j. differential geometry 57 (2001) 509-578

STABLE MORPHISMS TO SINGULAR SCHEMES AND RELATIVE STABLE MORPHISMS

JUN LI

Abstract

Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebro-geometric analogue of Donaldson-Floer theory in gauge theory. The construction of relative Gromov-Witten invariants and the degeneration formula of Gromov-Witten invariants will be treated in the subsequent paper.

0. Introduction

The goal of this paper is to develop a theory of relative Gromov-Witten invariants and prove a degeneration formula relating the Gromov-Witten invariants of smooth varieties with the relative Gromov-Witten invariants of pairs.

Let C be a connected smooth curve, 0 ∈ C be a fixed closed point and let W → C be a family of projective schemes so that the fibers W_{t} of W over t ≠ 0 ∈ C are smooth and the central fiber W_{0} is the union of two smooth varieties Y_{1} and Y_{2} intersecting transversally along a smooth irreducible divisor. We denote by D_{i} ⊂ Y_{i} the divisor Y_{1} ∩ Y_{2} ⊂ Y_{i}. We will call (Y_{i}, D_{i}) the decomposition pairs of W_{0}. As for t ≠ 0, since W_{t} are members of a connected family of smooth varieties, the Gromov-Witten invariants of W_{t} are all equivalent. The question is how to relate

Received May 7, 2001. The author was supported partially by NSF grants.

the Gromov-Witten invariants of W_{t} with those of the pairs (Y_{i}, D_{i}). In this project, we will construct the relative Gromov-Witten invariants of the pairs (Y_{i}, D_{i}). We then show that the Gromov-Witten invariants of W_{t} can be recovered from the relative Gromov-Witten invariants of the pairs (Y_{1}, D_{1}) and (Y_{2}, D_{2}), via a formula of the form

GW (W_{t}) = GW (Y_{1}, D_{1}) ∗ GW (Y_{2}, D_{2}).

Here we use GW (W_{t}) to denote the full GW-invariants of W_{t} and use GW (Y_{i}, D_{i}) to denote the full relative GW-invariants of (Y_{i}, D_{i}). The operator ∗ is an involution type product linear in both arguments.

Gromov-Witten invariants are essentially a virtual enumeration of algebraic curves in algebraic varieties (or pseudo-holomorphic curves in symplectic manifolds). It is a topological theory. Since their introduction, the Gromov-Witten invariants have become an invaluable tool in studying many mathematical problems, not to mention their significance to mathematical physics. A challenging problem of GWinvariants is the understanding of GW-invariants of singular varieties and the behavior of GW-invariants under the degeneration of target varieties. This line of research has been pursued by several groups. In [39], Tian first studied the Gromov-Witten invariants of symplectic sums for semi-positive symplectic manifolds and derived the decomposition formula of the Gromov-Witten invariants in this setting. Later, A. Li-Ruan [28, 37] worked out a version of degeneration formula of Gromov-Witten invariants in the general setting, along the line of Donaldson-Floer theory. A parallel theory was developed by Ionel-Parker around the same time [17, 18, 19]. A part of the SFT theory of Eliashberg-Givental-Hofer [5] can also be interpreted as research along this line. As is now well understood, one approach to degeneration of moduli spaces (in differential geometry) is to follow Floer’s original idea in his cohomology theory. This is the case for the Donaldson-Floer theory, and is the case for Gromov-Witten invariants as demonstrated by the works mentioned.

However, the algebro-geometric analogue of this approach to degeneration of moduli spaces has eluded algebraic geometers up until now. This in part explains why the Donaldson-Floer theory was never developed fully in algebraic geometry. This project fills in this gap. We have found the algebro-geometric approach to Floer’s original idea in treating degenerate objects in moduli spaces. Though this project is mainly about moduli of stable morphisms and Gromov-Witten invariants, the technique developed can be applied to many other moduli problems, including the moduli of stable sheaves.

