correspondingly W′ → W. Consider the stabilization (C, Σ) → W of the composition (C′, Σ′) → W′ → W (in the sense of [AV, Proposition 9.1.1]). It is easy to see that this defines a family of 𝔯-twisted stable maps over S, and that this construction commutes with base change; hence, it defines a morphism K′ → K.

(2) It is easy to check that the diagram is commutative. To construct a morphism from the fiber product to K′, assume that we are given a family of labeled accordions W′ → S (corresponding to a morphism S → T^{𝔯′}) and a family of twisted degenerate stable maps (C, Σ) → W over S, in such a way that W → S is induced by W′ → S as a family of labeled twisted accordions. We then define C′ := W′ ×_{W} C, and Σ′ the inverse image of Σ; again, it is easy to check the required properties.

(3) The diagram

\xymatrix{C′ \ar[r] \ar[d] & W′ \ar[d] \\ C \ar[r] & W}

is Cartesian. The isomorphism of obstruction theories now follows directly from their definition.

(4) The untwisted curve (C^{u′}, Σ′) is the pullback of (C^{u}, Σ). Both

claims easily follow.

♣

Proof of Theorem 4.4.1. By Remark 3.4.2 we may assume that 𝔯 ≺ 𝔯′. Using the notation in Proposition 4.4.2 we have that ϕ^{∗}𝔼_{K/𝔗_{0}^{𝔯}} → 𝔼_{K′/𝔗_{0}^{𝔯′}}. The morphism 𝔗_{0}^{𝔯′} → 𝔗_{0}^{𝔯} has degree 1 in the sense of [Co, Section 5] by Proposition 2.5.4. By [Co, Theorem 5.0.1] we have an equality of the associated virtual fundamental classes [K_{Γ}^{𝔯}(W)]^{vir} = [K_{Γ}^{𝔯}(W)]^{vir}. The equality of invariants follows by the projection formula. ♣

4.5. Invariance under twisting. The following is used in [ACW]:

Proposition 4.5.1. Let X = X(\sqrt[r]{D}) and D = \sqrt[r]{N/D}, and let π : X → X be the natural map. Then

⟨∏_{i∈N} τ_{m_{i}}(γ_{i})| ∏_{j∈M} γ_{j}⟩_{Γ}^{(X,D)} = ⟨∏_{i∈N} τ_{b_{i}}(π^{∗}γ_{i})| ∏_{j∈M} π^{∗}γ_{j}⟩_{Γ}^{(X ,D)}

Proof. For fixed Γ Let 𝔯 be a twisting choice giving a constant r_{0} for all contact orders appearing in K_{pd}, and 𝔯′ = r · 𝔯. Then K_{Γ}^{𝔯′}(X , D) and K_{Γ}^{𝔯}(X, D) are identical.

The result follows by the projection formula.

♣

Remark 4.5.2. A similar result holds for invariants of a singular variety.

4.6. Deformation invariance. Assume we are given a flat proper family π : W → B such that B is a smooth curve, b_{0} ∈ B is a point, π is smooth over B ∖ b_{0} and W_{0} := π^{−1}(b_{0}) is as in the previous section

Theorem 4.6.1. The Gromov–Witten invariants of W_{b} is independent of b.

Proof. This follows from [BF, Proposition 7.2 (2)], since the obstruction theory for K_{Γ}^{𝔯}(W/B) relative to 𝔗_{B,b_{0}} restricts to that of K_{Γ}^{𝔯}(W_{b}) relative to 𝔗_{0}. ♣

Remark 4.6.2. Deforamtion invairance can be stated more generally and apply also to relative invatiants. See [AF].

4.7. Comparison with Jun Li’s invariants. An immediate corollary of deformation invariance is the following:

Corollary 4.7.1. If in the situation of the theorem W is a scheme, then the Gromov–Witten invariants of W_{b} coincide with those defined in [Li2].

Proof. The invariants coincide on a smooth fiber. The theorem implies that they coincide on any fiber. ♣

The result also holds when W_{0} is not smoothable, and for relative invariants. We postpone showing these cases to [AF], as our proof is roundabout and requires some additional ground work. An outline is as follows:

4.7.2. One shows that orbifold virtual localization holds for group actions on moduli spaces induced by actions on W_{0} and (X, D), as it holds in Li’s case.

4.7.3. Applying the methods of Maulik-Pandharipande [MP], the degeneration formula and localization show that invariants of (X, D) are determined by the absolute invariants of X following precisely the same formal rules as for Li’s invariants.

4.7.4. It follows that out invariants of (X, D) coincide with those of Li.

4.7.5. Applying the degeneration formula again, it follows that out invariants of W_{0} coincide with those of Li.

An alternative and more direct approach goes by way of comparing our virtual fundamental classes to those defined via logarithmic geometry, and then comparing the logarithmic class with Jun Li’s virtual fundamental class. The last step - comparing the logarithmic class with Jun Li’s virtual fundamental class - requires a delicate explicit computation and has not yet been completed.

4.8. Disconnected maps and invariants. We fix an abelian semigroup H. In the application we have in mind, H will be the semigroup of effective curve classes.

Definition 4.8.1. An modular graph Ξ is a collection of vertices V (Ξ), edges E(Ξ) legs L(Ξ) and roots R(Ξ) with the usual relations and properties - here we divided the usual set of legs into two disjoint sets of legs and roots. These are weighted by the following data

(1) each vertex v ∈ V (Ξ) is assigned an integer g(v) ≥ 0 - the genus - and an element β(v) ∈ H -its weight.

(2) each leg l ∈ L(Ξ) is assigned an integer e(l) > 0 - its index.

(3) each root r ∈ R(Ξ) is assigned two integers f(l), c(l) > 0 - its index and contact order.

The total weight β(Ξ) of the graph is the sum of the weight of the vertices. The total genus g(Ξ) is given by the standard formula

2g(Ξ) − 2 = ∑_{v∈V (Ξ)}(2g(v) − 2) + 2#E(Ξ).

Note that we do not assume the graph to be connected or even nonempty. The letter Ξ is supposed to remind you of that.

A labeling of the legs and roots by disjoint sets M, N is the choice of a bijection M ↔ L(Ξ) and a bijection N ↔ R(Ξ).

4.8.2. Moduli stacks and their properties. The theory developed in the last two sections applies to disconnected curves as well. Consider a possibly disconected twisted curve C = ⊔_{ν=1}^{h}C_{ν} with C_{ν} its connected components of arithmetic genera g_{j}, each with marked points with labeling in disjoint subsets N_{ν} and M_{ν} forming partitions of given ordered sets N, M. We assume for simplicity that for each ν the set M_{ν} ∪ N_{ν} is nonempty. There is no loss of generality in assuming N = {1, …, n} and M = {n + 1, …, n + |M|}. We assign each C_{ν} with a target curve class β_{ν} and package the data in the notation of a modular graph Ξ = ⊔Γ_{ν} consisting of h vertices assigned genera g_{ν} and weights β_{ν}, no edges, legs indexed by N_{ν} ⊂ N with weights e_{i}, i ∈ N_{ν} corresponding to the indices of these markings, and roots indexed by M_{ν} ⊂ M with similar weights f_{j}, j ∈ M_{ν}. We further assume given contact orders {c_{j}|j ∈ N}.

An expanded 𝔯 twisted stable map of type Ξ into (X, D) is a morphism C → X of type Ξ into a half accordion, with finite automorphism group over X, which is transversal and 𝔯 twisted. We again denote by K_{Ξ}^{𝔯}(X, D) the moduli stack of expanded 𝔯 twisted stable map of type Ξ. These stacks are algebraic and proper by the same arguments as in the connected case.

4.8.3. Disconnected Gromov–Witten invariants. The stack K_{Ξ}^{𝔯}(X, D) admits a perfect obstruction theory relative to T^{tw} as before. This gives rise to a virtual fundamental class. We can use 4.3 to construct

Gromov–Witten invariants ⟨∏_{i∈N} τ_{b_{i}}(γ_{i})| ∏_{j∈M} γ_{j}⟩_{Ξ}^{(X,D)} with the exact same formula.

4.8.4. Contraction morphisms. Given an 𝔯-twisted stable map C → X of type Ξ, we have for each ν a map C_{ν} → X. This map is not necessarily stable, since some components of X require other C_{ν′} to stabilize them; it is not necessarily 𝔯-twisted since some contact orders are removed. There is a canonical way to stabilize C_{ν} → X as follows:

(1) Let X → X^{𝔯} the canonical partial untwisting associated to the contact orders appearing in C_{ν} and the twisting choice 𝔯. Let C_{ν}^{𝔯} → X^{𝔯} be the associated twisted stable map as in [AV, Corollary 9.1.2].

(2) By definition, any semistable component ℙ of X^{𝔯} has the same twisting on its two boundary divisors. It follows from [AV, Section 9.2] that there is a canonical contraction X^{𝔯} → X̄^{𝔯} of all semistable components. We again define C̄_{ν}^{𝔯} → X̄^{𝔯} as in [AV, Corollary 9.1.2].

This defines a contraction morphism K_{Ξ}^{𝔯}(X, D) → K_{Γ_{ν}}^{𝔯}(X, D). Combining these we obtain a morphism

ε : K_{Ξ}^{𝔯}(X, D) → ∏_{ν=1}^{h} K_{Γ_{ν}}^{𝔯}(X, D).

Proposition 4.8.5.

ε_{∗}[K_{Ξ}^{𝔯}(X, D)]^{vir} = ∏_{ν=1}^{h}[K_{Γ_{ν}}^{𝔯}(X, D)]^{vir}.

This immediately implies a result on Gromov–Witten invariants . Assume all classes γ_{i} have homogeneous parity and consider the

sign (−1)^{ϵ} determined formally by the equality

∏_{i∈N} γ_{i} · ∏_{j∈M} γ_{j} = (−1)^{ϵ} ∏_{ν=1}^{h} ( ∏_{i∈N_{ν}} γ_{i} · ∏_{j∈M_{ν}} γ_{j})

Corollary 4.8.6.