Out approach is geometric. We first construct a degeneration of the moduli spaces of stable morphisms, associated to the family W/C. Recall that to define the Gromov-Witten invariants of a projective scheme Z, one needs to apply the machinery of virtual moduli cycles, first developed by G. Tian and the author [24, 25]. (An alternative construction was achieved by K. Behrend and B. Fantechi [1, 2]. For related work using analytic techniques, see [7, 26, 35, 38].) It turns out that the moduli of stable morphisms to W, which can be viewed as a degeneration, is not suitable for studying Gromov-Witten invariants. Thus the first task is to construct a new degeneration in line with the construction of virtual moduli cycles. This is achieved in the present paper. In this paper, we first construct a stack 𝔚 of expanded degenerations of W/C. We then investigate the moduli of stable morphisms to this stack 𝔚 of expanded degenerations. Similarly, to each pair (Y, D) of a smooth divisor in a smooth variety, we construct the stack of expanded relative pairs 𝔜_{rel} and the moduli space of relative stable morphisms to the stack of expanded relative pairs 𝔜_{rel}. The main results of this paper are the following existence theorems.

Theorem 0.1. The moduli functor 𝔐(𝔚, Γ) of stable morphisms to 𝔚 of topological type Γ is a separated and proper Deligne-Mumford stack over C.

Theorem 0.2. The moduli functor 𝔐(ℨ^{rel}, Γ) of relative stable morphisms to ℨ^{rel} of topological type Γ is a separated and proper Deligne-Mumford stack.

In the subsequent paper [23], we will develop the obstruction theory of the moduli stacks 𝔐(𝔚, Γ) and 𝔐(ℨ^{rel}, Γ), we will show that they admit perfect-obstruction theories and thus have canonical virtual moduli cycles. We will construct the relative Gromov-Witten invariants of the pair (Y, D) and prove the degeneration formula of the Gromov-Witten invariants mentioned at the beginning of the introduction. We will address the application of this machinery in our future research.

As indicated and/or shown by the works of [16, 17, 29, 36], the degeneration machinery developed in this project will be useful in enumerating curves in varieties, including algebraic surfaces and Calabi-Yau manifolds. Some of these problems are crucial to the research in mathematical physics. The moduli of relative stable morphisms and the relative GW-invariants will help answering several outstanding conjectures raised by mathematical physicists [14]. The techniques developed in this project can be applied to the study of other moduli problems, including stable sheaves over algebraic varieties. Finally, we point out that based on this project the localization technique can be applied to the moduli of relative stable morphisms, which should be useful as suggested by [14]. We mention that the moduli space of relative stable morphisms and relative GW-invariants were constructed for a restricted class of varieties for genus 0 curves [9].

We now briefly describe the main idea of this paper. We let W → C be a projective family of schemes with general smooth fibers W_{t} for t ≠ 0 ∈ C and singular fiber W_{0} with normal crossing singularity, in the situation we mentioned at the beginning of the introduction. Let g, n be two integers and let b be a homology class. We consider the moduli space (stack) 𝔐_{g,k}(W_{t}, b) of stable morphisms from k-pointed arithmetic genus g curves to W_{t} of degree b. The union

𝔐_{g,k}(W_{C^{∘}}, b) = ∪_{t∈C^{∘}}𝔐_{g,k}(W_{t}, b), C^{∘} = C − 0,

is a proper family over C^{∘}. Inserting the central fiber 𝔐_{g,k}(W_{0}, b) gives one extension (which we will call a degeneration of moduli spaces). However, the natural obstruction theory of this new family is no longer perfect near degenerate stable morphisms. Here we say a stable morphism f : X → W_{0} is degenerate if some irreducible components of X are mapped entirely to the singular locus of W_{0}. Usually, it is impossible to avoid degenerate stable morphisms if the extension family is proper. This makes this choice of degeneration inappropriate to study Gromov-Witten invariants. In this paper, we will work out a new degeneration of 𝔐_{g,k}(W_{C^{∘}}, b) that will allow us to apply the machinery of virtual moduli cycles.

As a warming up to our construction, let us look at a simple case where degenerate stable morphisms arise. Let π : S → C be a flat morphism from a smooth curve to C with π^{−1}(0) consists of a single point s. We let f:X → W be a flat family of stable morphisms over S with f a C-morphism so that f_{s}: X_{s} → W_{0} is degenerate. We now let W^{∘} = W ×_{C} C^{∘}, let S^{∘} = S − s, let X^{∘} = X ×_{C^{∘}} S^{∘} and let f^{∘} = f|_{X^{∘}} be the restriction of f to over X^{∘}. Here we view W as a degeneration (extension) of the family W^{∘} to C and view f as an extension of f^{∘}. The fact that the family f is degenerate means that the degeneration W is not a suitable choice to extend f^{∘}. There are other degenerations of W^{∘}. For instance, we let (˜0, C̃) → (0, C) be a base change ramified at ˜0 ∈ C̃ with ramification index r. We let W̃ be the desingularization of the fiber product W ×_{C} C̃. Note that the central fiber of W̃ over ˜0 has (r + 1) irreducible components: Y_{1}, Y_{2} and (r − 1) copies of the ruled variety P_{D_{1}}(1_{D_{1}}⊕N_{D_{1}/Y_{1}}) over the divisor D_{1}. We now consider the new family f̃^{∘}:X̃^{∘} → W̃ over S̃^{∘} = S ×_{C} C̃, where X̃^{∘} = X^{∘} ×_{C} C̃. We can extend it to a family of stable morphisms f̃:X̃ → W̃ over S×_{C} C̃. It is conceivable that for some choice of r this new extension f̃ will be nondegenerate. It is less obvious, but will be proven, that there is a minimal choice of r so that f̃ is nondegenerate. Thus to avoid degenerate stable morphisms we will replace the original family W by some expanded degenerations of W^{∘}. The exact choice of the expanded degeneration, after imposing the minimality condition, depends uniquely on the individual family.

Using expanded degenerations to study degenerate stable morphisms is certainly not new. However, to obtain a moduli space that contains stable morphisms to various expanded degenerations, namely to 𝔚[r]_{0} for various r, we need to give the space of all expanded degenerations an algebraic structure. The new ingredient of this paper is that we can naturally give such space the structure of an (Artin) stack. This is the stack 𝔚 mentioned before the statement of Theorem 0.1. The moduli space of stable morphisms to all expanded degenerations of W is then defined to be the moduli space of stable morphisms to this Artin stack 𝔚. With appropriately defined stability, this moduli space is a proper Deligne-Mumford stack.

It turns out that this is not sufficient to construct the desired degeneration of moduli spaces. We need to build the notion of predeformable morphisms into the definition of the stable morphisms to 𝔚. Let f : X → W_{0} be a nondegenerate stable morphism so that there is a smooth point p ∈ X that is mapped to the singular locus D of W_{0}. Then a simple argument shows that there is no small deformation of f that moves it away from the singular fiber W_{0}. In other words, there is a local obstruction to deforming such morphisms away from the singular fiber W_{0}. As we are interested in the degeneration of the moduli spaces, we should exclude such morphisms from our total moduli space. To accommodate this we require all stable morphisms to 𝔚 to have vanishing local obstructions to deforming to smooth fibers of W/C. Those morphisms that have vanishing local obstructions will be called predeformable morphisms.

In the end, we define stable morphisms to 𝔚 of given topological type^{1} Γ to be those that are nondegenerate, are minimal among all possible target schemes W [n]_{0} and are predeformable. The main theorem of this paper is that the moduli of such stable morphisms form a separate and proper Deligne-Mumford stack over C, as stated in Theorem 0.1.