⟨∏_{i∈N} τ_{b_{i}}(γ_{i})| ∏_{j∈M} γ_{j}⟩_{Ξ}^{(X,D)} = (−1)^{ϵ} ∏_{ν=1}^{h} ⟨ ∏_{i∈N_{ν}} τ_{b_{i}}(γ_{i})| ∏_{j∈M_{ν}} γ_{j}⟩_{Γ_{ν}}^{(X,D)}

Proof. This follows from the projection formula. ♣

4.8.7. An auxiliary stack of expansions. We prove Proposition 4.8.5 by applying the main technical result, Theorem 5.0.1 of [Co]. This needs some preparation.

Let T′ be the stack defined as follows. T′(S) = {D × S ⊂ X →^{ε_{ν}}

X_{ν} →^{θ_{ν}} X × S} where:

(1) D × S → X_{i} → X × S is a half accordion for each ν = 1, …, h;

(2) there is a morphism ρ : X → X × S such that ρ is isomorphic to θ_{ν} ∘ ε_{ν} for each ν = 1 …, h;

(3) D × S → X →^{ρ} X × S is a half accordion;

(4) for each ν = 1, …, h the morphism ε_{ν} is a partial contraction ;

(5) for each ℙ_{i} in X there exists at least one 1 ≤ ν ≤ h such that ε_{ν}|_{ℙ_{i}} is an isomorphism with its image.

There is a natural forgetful morphism T′ → (T^{tw})^{h} given by

{D × S ⊂ X → X_{ν} → X × S} ↦ {D × S ⊂ X_{ν}→X × S}_{ν=1,...,h}.

This map has degree 1 in the sense of [Co, Section 5], since both source and target have Spec ℂ as a dense open substack.

Lemma 4.8.8. There is a natural cartesian diagram

\xymatrix{K_{Ξ} \ar[r] \ar[d] & ∏ K_{Γ_{ν}} \ar[d] \\ T′ \ar[r] & T^{h}}

Proof. That the diagram is commutative is obvious. To prove that it is cartesian, let S be a base scheme. Let {(C_{ν}, f_{ν}, D × S → X_{ν} → X × S}_{ν=1...,h} be an object of ∏ K_{Γ_{ν}}(S), and {D × S ⊂ X → X_{ν} → X × S} an object in T′(S). Define curves C′_{ν} by C′_{ν} := C_{ν}×_{X_{ν}} X; lift the marked gerbes of C′_{ν} to C̃′_{ν} in the unique possible way if they do not map to D × S and respecting the map to D × S otherwise. Let C := ⊔C′_{ν} and f : C → X be the morphism such that f|_{C′_{ν}} is induced by the fiber product. We leave it to the reader to check that this provides an inverse to the natural map from K_{Ξ} to the fiber product of T′ and ∏ K_{Γ_{ν}} over T^{h}. ♣

Let T′ → T be the forgetful morphism defined by

{D × S ⊂ X → X_{ν} → X × S} ↦ {D × S ⊂ X →X × S}.

Lemma 4.8.9. The morphism T′ → T so defined is étale.

Proof. This is identical to [ACW] . ♣

The construction in Section C give relative obstruction theories for the morphisms K_{Γ_{ν}} → T , hence for the morphism ∏ K_{Γ_{ν}} → T^{h}, and for K_{Ξ} → T′.

Lemma 4.8.10. The obstruction theory for K_{Ξ}/T′ is the pullback of the obstruction theory ∏ K_{Γ_{ν}} → T^{h}.

Proof. Write K′ for K_{Ξ}, and fix an index ν. We consider the commutative diagram

\xymatrix{C′_{ν} \ar[d]_{p} \ar[r]^{g} & X \ar[d]^{q} \\ C_{ν} \ar[d] \ar[r]^{f} & X_{ν} \\ K′}

where C_{ν} is the pullback to K′ of the universal curve over K_{Γ_{ν}} and C′_{ν} is the corresponding component of the universal curve over K′, together with the structure maps. Note that C′_{ν} → C_{ν} is a partial stabilization map, i.e. it is locally a base change of some forgetful morphism M̄_{g,n+k}^{tw} → M̄_{g,n}^{tw} . This implies that ℝπ_{∗}O_{C′} = O_{C}, and therefore that ℝπ_{∗} ∘ Lπ^{∗} : D(C) → D(C) is the identity morphism. On the other hand, the fact that the square in the diagram is cartesian shows that the pullback of the complex 𝕃 from C_{ν} is the corresponding complex for C′_{ν}. ♣

Proof of Proposition 4.8.5. This is now immediate from [Co, Theorem 5.0.1], as we have shown that the obstruction theories are compatible and the map T′ → (T^{tw})^{h} is of pure degree 1. ♣

5. Degeneration formula

5.1. Setup. We fix a variety W_{0} = X_{1} ⊔_{D} X_{2} with first-order smoothable singularity along D dividing it in two smooth pairs (X_{1}, D) and (X_{2}, D). We let H be the monoid of curve classes on W_{0}, and H_{1}, H_{2} the coresponding monoids on X_{1}, X_{2}. We view H_{1}, H_{2} as submonoids of H.

We Notice that, although H_{orb}^{∗}(W_{0}) = H^{∗}(\overline{I}(W_{0})) has a rational degree shifting, when we consider parity we always refer to the unshifted grading.

In this section we will keep fixed the notation introduced in Subsection 4.2. In particular we fix Γ = (g, N, β, e) as in Convention 3.1.1, which we may view as a modular graph with one vertex with genus g and weight β, a curve class on W_{0} and legs labelled by an ordered set N and marked with indices e_{i}. We will also fix cohomology classes

γ_{i} ∈ H^{∗}(\overline{I}(W_{0})), i ∈ N

with homogeneous parity, and nonnegative integers m_{1}, …, m_{n}. These will be used for insertions

in Gromov–Witten invariants. In this section we may take N = {1, …, n}.

Definition 5.1.1. Asplitting η of Γ is an ordered pair η = (Ξ_{1}, Ξ_{2}) where

(1) Ξ_{1} and Ξ_{2} are modular graphs as in Definition 4.8.1 with no edges or loops,

(2) The labelling of legs L(Ξ_{1}) ↔ N_{1} and L(Ξ_{2}) ↔ N_{2} form a partition N_{1} ⊔ N_{2} = N.

(3) The labelling of roots R(Ξ_{1}) ↔ M and R(Ξ_{2}) ↔ M are in the same ordered set disjoint from N, which can be safely taken as {n + 1, …, n + |M|}.

(4) A leg l ∈ L(Ξ_{1}) ∪ L(Ξ_{2}) corresponding to i ∈ N is assigned the corresponding index e_{i}.

(5) A root r ∈ R(Ξ_{1}) corresponding to j ∈ M is assigned index f_{j} and contact order c_{j}. A root r ∈ R(Ξ_{2}) is assigned the same corresponding index f_{j} and contact order c_{j}.

(6) A vertex v ∈ V (Ξ_{1}) is assigned genus g(v) and weight β(v) ∈ H_{1}; similarly the weight of v ∈ V (Ξ_{2}) is a curve class β(v) ∈ H_{2}.

We define d = (d_{j}, j ∈ M) by d_{j} = c_{j}/f_{j}. These are in general rational numbers, which we call intersection multiplicities. We denote the set of roots incident to a vertex v by R(v) and identify it with a subset of M.

These data must satisfy the following conditions: A. The graph Γ obtained by gluing Ξ_{1} and Ξ_{2} along the legs labeled

by {n+1, …, n+n_{D}} is connected of genus g and total weight^{1} β.

B. For every vertex v ∈ V (Ξ_{1}), one has

∑_{j∈R(v)} d_{j} = (β(v) · D)_{X_{1}}.

Similarly if v ∈ V (Ξ_{2}) then

∑_{j∈R(v)} d_{j} = (β(v) · D)_{X_{2}}.

Remark 5.1.2. Let β_{1} be the total weight of Ξ_{1} and β_{2} the total weight of Ξ_{2}. Then (B) implies that (β_{1} · D)_{X_{1}} = (β_{2} · D)_{X_{2}}.

Remark 5.1.3. The distinction between intersection multiplicities d_{j} and contact orders c_{j} is a feature of the orbifold situation, the ratios f_{i} = c_{i}/d_{i} being the indices of the corresponding marked points mapping to D. We see in 5.2.3 that one can avoid the need for the c_{i} in the forumla as stated in 0.4.1, but our proof requires using them.

1Of course here we are silently identifying H_{2}(X_{e}) with its image in H_{2}(X).

Definition 5.1.4. An isomorphism of splittings (Ξ_{1}, Ξ_{2}) → (Ξ′_{1}, Ξ′_{2}) is an isomorphism of modular graphs respecting the labellings, in particular the orders of N and M.

We denote by Ω(Γ) the set of isomorphism classes of splittings of Γ.

Remark 5.1.5. Passing to isomorphism classes is harmless: since by assumption the glued graph is connected, every vertex in Ξ_{1}, Ξ_{2} is incident to at least one root, and since the roots are labelled by an ordered finite set, the automorphism group of a splitting is trivial. So the groupoid of splittings is rigid and therefore equivalent to the set Ω(Γ).

Definition 5.1.6. The symmetric group S(M) acts on Ω by its action on M. Two splittings are said to be equivalent if they belong to the same S(M)-orbit. We let \overline{Ω} be the set of equivalence classes.

Definition 5.1.7. Fix a twisting choice 𝔯. For each η ∈ Ω, we define

r(η) := 𝔯(c).