As I mentioned earlier in the introduction, this construction can be viewed as an algebro-geometric adaptation of Floer’s pioneering work [6]. I should point out that in early 1980’s, Gieseker [10] (see also Gieseker-Morrison [13]) constructed a degeneration of the moduli spaces of stable vector bundles over a family of smooth curves degenerating to a nodal curve. The moduli space of vector bundles over the singular curve D consists of the stable vector bundles over D and the stable vector bundles over D̃ that are the result of replacing the node of D by a rational curve. Bundles over D̃ are exactly the substitute of those stable non-locally free sheaves over D. The current work can be viewed as a realization of Gieseker’s construction to the general cases. I should also mention that the notion of admissible cover introduced by Harris-Mumford [15] in early 80’s utilized similar idea to deal with singular objects. A more recent work related to this construction is Caporaso’s construction of the universal Picard scheme over \overline{M}_{g} [3].

This technique can be applied to construct the relative stable morphisms of pairs. Let (D, Z) be a pair consisting of a smooth divisor D in a smooth projective variety Z. We consider the moduli of stable morphisms f:X → Z. We call a stable morphism f:X → Z nondegenerate (relative to the divisor D) if f^{−1}(D) is a proper divisor in X and is away from the nodal and the marked points of X. As before, the issue is to construct a complete moduli space that contains no degenerate stable morphisms relative the to D. As for W, we should consider stable morphisms to expanded relative pairs. An expanded relative pair of (Z, D) is a reducible scheme that is the union of Z with several copies of the ruled variety P_{D}(1_{D} ⊕ N_{D/Z}). To define the moduli of relative stable morphisms we first construct the (Artin) stack ℨ^{rel} of all expanded relative pairs. We then introduce the notion of relative stable morphisms to ℨ^{rel}, similar to the case of stable morphisms to 𝔚. After that we can define the moduli functor of relative stable morphisms to ℨ^{rel} of topological type Γ. In the end we prove the existence Theorem 0.2.

The paper is organized as follows. In Section 1 we will construct the stack 𝔚 of all expanded degenerations of W. The notion of predeformable morphisms and related properties will be worked out in Section 2. In Section 3 we will define the notion of stable morphisms to 𝔚 and prove that the moduli functor of stable morphisms to 𝔚 is a separated and proper algebraic stack over C. Section 4 is devoted to study the moduli of relative stable morphisms 𝔐(ℨ^{rel}, Γ). The relation of stable morphisms to 𝔚_{0} = 𝔚 ×_{C} 0 and relative stable morphisms to pairs (Y_{i}, D) will be investigated there as well.

1Γ contains all the topological restrictions on the stable morphisms, including the genus, the number of marked points and the degree, etc.

0.1 Conventions

We provide a selected list of conventions used throughout this paper.

W [n]. It is the expanded degeneration (of W/C) constructed in Section 2.

C[n]. C[n] = C×_{A^{1}}A^{n+1}. It has two tautological morphisms C[n] → A^{1} and C[n] → A^{n+1}. It is defined in Section 2.

G[n]. It is ≅ G_{m}×⋯×G_{m}, n copies. For notation related to this group see (1.2) and (1.3).

(Y_{i}, D_{i}). They are the pairs (of irreducible components with singular locus) derived from the decomposition of W_{0}.

X_{1} ⊔ X_{2}. This is the gluing of two schemes X_{1} and X_{2}. Let A_{1} ⊂ X_{1} and A_{2} ⊂ X_{2} be two pairs of closed subschemes in schemes. Assume A_{1} is isomorphic to A_{2}. Then we can define a new scheme, denoted by X_{1} ⊔ X_{2}, that is the result of gluing X_{1} and X_{2} along A_{1} and A_{2}. X_{1} ⊔ X_{2} is the scheme that satisfies the universal property of the push-out.

C[n]_{[l^{∨}]}. H_{l}^{n+1} is the coordinate hyperplane given by the vanishing of the l-th coordinate axis of A^{n+1}. We will call it the l-th coordinate hyperplane. C[n]_{[l^{∨}]} = C[n] ×_{A^{n+1}} H_{l}^{n+1}.

t. t = (t_{1}, …, t_{n+1}) and hence k[t] = k[t_{1}, …, t_{n+1}].

[n]. This is the set of all integers between 1 and n.