Definition 5.1.8. Consider the standard pairing

\begin{matrix}H^{∗}(\overline{I}(D)) × H^{∗}(\overline{I}(D)) & → & ℚ \\ (θ_{1}, θ_{2}) & ↦ & ∫_{\overline{I}(D))} θ_{1} · θ_{2}. \end{matrix}

Let F be a basis of H^{∗}(\overline{I}(D)) of classes with homogeneous parity. For each δ ∈ F we denote by δ^{∨} be the dual element in the dual basis with respect to this pairing. In order to avoid issues of signs we define δ^{∨} to be dual to δ if ∫_{\overline{I}(D))} δ^{∨} · δ = 1 in this order - this ensures that the Poincaré dual class of the diagonal of \overline{I}(D) is ∑_{δ∈F} δ × δ^{∨}.

5.2. Statement of the formula. Here is the degeneration formula the way it naturally arises in our proof:

Theorem 5.2.1. For any choice of nonegative integers m_{1}, …, m_{n}, and cohomology classes γ_{i} ∈ H^{∗}(I(W_{0})), the following degeneration formula holds:

⟨∏_{i=1}^{n} τ_{m_{i}}(γ_{i})⟩_{β,g}^{W_{b_{0}}}

= ∑_{η∈Ω} ^{\frac{∏_{j∈M} c_{j}}{|M|!}} ∑_{δ_{i}∈F} (−1)^{ϵ} ⟨ ∏_{i∈N_{1}} τ_{m_{i}}(γ_{i})| ∏_{i∈M} δ_{i}⟩_{Γ_{1}}^{(X_{1},D)}

· ⟨ ∏_{i∈N_{2}} τ_{m_{i}}(γ_{i})| ∏_{i∈M} ι^{∗}δ_{i}^{∨}⟩_{Γ_{2}}^{(X_{2},D)} .

The sign (−1)^{ϵ} is fixed in terms of the parity of the classes so that formally the following holds:

∏_{i∈N} γ_{i} · ∏_{j∈M} δ_{j}δ_{j}^{∨} = (−1)^{ϵ} ∏_{i∈N_{1}} γ_{i} ∏_{i∈M} δ_{i} ∏_{i∈N_{2}} γ_{i} ∏_{i∈M} δ_{i}^{∨}.

Remark 5.2.2. In [Li2], one sums over the set of equivalence classes \overline{Ω} of splitting types, and therefore the factor |M|! in the denominator is replaced by |Eq(η)|, the stabilizer of η inside S_{|M|}, introduced in [Li1, p. 574], [Li2, p. 203].

5.2.3. The Chen–Ruan pairing and Theorem 0.4.1. As in [AGV], Section 6.4, or [CR], the formalism becomes a bit more elegant if one uses the Chen–Ruan pairing. Here one treats the evaluation maps ev_{j}, j ∈ M as if their target is I(D) rather than \overline{I}(D), and further includes ι in the pairing. In our situation, if we identify δ ∈ F with its pullback in H^{∗}(I(D)), the dual element with respect to the standard pairing of H^{∗}(I(D)) becomes r·δ^{∨}. Further, we can change this pairing by applying ι on the right element, namely use

\begin{matrix}H^{∗}(I(D)) × H^{∗}(I(D)) & → & ℚ \\ (θ_{1}, θ_{2}) & ↦ & ∫_{I(D))} θ_{1}ι^{∗}θ_{2}, \end{matrix}

equivalently

\begin{matrix}H^{∗}(\overline{I}(D)) × H^{∗}(\overline{I}(D)) & → & ℚ \\ (θ_{1}, θ_{2}) & ↦ & ∫_{\overline{I}(D))} \frac{1}{r}θ_{1}ι^{∗}θ_{2}, \end{matrix}

obtaining the Chen Ruan pairing. Then the dual element of δ with respect to the Chen–Ruan pairing is δ̃^{∨} = r · ι^{∗}δ^{∨}. Note again that the duality is defined so that ∫_{I(D))} ι^{∗}δ̃^{∨}δ = 1 to avoid signs in the decomposition of the class of the diagonal.

Theorem 0.4.1 follows as a version of the Theorem 5.2.1 above, in which the contact orders c_{i} are not used but the more invariant intersection multiplicities d_{i} = c_{j}/f_{j} instead. Indeed the pullback under the evaluation map ev_{j} : K_{Ξ_{i}}^{𝔯}(X_{i}, D) → \overline{I}(D) of the involution-invariant locally constant factor r is the index f_{j}. This gives

∏_{j∈M} c_{j}ev_{j}^{∗}ι^{∗}δ_{j}^{∨} = ∏_{j∈M} d_{j}ev_{j}^{∗}ι^{∗}δ̃_{j}^{∨}

as required.

5.3. Outline of proof of Theorem 5.2.1. Fix a twisting choice r and write K for K_{Γ}^{𝔯}(W, π)_{b_{0}}. The proof goes in several steps. These will be completed in the next sections, as follows: Sections 5.4, 5.5, 5.6: for η ∈ Ω we define a proper Deligne– Mumford stack K_{η} parametrizing maps to a twisted accordion with a fixed splitting divisor of type η, together with a morphisms st_{η} : K_{η} → K. We prove (Proposition 5.6.1)

[K]^{vir} = ∑_{η∈Ω} \frac{r(η)}{|M|!}st_{η∗}[K_{η}]^{vir}.

Section 5.7: fix η = (Ξ_{1}, Ξ_{2}) ∈ Ω, and let K_{Ξ_{1}} and K_{Ξ_{2}} be the moduli stacks of relative stable maps corresponding to Ξ_{1} and Ξ_{2} respectively. On K_{Ξ_{1}} × K_{Ξ_{2}} there is a canonical gerbe banded by μ_{r(η)}, which we denote u_{η} : K_{1,2} → K_{Ξ_{1}} × K_{Ξ_{2}}, which parametrizes pairs of twisted stable maps together with the data of a glued target. Sections 5.8-5.9: We construct a commutative diagram with cartesian square

\xymatrix{K_{η} \ar[r]^{q_{η}} \ar[dr] & K_{η}^{∗} \ar[r] \ar[d] & K_{1,2} \ar[d] \\  & \overline{I}(D)^{M} \ar[r]_{Δ} & (\overline{I}(D) × \overline{I}(D))^{M}}

where the morphism on the right is the product of the two evaluation maps in M, and Δ is the diagonal with the second factor composed with ι; In Proposition 5.9.1 we prove that

q_{η∗}[K_{η}]^{vir} = ( ∏_{j∈M} c_{j}) · Δ^{!}[K_{1,2}]^{vir} ∈ A_{∗}(K_{η}^{∗}).

Section 5.10: The degeneration formula follows by another application of the projection formula.

5.4. Splitting the coarse target. We form the following cartesian diagram:

\xymatrix{K_{𝔔} \ar[r]^{s} \ar[d] & K \ar[d] \\ 𝔔 \ar[r] \ar[d] & 𝔗_{0}^{tw} \ar[d] \\ 𝔗_{0}^{u,spl} \ar[r] & 𝔗_{0}^{u}}

The stack 𝔗_{0}^{u,spl} of untwisted accordions with a choice of splitting divisor is defined in Section 2.4. It is nonsingular, and coincides with the normalization of 𝔗_{0}^{u}.

The stacks 𝔔 and K_{𝔔} are formed as the fibered products making the diagram cartesian. Therefore the perfect obstruction theory of 𝔼_{K/𝔗_{0}^{tw}}^{•} pulls pack to a perfect obstruction theory 𝔼_{K_{𝔔}/𝔔}^{•} defining a virtual fundamental class which we denote [K_{𝔔}]^{vir}.

Since 𝔗_{0}^{u,spl} → 𝔗_{0}^{u} is the normalization of a reduced normal crossings stack, it has pure degree 1 in the sense of [Co, Section 5]. Since 𝔗_{0}^{tw} → 𝔗_{0}^{u} is flat, it follows that the morphism 𝔔 → 𝔗_{0}^{tw} is of pure degree 1 in the same sense as well. We have the following:

Lemma 5.4.1.

s_{∗}[K_{𝔔}]^{vir} = [K_{b_{0}}]^{vir}

Proof. This follows from [Co, Theorem 5.0.1], see also [Ma, Proposition 2, Section 4.3]. ♣

5.5. Splitting the stack target. In Section 2.4 we introduced a natural decomposition of 𝔔 = 𝔗_{0}^{u,spl} ×_{𝔗_{0}^{u}} 𝔗_{0}^{tw} into open and closed loci according to the twisting index of the twisted accordion along the chosen singular component:

𝔔 = ∐_{r≥1} 𝔔_{r}

and accordingly we have a decomposition

K_{𝔔} = ∐_{r≥1} K_{𝔔_{r}}.

The stack 𝔔_{r} is nonreduced. The reduced substack is the smooth stack 𝔗_{0}^{r,spl}, the stack of twisted accordions with splitting divisor of index r.

By Lemma 2.4.1 the morphism 𝔗_{0}^{r,spl} → 𝔔_{r} is of degree 1/r, in the sense that the image of [𝔗_{0}^{r,spl}] is r^{−1}[𝔔_{r}]. This is sufficient for applying [Co, Theorem 5.0.1] in Manolache’s version [Ma, Proposition 2, Section 4.3]. We therefore obtain the following:

Lemma 5.5.1. Consider the fiber diagram

\xymatrix{K_{r}^{spl} \ar[r]^{t_{r}} \ar[d] & K_{𝔔_{r}} \ar[d] \\ 𝔗_{0}^{r,spl} \ar[r] & 𝔔_{r}}

Then

[K_{𝔔_{r}}]^{vir} = r · (t_{r})_{∗}[K_{r}^{spl}]^{vir}.

The multiplicity r in this lemma depends on the twisting choice, since the formation of the moduli spaces does. It is important to notice that at the end it will be cancelled by that appearing in Lemma 5.7.2 below.

5.6. Decomposing the moduli space with split target. Recall that we denote by \overline{Ω} = Ω/∼ the set of equivalence classes of splitting types under the action of the symmetric group S(M), and by η̄ the equivalence class of η ∈ Ω.

Given a positive integer r, denote by Ω_{r} the set of isomorphism classes of types η satisfying r(η) = r, and by Ω̄_{r} the set of equivalence classes. We can now refine the decomposition as follows:

K_{r}^{spl} = ∐_{η̄∈Ω̄_{r}} Kspl ; η̄.

Denote by tη̄ : Kspl ; η̄ → K_{𝔔_{r}} the restricted morphism. On the level of virtual fundamental classes, Lemma 5.5.1 gives

[K_{𝔔_{r}}] = r · (tη̄)_{∗} ∑_{η̄∈Ω̄_{r}}[Kspl ; η̄].

Now denote by K_{η} → Kη̄ the cover obtained by labeling the distinguished nodes of the source curve by the set M. This is clearly an S_{|M|}-bundle, and therefore it has an associated perfect obstruction theory and virtual fundamental class. Denote by t_{η} : K_{η} → K_{𝔔_{r}} the composite map.

Putting Lemmas 5.4.1 and 5.5.1 together we obtain Proposition 5.6.1.

[K] = ∑_{η∈Ω} \frac{r(η)}{|M|!} · (s ∘ t_{η})_{∗}[K_{η}^{spl}].

5.7. Gluing the target. Recall from Definition 2.1.5 that for an integer r (not to be confused by the implicit twisting choice) we denote by T_{r}^{tw} ⊂ T^{tw} the substack of relative twisted expanded degenerations with twisting index r along D. In Lemma 2.4.2 we considered the natural morphism 𝔗_{0}^{r,spl} → T_{r}^{tw} ×T_{r}^{tw} corresponding to the two components of the partial normalization of the universal family W_{0,univ}^{tw} and showed that it is a gerbe banded by μ_{r}.

We begin approaching K_{η}^{spl} from the other direction, namely from stacks of relative stable maps to the components of W_{0}. Given η = (Ξ_{1}, Ξ_{2}) we denote r = r(η), and use the shorthand notation K_{Ξ_{1}} = K_{Ξ_{1}}^{𝔯}(X_{1}, D) and K_{Ξ_{2}} = K_{Ξ_{2}}^{𝔯}(X_{2}, D).

Definition 5.7.1. We define K_{1,2} by the following fiber diagram:

\xymatrix{K_{1,2} \ar[r]^{u_{η}} \ar[d] & K_{Ξ_{1}} × K_{Ξ_{2}} \ar[d] \\ 𝔗_{0}^{r,spl} \ar[r] & T_{r}^{tw} × T_{r}^{tw}.}

The stack K_{1,2}, which depends on η, parametrizes a glued twisted target, along with a pair of relative stable maps of types Ξ_{1} and Ξ_{2} to the two parts of the twisted target.

Composing with the projections, we have morphisms u_{η1} : K_{1,2} → K_{Ξ_{1}} and u_{η2} : K_{1,2} → K_{Ξ_{2}}

Recall (Lemmas C.2.1, C.3.3) that we have perfect obstruction theories 𝔼_{K_{Ξ_{1}}/T_{r}^{tw}} → 𝕃_{K_{Ξ_{1}}/T_{r}^{tw}} and 𝔼_{K_{Ξ_{2}}/T_{r}^{tw}} → 𝕃_{K_{Ξ_{2}}/T_{r}^{tw}}. These are defined as follows: consider the universal relative twisted stable map

\xymatrix{C_{1} \ar[d]_{p_{1}} \ar[r]^{f_{1}} & 𝔛_{1} \\ K_{Ξ_{1}}}

Denote by P_{1} ⊂ C the divisor given by the leg markings.

Consider the complex 𝕃_{□_{1}} := [f_{1}^{∗}𝕃_{𝔛_{1}/T_{r}^{tw}} → Ω_{C_{1}/K_{Ξ_{1}}}^{1}(log P_{1})]. We have a perfect relative obstruction theory on K_{Ξ_{1}}/T_{r}^{tw} given by taking the complex

𝔼_{K_{Ξ_{1}}/T_{r}^{tw}}^{•} = (Rp_{1∗}(𝕃_{□_{1}}^{∨}))^{∨}[−1]

with its natural map to 𝕃_{K_{Ξ_{1}}/T_{r}^{tw}}. The construction for 𝔼_{K_{Ξ_{2}}/T_{r}^{tw}}^{•} is identical.

Combining these, we have a perfect obstruction theory 𝔼_{K_{Ξ_{1}}/T_{r}^{tw}}^{•} ⊕

𝔼_{K_{Ξ_{2}}/T_{r}^{tw}}^{•} on K_{Ξ_{1}} ×K_{Ξ_{2}}/T_{r}^{tw}×T_{r}^{tw}. As the morphism 𝔗_{0}^{r,spl} → T_{r}^{tw}×T_{r}^{tw} is étale, so is the morphism K_{1,2} → K_{Ξ_{1}} × K_{Ξ_{2}}, and the pullback of the same complex gives a perfect obstruction theory for K_{1,2}/𝔗_{0}^{r,spl}. We denote by [K_{1,2}]^{vir} and [K_{Ξ_{1}} ×K_{Ξ_{2}}]^{vir} = [K_{Ξ_{1}}]^{vir} ×[K_{Ξ_{2}}]^{vir} the associated virtual fundamental classes. Since the degree of u_{η} : K_{1,2} → K_{Ξ_{1}} × K_{Ξ_{2}} is r^{−1} we obtain the following:

Lemma 5.7.2.

[K_{Ξ_{1}} × K_{Ξ_{2}}]^{vir} = r · (u_{η})_{∗}[K_{1,2}]^{vir}.

Notice that the multiplicity r = r(η) obtained here, which depends on the twisting choice, coincides with the multiplicity appearing in Lemma 5.5.1. In the comparison of invariants this multiplicity cancels out.

Denote by D the universal boundary divisor over K_{1,2}. It is a gerbe banded by μ_{r} over the coarse boundary divisor K_{1,2} × D.

5.8. Gluing the source. There is a natural morphism

v_{η} : K_{η} → K_{1,2}

obtained by associating to a map C → W_{0}^{tw} with splitting of type η the two maps C_{1} → X_{1} ↪ W_{0}^{tw} of type Ξ_{1} and C_{2} → X_{2} ↪ W_{0}^{tw} of type Ξ_{2} with source curves determined by the splitting. We now put this in a fiber diagram and demonstrate the compatibility of the given perfect obstruction theories.

Recall from Definition 4.3.1 that the restriction of a stable map f_{1} : C_{1} → X_{1} to Σ_{1j} gives rise to the evaluation map ev_{j}^{𝔯}_{Ξ_{1}} : K_{Ξ_{1}} → \overline{I}(D/T^{tw}).

Composing with u_{η1} ∘ v_{η} : K_{η} → K_{Ξ_{1}} denote the product morphisms

ev_{η}^{𝔯} = ∏_{j∈M} ev_{j}^{𝔯}_{Ξ_{1}} ∘ u_{η1} ∘ v_{η} : K_{η} → \overline{I}(D/T^{tw})_{T^{tw}}^{M}.

This notation means that we are taking the M-th fibered product over T^{tw}. Since this notation is cumbersome we use the shorthand

\overline{I}^{M} := \overline{I}(D/T^{tw})_{T^{tw}}^{M}.

Also denote

ev_{1,2}^{𝔯} = ∏_{j∈M}(ev_{j}^{𝔯}_{Ξ_{1}} ∘ u_{η1}) × (ev_{j}^{𝔯}_{Ξ_{2}} ∘ u_{η2}) : K_{1,2} → (\overline{I} × \overline{I})^{M}.

On the right we again use shorthand where \overline{I} stands for \overline{I}(D/T^{tw}) and all products are fibered over T^{tw}. As in [AGV, Section 5], we have a cartesian diagram

\xymatrix{K_{η} \ar[r]^{v} \ar[d]_{_{ev_{𝔯}_{η}}} & K_{1,2} \ar[d]^{_{ev{𝔯 \atop 1,2}}} \\ _{\overline{I}_{M}} \ar[r]^{Δ̃} & (\overline{I} × \overline{I})^{M}.}

Here the map Δ̃ sends \overline{I}(D/T^{tw}) to itself by the identity map on the left component, and by the map ι : \overline{I}(D/T^{tw}) → \overline{I}(D/T^{tw}) inverting the band on the right. Indeed an object of the fibered product consists of a pair of maps to the glued target along with an isomorphism of the restricted maps on the gerbe with band inverted. Since the glued curve is a pushout, such a pair of maps with isomorphism is precisely the data of a map from the glued curve, hence an object of K_{η}. This works for arrow as well.

We now have the following:

Proposition 5.8.1.

[K_{η}]^{vir} = Δ̃^{!}[K_{1,2}]^{vir}.

Proof. Recall the perfect obstruction theory 𝔼_{K_{η}/𝔗_{0,r}^{tw,spl}}^{•} → 𝕃_{K_{η}/𝔗_{0,r}^{tw,spl}}^{•}

defined in C.2.1. By [BF, Proposition 5.10] it suffices to produce a diagram of distinguished triangles

[figure: v^{∗}𝔼_{K_{1,2}/𝔗_{0,r}^{tw,spl}}^{•}; 𝔼_{K_{η}/𝔗_{0,r}^{tw,spl}}^{•}; ev_{η}^{𝔯 ∗} 𝕃Δ̃; [1]; id; v^{∗}𝕃_{K_{1,2}/𝔗_{0,r}^{tw,spl}}^{•}; 𝕃_{K_{η}/𝔗_{0,r}^{tw,spl}}^{•}; ev_{η}^{𝔯 ∗} 𝕃Δ̃; [1]; .]

Since Δ̃ is a regular embedding 𝕃Δ̃ ≃ N ∨ ; Δ̃[1].

Consider the cartesian and co-cartesian square

\xymatrix{G \ar[r]^{ι_{1}} \ar[d]_{ι_{2}} & C_{1} \ar[d]^{ν_{1}} \\ C_{1} \ar[r]^{ν_{2}} & C}

where G is the disjoint union of the marking corresponding to the roots of Ξ_{1} or Ξ_{2}. Also denote the normalization map ν : C_{1} ⊔C_{2} → C and the embedding ι : G → C. We have the standard normalization triangle

[figure: 𝕃_{□}^{∨}; ν_{∗}Lν^{∗}𝕃_{□}^{∨}; ι_{∗}Lι^{∗}𝕃_{□}^{∨}; [1]]

and a natural decomposition

ν_{∗}Lν^{∗}𝕃_{□}^{∨} = ν_{1∗}Lν_{1}^{∗}𝕃_{□}^{∨} ⊕ ν_{2∗}Lν_{2}^{∗}𝕃_{□}^{∨}.

Lemma 5.8.2.

Lν_{1}^{∗}𝕃_{□}^{∨} = 𝕃_{□_{1}}^{∨}, Lν_{2}^{∗}𝕃_{□}^{∨} = 𝕃_{□_{2}}^{∨},

and

Lι^{∗}𝕃_{□}^{∨} = (f ∘ ι)^{∗}T_{D}.

Proof of Lemma. The commutative diagram

\xymatrix{C_{1} \ar[r] \ar[d] & X_{1} \ar[d] \\ C \ar[r] & W}

induces a canonical arrow Lν_{1}^{∗}𝕃_{□} → 𝕃_{□_{1}}, and similarly for Lν_{2}^{∗}𝕃_{□} → 𝕃_{□_{2}}. We can check that this is an isomorphism locally. Away from D nothing is changed. Near D, the complex 𝕃_{□} is the conormal to C → W since f is transversal, and it restricts to 𝕃_{□_{1}}, the conormal to C_{1} → X_{1}. For the same reason the conormal to C → W restricts on G to the conormal of G → D. ♣

The triangle now looks as follows:

[figure: 𝕃_{□}^{∨}; ν_{1∗}𝕃_{□_{1}}^{∨} ⊕ ν_{2∗}𝕃_{□_{2}}^{∨}; ι_{∗}(f ∘ ι)^{∗}T_{D}; [1]]

Since D is a gerbe, the tangent sheaf T_{D} is the pullback of T_{D}, and it follows from the Tangent Bundle Lemma (see [AGV, Lemma 3.6.1]) that

p_{∗}ι_{∗}(f ∘ ι)^{∗}T_{D} = ev_{η}^{𝔯 ∗} NΔ̃.

Therefore when applying Rπ_{∗}, dualizing and rotating the above triangle we get

[figure: (Rp_{1∗}𝕃_{□_{1}}^{∨})^{∨} ⊕ (Rp_{2∗}𝕃_{□_{2}}^{∨})^{∨}; (Rp_{∗}𝕃_{□}^{∨})^{∨}; ev_{η}^{𝔯 ∗} N ∨ ; Δ̃[1]; [1]]

which is clearly compatible with the triangle of cotangent complexes, as required. (A detailed verification of such compatibility is found in [ACW, Appendix].) ♣ 5.9. Comparison with Δ. We now translate Proposition 5.8.1 into a result involving Δ : \overline{I}(D)^{M} → (\overline{I}(D) × \overline{I}(D))^{M} instead of \overline{I}^{M} = \overline{I}(D/T^{tw})_{T^{tw}}^{M} and Δ̃.

We have a cartesian diagram

\xymatrix{K_{η} \ar[r]^{q_{η}} \ar[d] & K_{η}^{∗} \ar[r] \ar[d] & K_{1,2} \ar[d] \\ \overline{I}(D)^{M} \ar[r]^{q} & ∗ \ar[r] \ar[d] & (\overline{I}(D) × \overline{I}(D))^{M} \ar[d] \\  & T^{tw} × \overline{I}(D)^{M} \ar[r]^{Id×Δ} \ar[d] & T^{tw} × (\overline{I}(D) × \overline{I}(D))^{M} \ar[d] \\  & \overline{I}(D)^{M} \ar[r]^{Δ} & (\overline{I}(D) × \overline{I}(D))^{M}.}

The arrow Δ is again the diagonal composed with ι on the right.

By Lemma 1.3.3, the component Z^{j} of \overline{I}(D/T^{tw}) where ev_{j}^{𝔯} maps is a gerbe over the corresponding component Z^{j} of T^{tw} × \overline{I}(D), and this gerbe is banded by μ_{c_{j}}. It follows that the arrows q and q_{η} are étale surjective of pure degree ∏_{j∈M} c_{j}: the arrow q is the product of the étale surjective morphisms Z^{j} → Z^{j} ×_{Z^{j}} Z^{j} = Z^{j} × Bμ_{c_{j}}. Therefore we have the following:

Proposition 5.9.1.

(q_{η})_{∗} [K_{η}]^{vir} = ( ∏_{j∈M} c_{j}) · Δ^{!}[K_{1,2}]^{vir}.

We can now use the projection formula. The composite top morphism in the last diagram is v_{η} : K_{η} → K_{1,2}. We can compose the vertical arrow on the right and obtain the “untwisted” evaluation morphism ev_{1,2} : K_{1,2} → (\overline{I}(D) × \overline{I}(D))^{M}. Denoting by [Δ] the class (Δ)_{∗}[\overline{I}(D)^{M}], we have that

v_{η∗} [K_{η}]^{vir} = ( ∏_{j∈M} c_{j}) · ev_{1,2}^{∗}[Δ] ∩ [K_{1,2}]^{vir}.

But [Δ] = ∏_{j∈M} (∑_{δ_{j}∈F} δ_{j} × ι^{∗}δ_{j}^{∨}) . We thus obtain Corollary 5.9.2.

v_{η∗} [K_{η}]^{vir} = ∏_{j∈M} (c_{j} ∑_{δ_{j}∈F} ev_{j}_{Ξ_{1}}^{∗}δ_{j} × ev_{j}_{Ξ_{2}}^{∗}ι^{∗}δ_{j}^{∨}) ∩ [K_{1,2}]^{vir};

combining with Lemma 5.7.2, with a slight abuse of notation we have

(v_{η} ∘ u_{η})_{∗} [K_{η}]^{vir}

= r(η) · ∏_{j∈M} (c_{j} ∑_{δ_{j}∈F} ev_{j}_{Ξ_{1}}^{∗}δ_{j} × ev_{j}_{Ξ_{2}}^{∗}ι^{∗}δ_{j}^{∨}) ∩ [K_{Ξ_{1}} × K_{Ξ_{2}}]^{vir}

5.10. End of proof. The stack K_{η} carries two universal families of contracted curves: a disconnected family C′ → K_{η} pulled back from C_{1} ⊔ C_{2} → K_{Ξ_{1}} × K_{Ξ_{2}} inducing evaluations ev′_{i} with coarse curve \underline{C}′ → K_{η} having sections s′_{i}; and a connected family C_{η} → K_{η} coming from C → K inducing evaluations ev_{i}, with coarse curve having sections s_{i}. These families differ only where they meet the splitting divisor. In particular the pullback of the classes ψ_{i} of the sheaves s_{i}^{∗}ω_{\underline{C}/K} coincides with that of the class ψ′_{i} corresponding to s′_{i}^{∗}ω_{\underline{C}_{2}⊔\underline{C}_{2}/K_{Ξ_{1}}×K_{Ξ_{2}}}, and similarly for the pullbacks of γ_{i} via evaluation maps. We compute:

⟨∏_{i∈N} τ_{m_{i}}(γ_{i})⟩_{Γ}^{W_{0}}

= ∑_{η∈Ω} \frac{r(η)}{|M|!} deg ((s ∘ t_{η})^{∗} (∏_{i∈N} ψ_{i}^{m_{i}} · ev_{i}^{∗}γ_{i}) ∩ [K_{η}]^{vir})

(by the projection formula and Lemma 5.6.1)

= ∑_{η} \frac{r(η)}{|M|!} deg ((u ∘ v_{η})^{∗} (∏_{i∈N} ψ′_{i}^{m_{i}} · ev′_{i}^{∗}γ_{i}) ∩ [K_{η}]^{vir})

(by the discussion above)

= ∑_{η∈Ω} \frac{r(η)}{|M|!} \frac{∏_{j∈M} c_{j}}{r(η)}

deg ((∏_{i∈N} ψ′_{i}^{m_{i}} · ev′_{i}^{∗}γ_{i})

· ∏_{j∈M} (c_{j} ∑_{δ_{j}∈F} ev_{j}_{Ξ_{1}}^{∗}δ_{j} × ev_{j}_{Ξ_{2}}^{∗}ι^{∗}δ_{j}^{∨}) ∩ [K_{Ξ_{1}}]^{vir} × [K_{Ξ_{2}}]^{vir})

(by the projection formula and Corollary 5.9.2)

= ∑_{η∈Ω} \frac{∏_{j} c_{j}}{|M|!} ∑_{δ_{j}∈F ∀j∈M}(−1)^{ϵ} ⟨ ∏_{i∈N_{1}} τ_{m_{i}}(γ_{i})| ∏_{j∈M} δ_{j}⟩_{Ξ_{1}}^{(X_{1},D)}

· ⟨ ∏_{i∈N_{2}} τ_{m_{i}}(γ_{i})| ∏_{j∈M} δ̃_{j}^{∨}⟩_{Ξ_{2}}^{(X_{2},D)}

as required.

Appendix A. Pairs and nodes

A.1. Smooth and locally smooth pairs. A smooth pair is a pair (X, D) where X is a smooth algebraic stack and D is a smooth divisor. A locally smooth pair is obtained if we only require X to be smooth near the smooth divisor D. We sometimes call X the ambient scheme/stack and D the boundary divisor.

Let A := [𝔸^{1}/𝔾_{m}] - this notation will be kept throughout the paper. A morphism f : X → A is equivalent to the data (L, s) of a line bundle L on X with a section s, as explained in [ACW]. The morphism f is dominant if and only if the section s is nonzero; in particular, every pair (X, D) with X an algebraic stack and D an effective Cartier divisor defines such a dominant morphism. The pair is smooth (respectively locally smooth) if and only if the morphism to A is smooth (respectively smooth over the divisor B𝔾_{m} = [0/𝔾_{m}]).

A morphism of locally smooth pairs ϕ : (X, D) → (X′, D′) is a morphism ϕ : X → X′ such that ϕ^{−1}(D′)_{red} ⊂ D. If D′ is empty, every morphism X → X′ defines a morphism of pairs (X, D) → (X′, ∅).

A family of locally smooth pairs over a base stack S is the datum of a flat morphism X → S and an S-flat closed substack D ⊂ X such that for every point s ∈ S the fiber (X_{s}, D_{s}) is a locally smooth pair.

Given two families (X, D) and (X′, D′) of locally smooth pairs over the same base S, a log morphism is a morphism ϕ : X → X′ such that

(1) for every s ∈ S, ϕ_{s} : (X_{s}, D_{s}) → (X′_{s}, D′_{s}) is a morphism of locally smooth pairs;

(2) the morphism ϕ^{∗}Ω_{X′} → Ω_{X} induces a morphism ϕ^{∗}(Ω_{X′}(log D′)) → Ω_{X}(log D).

Note that:

(a) The first condition implies the second if S is reduced, but not in general;

(b) Assume that S is connected, and that D has connected components D_{i} such that D_{i} ∩ D_{s} is also connected for every i and every s ∈ S (this is true e.g. if S is simply connected). Then ϕ : X → X′ is a log morphism if and only if there exist nonegative integers c_{i} such that ϕ^{∗}D′ = ∑ c_{i}D_{i}.

Locally smooth pairs their morphisms are classical special cases of logarithmic structures in the sense of [Ka].

A.2. Transversality for nodal singularities and pairs. A morphism of locally smooth pairs (C, Σ) → (X, D) is transversal to the boundary divisor (or just transversal) if the scheme theoretic inverse image of D is smooth (and hence a union of connected components of Σ).

Let X be a complex algebraic stack; we say that it has nodal codimension one singularities - or just nodal singularities for brevity - if it is locally isomorphic in the f.p.p.f. topology to {xy = 0} × 𝔸^{n}; in particular its singular locus D is smooth. Let ν : X̃ → X be the normalization and D̃ = ν^{−1}(D). Then (X̃, D̃) is a smooth pair, and D̃ → D is an étale double cover.

Definition A.2.1. A morphism between nodal algebraic stacks f : C → X is called transversal to the singular locus if

(1) the induced morphism C̃ → X̃ defines a morphism of locally smooth pairs which is transversal to the boundary divisor;

(2) for every point p ∈ f^{−1}(D) its two inverse images in C̃ map to different points of D̃ via f̃.

This means that we have smooth charts C̃ → C and X̃ → X, lifting C̃ → X̃ of C → X and smooth morphisms C̃ → {xy = 0} and X̃ → {xy = 0} making the following diagram commutative

\xymatrix{C̃ \ar[rr] \ar[dr] &  & X̃ \ar[dl] \\  & \{xy = 0\},}

so on the charts C̃ and X̃ the coordinates x, y with xy = 0 are the same.

Suppose now (C, Σ) → S and (X, D) → S are flat families of locally smooth pairs with at most nodal singularities and f : C → X a map. The following is evident:

Lemma A.2.2. The locus S^{tr} ⊂ S where the fibers are transversal is open

A.3. First-order smoothability of nodal singularities. If X is a stack with codimension-1 singular locus D, we say that X is first-order smoothable if the line bundle Ext^{1}(Ω_{X}, O_{X}) on D is trivial. If X is the union of two smooth components X_{1} and X_{2} meeting transversally along D, then it is first-order smoothable if and only if N_{D/X_{1}} ⊗ N_{D/X_{2}} is isomorphic to O_{D}. Note that if there is a one-parameter smoothing of X with smooth total space then X is first-order smoothable, while the converse is in general not true.

Appendix B. Stack constructions

B.1. Using 2-stacks to define stacks. Occasionally we define a 2– groupoid 𝔛 by giving objects, 1-morphisms and 2-morphisms, and then we show that every 1-morphism in 𝔛 is rigid (i.e., it has only the identity as 2-automorphism); equivalently, for any two objects X and Y of 𝔛, the groupoid Mor(X, Y) is equivalent to a set. In this case we say that the 2-groupoid 𝔛 is 1-rigid. We can then consider the associated groupoid 𝔛^{[1]}, where objects are unchanged, and morphisms are isomorphism classes of 1-morphisms of the given 2-groupoid 𝔛. Since 𝔛 is 1-rigid, it is equivalent to 𝔛^{[1]} (in the appropriate lax sense). We might as well replace 𝔛, which may arise naturally but is likely to intimidate us with its dæmonic 2-arrows, by the more friendly, yet equivalent, groupoid 𝔛^{[1]}.

We will use this particularly in the definition of algebraic stacks. In particular if X is a stack then stacks with a representable morphism to X form a 1-rigid 2-groupoid, see Lemma 3.3.3 in [AGV]; also, the 2groupoid of stacks with a dense open algebraic space and isomorphisms as 1-morphisms is also 1-rigid, see Lemma 4.2.3 on page 42 of [AV]. Both cases are generalized using the following lemma.

Lemma B.1.1. Let X be a stack with separated diagonal, and U a scheme-theoretic dense open substack. Let β : id_{X} → id_{X} be a 2– morphism such that β|_{U} is the identity of id_{U}. Then β is the identity 2–morphism.

Proof. The fact that the diagonal is separated implies that the natural projection π : I(X) → X is separated. The automorphisms of id_{X} are the sections of I(X) → X. Since we assumed that this section is the identity on a scheme-theoretically dense substack, it coincides with the identity on X. ♣

Let p : X → Y be a morphism of stacks, and assume that there is a scheme-theoretic dense open substack U of X such that p|_{U} : U → p(U) is an isomorphism. Let g : Y → Y be an isomorphism. A lifting of g to X is a pair (f, α) such that f : X → X is an isomorphism and α : p∘f → g ∘p a 2-morphism. A morphism of liftings is a 2-morphism γ : f → f′ such that γ and α′ induce

α.

Lemma B.1.2. The groupoid of liftings of g is rigid, i.e., equivalent to a set.

Proof. Let (f, α) and (f′, α′) be two liftings of g to X. We want to show that a 2-morphism γ : f → f′ such that γ and α′ induce α is unique if it exists. Let γ and γ′ be two such two-morphisms, and let β := γ^{−1} ∘ γ′ : f → f. Then β|_{U} : f_{U} → f_{U} is the identity 2–morphism. Let β̄ : id_{X} → id_{X} be the composition of β with the identity of f^{−1}; then β̄|U is also the identity 2-morphism of id_{U}. Therefore β̄ must be the identity 2–morphism by Lemma B.1.1, and hence β must be the identity 2–morphism, hence γ = γ′. ♣

Convention B.1.3. In this case, we will refer to the liftings of g as a set, meaning the set of equivalence classes of the corresponding groupoid.

B.2. Constructing moduli over a base stack. Numerous constructions of stacks have been given over base schemes. For instance, if S is a scheme, and X → S and Y → S are algebraic stacks satisfying suitable assumptions, then Olsson constructed an algebraic stack Hom_{S}(X , Y), see [Ol06]; also if X → S is an algebraic stack satisfying suitable assumptions, then in [AV] the stack K_{g,n}(X , d) is constructed. However we will need these constructions when the base S is an algebraic stack and not a scheme. The fact that this can be done is well known to experts and appears time and again in the literature, where the authors did not find it necessary to justify. What is completely evident in these situations is that we have a stack (though sometimes an argument as in Subsection B.1 above is needed). The fact that the stack is algebraic follows from the following lemma.

Lemma B.2.1. Consider a morphism X → Y of stacks. Assume Y is algebraic, and assume there is a scheme U and a faithfully flat U → Y such that the stack X_{U} = U ×_{Y} X is algebraic. Then X is algebraic.

Proof. Step 1: representability of the diagonal. By assumption Y → Y × Y is representable, therefore X ×_{Y} X → X × X is representable. It remains to show that X → X ×_{Y} X is representable. By assumption X_{U} → X_{U} ×_{U} X_{U} is representable. Take a scheme S and a morphism S → X ×_{Y} X. We need to show that X′ = S ×_{X×_{Y} X} X is representable. Consider S_{U} = S ×_{Y} U, which is representable since Y is algebraic. Then X′_{U} = S_{U} ×_{X_{U} ×_{U} X_{U}} X_{U} is representable. The fiber diagram

\xymatrix{X′_{U} \ar[r] \ar[d] & X′ \ar[d] \\ S_{U} \ar[r] & S}

shows that X′ → S is representable, hence X′ is representable, as needed.

Step 2: Existence of a smooth cover. We may assume U → Y is smooth surjective. Let V be a scheme and V → X_{U} be smooth surjective. Then the composite V → X is smooth surjective, as required. ♣

B.3. Inertia stacks of various flavors.

B.3.1. The inertia stack. Let X be an algebraic stack. Its inertia stack I(X) is the stack whose objects over a scheme S are pairs (x, g) with x ∈ X(S) and g ∈ Aut(x). Arrows are given by pullback diagrams.

The inertia stack can be identified as I(X) = X ×_{X×X} X, with both arrows given by the diagonal. Since the diagonal is representable, the morphism I(X) → X given by the first projection is representable. This is simply the forgetful morphisms which sends an object (x, g) to x.

Let Bℤ be the classifying prestack of ℤ. Then we have a canonical isomorphism of prestacks I(X) ≃ Hom(Bℤ, X). This in particular implies that forming the inertia stack is compatible with fiber products: given a fiber product of algebraic stacks X = X_{1}×_{Z} X_{2} we have I(X) = I(X_{1}) ×_{I(Z)} I(X_{2}) (an observation due to Tom Bridgeland).

B.3.2. Inertia of Deligne–Mumford stacks. Suppose now X is a Deligne– Mumford stack, and let r be a positive integer such that the exponent of any automorphism group in X divides r. In this case we have I(X) = Hom(B(ℤ/rℤ), X). The stack I(X) has an evident decomposition I(X) = ⊔_{d|r}I_{d}(X), where I_{d}(X) is the stack of (x, g) with g of order d. Then I_{d}(X) = Hom^{rep}(B(ℤ/dℤ), X), the substack of representable morphisms, see [AGV]

.

B.3.3. Rigidified inertia. The automorphism group of an object (x, g) of I_{d}(X) has the subgroup ℤ/dℤ ≃ ⟨g⟩ sitting in its center. We can therefore rigidify by removing this subgroup and obtain the rigidified stack \overline{I}_{d}(X) = I_{d}(X) (ℤ/dℤ). It is canonically isomorphic to the stack whose objects over S are G → X, where G is a gerbe banded by ℤ/dℤ and G → X is representable. The rigidified inertia stack is \overline{I}(X) = ⊔_{d|r}\overline{I}_{d}(X). The morphism I(X) → \overline{I}(X) is the universal gerbe, with universal representable morphism I(X) → X. We stress that the data of the band is important - without it we would get a different tack, a rigidification of the stack of cyclic subgroups (without choice of generator) of inertia.

B.3.4. Cyclotomic inertia and rigidified inertia. In the theory of twisted stable maps, a cyclotomic twist of these stacks arises naturally. Since in this paper we work over ℂ, it is safe to choose the generator exp(2π i/d) of μ_{d}, so the distinction is not crucial. Let us mention the appropriate identification of stacks: we have I(X) ≃ Hom(Bμ_{r}, X), the cyclotomic inertia stack; I_{d}(X) ≃ Hom^{rep}(Bμ_{d}, X); and \overline{I}_{d}(X) ≃ I_{d}(X) (μ_{d}) is canonically isomorphic to the stack whose objects over S are G → X, where G is a gerbe banded by μ_{d} and G → X is representable. The stack \overline{I}(X) = ⊔_{d|r}\overline{I}_{d}(X) is then identified as the cyclotomic inertia stack, see [AGV]

B.4. Deformations and obstructions for Artin stacks. A key technical tool for deformation theory is the cotangent complex of a morphism: we refer the reader to [Ol07b] for the correct definition of cotangent complex 𝕃_{f} for a morphism f : X → Y of Artin stacks and for the relevant results in deformation theory, see also [Ao]. Note that in [Ol07b] Olsson’s cotangent complex 𝕃_{f} is actually not defined as an object in the derived category: its right truncations τ_{≥n}𝕃_{f} are for n ∈ ℤ, and 𝕃_{f} is defined as an object of a filtered category. This issue is removed in [LO, 2.2.ix], specifically the equivalence at the end of page 119 between the appropriate derived categories of quasi-coherent sheaves on the stack and on a symplicial resolution.

In particular, to any morphism of Artin stacks f : X → Y we can after all associate its cotangent complex 𝕃_{f} ∈ D_{coh}^{≤1}(X). This is functorial, in the sense that for any composable morphisms of Artin stacks f : X → Y and g : Y → Z, there is a distinguished triangle in D^{−}(X):

f^{∗}𝕃_{g} → 𝕃_{g∘f} → 𝕃_{f} →^{+1} .

The morphism f is étale if and only if 𝕃_{f} = 0; it is smooth if and only if its cotangent complex 𝕃_{f} is perfect of perfect amplitude contained in [0, 1].

Recall that f is said to be of Deligne–Mumford type if for any morphism S → Y with S a Deligne–Mumford stack, the stack X ×_{Y} S is also Deligne–Mumford. Then f is Deligne–Mumford type if and only if h^{1}(𝕃_{f}) is the zero sheaf, or equivalently if 𝕃_{f} ∈ D^{≤0}(X).

For any stack X we write 𝕃_{X} for 𝕃_{X→Spec k}; the complex 𝕃_{X} is perfect of perfect amplitude in [−1, 0] if and only if X is a Deligne–Mumford stack with l.c.i. singularities.

If f : X → Y is a morphism of Deligne–Mumford type, an obstruction theory for f is a morphism ϕ : 𝔼 → 𝕃_{f} in D^{≤0}(X) such that h^{0}(ϕ) is an isomorphism, and h^{−1}(ϕ) is surjective. We say that it is a perfect obstruction theory if 𝔼 is a perfect complex, of perfect amplitude contained in [−1, 0] (i.e., locally isomorphic to a morphism E^{−1} → E^{0} of locally free sheaves).

We define the cotangent complex of a locally smooth pair (X, D) to be 𝕃_{X/A} (where X → A is the morphism associated to the pair, see §A.1); we sometimes denote it by 𝕃_{X(log D)}. Note that if (X, D) is a smooth pair with X a scheme, or, more generally, a smooth DM stack, then 𝕃_{X(log D)} is concentrated in degree zero and isomorphic to the classically defined locally free sheaf Ω_{X}(log D). It is easy to see that a morphism of locally smooth pairs induces a morphism of log cotangent complexes, which has the usual deformation-theoretic properties (see [Ol05]).

Appendix C. Stacks of maps and their obstruction theory

C.1. Stacks of maps. We define a relative obstruction theory on certain algebraic stacks parametrizing stable maps. This includes the obstruction theories needed in the singular and in the relative case, see Section C.2; in fact, a common generalization is possible.

Convention C.1.1. In this section, we will fix an algebraic stack T, and a family of locally smooth pairs (W, D) → T such that W → T is of Deligne-Mumford type (note that the case D = ∅ is possible, in which case we are just assuming W → T to be flat). Fix nonnegative integers g, n and a curve class β in the fibers of W → T. Fix n-tuples e of positive integers e_{i} and c of nonnegative integers c_{i} such that ∑ c_{i} · e_{i}^{−1} = β · D. In particular if D = ∅, we must have c_{i} = 0. We combine the data under the shorthand notation Γ = (g, n, e, c, β)

Definition C.1.2. Let 𝔎̃_{Γ}((W, D)/T) be the stack

of representable maps f from a twisted prestable n-pointed curve (C, Σ) to fibers of (W, D) → T such that f^{∗}D = ∑ c_{i}Σ_{i} and such that Σ_{i} is twisted with index e_{i}. If D is empty, we write 𝔎̃_{Γ}(W/T).

Remark C.1.3. The condition on f^{∗}D can be rephrased as saying that we consider the stack of log morphisms; see section A.1 for details.

Convention C.1.4. We will write just 𝔎̃ for 𝔎̃_{Γ}((W, D)/T) within this section.

Lemma C.1.5. The stack 𝔎̃ is an algebraic stack in the sense of Artin.

Proof. We first do the case D = ∅. Consider the stack of twisted curves 𝔐 := 𝔐_{g,n}^{tw} and its universal family ℭ → 𝔐. Over 𝔐 × T we have two families, ℭ′ := ℭ × T → 𝔐 × T and W′ := 𝔐 × W → 𝔐 × T. We first prove that Hom_{𝔐×T}^{rep}(ℭ′, W′) is algebraic. By [Ol06], the pullback of 𝔎̃ to a scheme by any arrow S → 𝔐 × T is an algebraic stack in the sense of Artin. Also 𝔎̃ is a stack, since this property is tested over a base scheme. By Lemma B.2.1, the stack 𝔎̃ is an algebraic stack, as required.

If D is nonempty, let 𝔎̃′ be the stack obtained by assuming D empty. The stack 𝔎̃ is obtained by first passing to the open substack where f^{∗}D is a divisor on C, and then to the closed substack where the two divisors ∑ c_{i}Σ_{i} and f^{∗}D coincide. ♣

Notation C.1.6. Let us now denote by 𝔎 ⊂ 𝔎̃ the maximal open substack of 𝔎̃ such that the morphism 𝔎 → T is of Deligne–Mumford type. We will reserve the notation K for the substack which is Deligne– Mumford in the absolute sense.

C.1.7. Base change. The construction of 𝔎̃ and 𝔎 behaves well under base change in the following sense. Assume that (W, D) → T satisfies the assumption in Convention C.1.1. Let a_{T} : T′ → T be any morphism, and write W′ := W ×_{T} T′ and D′ := D ×_{T} T′. Let β′ be the homology class in the fibers of W′ → T′ induced by β, let Γ′ e obtained by replacing β by β′ in Γ, and a_{W} : W′ → W the natural morphism. Then (W′, D′) → T′ satisfies the same assumptions, and there is a natural cartesian diagram

\xymatrix{𝔎̃_{Γ′}((W′, D′)/T′) \ar[r]^{a𝔎̃} \ar[d] & 𝔎̃_{Γ}((W, D)/T) \ar[d] \\ T′ \ar[r]_{a_{T}} & T}

where a𝔎̃ is given by mapping an object (C, Σ, f′) to ((C, Σ, a_{W} ∘ f). Since the property of being of DM type is stable under base change, one gets an analogous cartesian diagram by replacing 𝔎̃ by K.

C.1.8. Change of family. Assume that we are given a proper morphism θ : W_{1} → W and a closed substack D_{1} ⊂ W_{1} such that

(1) the composite morphism (W_{1}, D_{1}) → T satisfies the assumption in Convention C.1.1;

(2) one has θ^{−1}(D) = D_{1} as a closed subscheme, and θ|_{D_{1}} : D_{1} → D is an isomorphism.

Let β_{1} be a class in the fibers of W_{1} → T, and β = θ_{∗}β_{1}, and Γ_{1}, Γ the corresponding discrete data. If β_{1} = 0, assume moreover that 2g − 2 + n > 0. Then there is a natural induced proper morphism of T –stacks

𝔎̃_{Γ_{1}}((W_{1}, D_{1})/T) → 𝔎̃_{Γ}((W, D)/T)

defined by (C, Σ, f) ↦ (C, Σ, θ ∘ f)^{stab}. This follows by applying [AV], Corollary 9.1.3, where we replace the base scheme 𝕊 by T using Lemma B.2.1.

C.2. Obstruction theory on stacks of maps. For simplicity we now restrict to the open substack 𝔎^{⋔} ⊂ 𝔎 parametrizing maps which are transversal to the boundary divisor in the sense of A.2. One can avoid this simplifying assumption using logarithmic structures, but we will not need this generality in this paper.

The aim of this section is to define a relative obstruction theory for 𝔎^{⋔} → T, and to give conditions so that it is perfect in [−1, 0]. The construction works for 𝔎̃^{⋔} instead of 𝔎^{⋔} if we allow obstruction theories for morphisms which are not of Deligne–Mumford type using the work [No], requiring 𝔼_{𝔎̃^{⋔}/T} → 𝕃_{𝔎̃^{⋔}/T} to also be an isomorphism in degree Consider the structure commutative diagram

\xymatrix{C \ar[drr]^{f} \ar[ddr]_{p} \ar[dr] \\  & W_{𝔎} \ar[d] \ar[r]_{u} & W \ar[d] \\  & 𝔎 \ar[r] & T}

where C → 𝔎^{⋔} is the universal curve, f is the universal map and Σ := ∪Σ_{i} is the union of the marked gerbes. Also denote Σ′ = Σ ∖ f^{−1}D. Since we are assuming the maps are transversal to D, deforming f : (C, Σ) → (W, D) is equivalent to deforming f : (C, Σ′) → W, which is in turn equivalent to deforming the diagonal map f̄: (C, Σ′) → W_{𝔎^{tw}}.

Let 𝕃_{□} be the cotangent complex to the morphism f̄ : (C, Σ′) → W_{𝔎^{⋔}}): it is canonically isomorphic in the derived category to the cone of the canonical morphism of cotangent complexes

f^{∗}𝕃_{W/T} ⟶ 𝕃_{C(log Σ′)/𝔎}

induced by the structure morphisms u^{∗}𝕃_{W/T} → 𝕃_{W_{𝔎^{⋔}} /𝔎^{⋔}} - which is an isomorphism since W → T is flat - and f̄^{∗}𝕃_{W_{𝔎^{⋔}} /𝔎^{⋔}} → 𝕃_{C(log Σ′)/𝔎^{⋔}}. See [Ol07b, Theorem 8.1].

By the same argument, the object 𝕃_{□} is also isomorphic to the cone of the morphism

p^{∗}𝕃_{𝔎^{⋔}/T} ⟶ 𝕃_{C(log Σ′)/W}

and therefore there is a natural morphism

𝕃_{□}[−1] ⟶ p^{∗}𝕃_{𝔎/T}.

The morphism p is proper. It is also l.c.i., therefore Gorenstein, and its dualizing complex ω_{p} is a line bundle positioned in degree −1. By [LN]

the functor Rp_{∗} : D(C) → D(𝔎) has a right adjoint p^{!} which is isomorphic to the functor 𝔽 ↦ Lp^{∗}(𝔽) ⊗ ω_{p}.

We denote by 𝔼_{𝔎/T} the object Rp_{∗}(𝕃_{□} ⊗ ω_{p}); there is a natural morphism 𝔼_{𝔎/T} → 𝕃_{𝔎/T} induced by adjunction from the morphism 𝕃_{□}[−1] → p^{∗}𝕃_{𝔎/T} defined above. Note that we have a canonical isomorphism 𝔼_{𝔎/T} ≃ (Rp_{∗}𝕃_{□}^{∨})^{∨}[−1].

Lemma C.2.1. (1) The morphism 𝔼_{𝔎/T} → 𝕃_{𝔎/T} is an obstruction theory.

(2) Its formation commutes with base change on T in the sense of Remark C.1.7.

Proof. We first give a proof in the case where Σ = ∅. (1) We use the criterion in [BF], Lemma 4.5, see also p. 85 there for the relative case - a detailed proof is available in [ACW]

. The result than follows from [Il], Theorem

III 2.1.7.

(2) Given a morphism ϕ : 𝔎′ → 𝔎 with C′ → 𝔎′ the pullback of C → 𝔎, we have a canonical isomorphism Lϕ^{∗}𝔼_{𝔎/T} ≃ 𝔼_{𝔎′/T}, such that the composite morphism Lϕ^{∗}𝔼_{𝔎/T} → Lϕ^{∗}𝕃_{𝔎/T} → 𝕃_{𝔎′/T} coincides with the composition Lϕ^{∗}𝔼_{𝔎/T} → 𝔼_{𝔎′/T} → 𝕃_{𝔎′/T}. In particular, given a morphism ψ : T′ → T we can pull back the entire diagram. Denote by ϕ : 𝔎′ → 𝔎 the pullback via ϕ. Again we have an isomorphism Lϕ^{∗}𝔼_{𝔎/T} ≃ 𝔼_{𝔎′/T} ≃ 𝔼_{𝔎′/T′}, and the compatibility above lifts to 𝕃_{𝔎′/T′}.

For the general case, we remark that the above proof remains valid by replacing the cotangent complex of C by the logarithmic cotangent complexes. Treatment of this can be found in [La, Ra89]. This also follows from [Il, III, SS2.3 and §4] by using the cotangent complex of the topos Σ′C, or by using the morphism C → A associated to Σ′. ♣

C.3. Perfect amplitude.

Definition C.3.1. Let t be a geometric point of T, W_{t} the fiber of W over t. We say that a prestable map f : (C, Σ_{i}) → W_{t} (i.e., a point in 𝔎) is nondegenerate if

no irreducible component of C maps to the singular locus of W_{t}. An irreducible component which does map to the singular locus of W_{t} is callled degenerate.

Remark C.3.2. The points corresponding to nondegenerate maps form an open substack 𝔎_{nd} of 𝔎, which commutes with base change in the sense of Remark C.1.7.

Lemma C.3.3. Assume that the morphism W → T is l.c.i. (1) The obstruction theory 𝔼_{𝔎^{⋔}/T} is perfect in [−2, 0];

(2) It is perfect in [−1, 0] over the open substack 𝔎_{nd}^{⋔}.

Proof. (1) Since W → T and C → 𝔎 are l.c.i., both 𝕃_{W/T} and 𝕃_{C(log Σ′)/𝔎} are perfect in [−1, 0]. Therefore 𝕃_{□} is perfect in [−2, 0]; hence 𝕃_{□} ⊗ ω_{π}[−1] is also perfect in [−2, 0], and since p is proper and flat of relative dimension 1, one has that 𝔼_{𝔎^{⋔}/T} = Rp_{∗}(𝕃_{□} ⊗ ω_{p}[−1]) is perfect in [−2, 1]. Since 𝔼_{𝔎^{⋔}/T} is an obstruction theory, it has vanishing h^{1}, hence it is perfect in [−2, 0].

(2) It is enough to prove that, for each point x ∈ 𝔎_{nd}^{⋔}, h^{−2}(x^{∗}𝔼_{𝔎_{nd}^{⋔}/T}) = 0. Assume the point x corresponds to a prestable map f : (C, Σ) → W.

We want to show that Ext^{2}(𝕃_{□}|_{C}, O_{C}) = 0; by the local-to-global spectral sequence of Ext, we reduce to showing that H^{1}(C, h^{1}((𝕃_{□}|_{C})^{∨})) = 0. Remark that 𝕃_{□}|_{C} is the mapping cone of the morphism f^{∗}𝕃_{W/T} → 𝕃_{C(log Σ)}. Note that the support of h^{1}((𝕃_{□}|_{C})^{∨}) is contained in the locus of points in C which map to the singular locus of W → T (i.e., the support of h^{1}((f^{∗}𝕃_{W/T})^{∨})) which by assumption is zero-dimensional. ♣

Remark C.3.4. In fact, both in this subsection and in the following one we could replace the moduli stack of twisted prestable curves with any other moduli stack of d-dimensional proper Deligne–Mumford locally smooth pairs; Lemma C.2.1 would still hold, and Lemma C.3.3 would hold with [−2, 0] (respectively [−1, 0]) replaced by [−(d + 1), 0] (respectively [−d, 0]).

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Appendix D. Things to do

(1) Add details on curve classes and cohomology classes in invariant section 4. This includes some data on cohomology of types of inertia and the way we identify; the part of cohomology of a nodal stack that is relevant; etc.

(2) Transform degeneration formula (and final steps in proof) to a result in terms of push forward of fundamental classes.

(3) Explicitly explain the reason for the appearance of contact order or intersection multiplicities in the degeneration formula. Maybe this goes in a more expository article?

(4) Rejuvenate notation table.

(5) Pictures - maybe in an expository article?

Department of Mathematics, Box 1917, Brown University, Providence, RI, 02912, U.S.A

E-mail address: abrmovic@math.brown.edu

SISSA, Via Beirut 4, 34014 Trieste, Italy

E-mail address: fantechi@sissa.it