1. The stack of expanded degeneration

We first fix the notation that will be used throughout this paper. In this paper, we will work with an algebraically closed field k with characteristic 0. We let C be a smooth irreducible curve, 0 ∈ C a closed point and π : W → C a flat and projective family of schemes satisfying the following condition: The morphism π is smooth away from the central fiber W_{0} = W ×_{C} 0 and the central fiber W_{0} is reducible with normal crossing singularity and has two smooth irreducible components Y_{1} and Y_{2} intersecting along a smooth divisor D ⊂ W_{0}. When we view D as a divisor in Y_{i}, we will denote it by D_{i} ⊂ Y_{i}.

We now construct the class of expanded degenerations of W mentioned in the introduction of this paper. We let Δ be the projective bundle over D:

Δ = P(1_{D} ⊕ N_{D_{2}/Y_{2}}),

where 1_{D} is the trivial holomorphic line bundle on D and N_{D_{2}/Y_{2}} is the normal bundle of D_{2} in Y_{2}, viewed as a line bundle on D. Δ has two distinguished divisors

D_{−} = P(1_{D} ⊕ 0) and D_{+} = P(0 ⊕ N_{D_{2}/Y_{2}}).

For convenience, we call D_{−} the left distinguished and D_{+} the right distinguished divisor. Of course both D_{−} and D_{+} are canonically isomorphic to D. Note that

N_{D_{−}/Δ} ≅ N_{D_{2}/Y_{2}} and N_{D_{+}/Δ} ≅ N_{D_{1}/Y_{1}} ≅ N_{D_{2}/Y_{2}}^{−1}.

Using this identification, we can glue^{2} an ordered chain of n Δ’s by identifying the right distinguished divisor (i.e., D_{+}) in the k-th Δ with the left distinguished divisor (i.e., D_{−}) of the (k + 1)-th Δ, for k = 1, …, n − 1. We denote the resulting scheme by Δ[n]. It is connected, has normal crossing singularity and has n irreducible components all isomorphic to Δ. We keep the ordering of these n Δ’s. We then glue Y_{1} to Δ[n] by identifying D_{1} in Y_{1} with the left distinguished divisor D_{−} in the first Δ of Δ[n], and then glue Y_{2} to this scheme by identifying the right distinguished divisor D_{+} of the last Δ in Δ[n] with D_{2} in Y_{2}. We denote the resulting scheme by W [n]_{0}. Note that W [n]_{0} has (n + 2) irreducible components. These (n+2) components form a chain ordered from left to right according to their intersection pattern:

W [n]_{0} = Y_{1} ⊔ Δ ⊔ ⋯ ⊔ Δ ⊔ Y_{2}.

2See the convention in the Introduction.

Often, we will denote these n + 2 components by Δ_{1}, …, Δ_{n+2}, according to this ordering. As a result, W [n]_{0} has (n + 1) nodal divisors, ordered so that the k-th nodal divisor is D_{k} ≜ Δ_{k} ∩ Δ_{k+1}. Note that D_{k} ⊂ Δ_{k} (resp. D_{k} ⊂ Δ_{k+1}) is the right (resp. left) distinguished divisor. We agree W [0]_{0} is W_{0}, the fiber of W over 0 ∈ C.

The purpose of this paper is to construct a stack 𝔚 representing all degenerations W̃ → C̃, where ρ: C̃ → C are base changes, so that the fiber of W̃ over t is W_{ρ(t)} in case ρ(t) ≠ 0 and is one of W [n]_{0} when ρ(t) = 0.

1.1 Construction of the standard model

We begin with the construction of the standard models W [n]. By replacing C with an open neighborhood of 0 ∈ C we can assume that there is an étale morphism C → A^{1} so that 0 ∈ C is mapped to 0_{A^{1}} ∈ A^{1} and 0 ∈ C is the only point in C lies over 0_{A^{1}} ∈ A^{1}. We fix such a map C → A^{1} once and for all. We let G_{m} be the general linear group GL(1) and let G[n] be the group scheme

G[n] = G_{m} × ⋯ × G_{m} n copies (1.1)

which acts on A^{n+1} via

t^{σ} = (σ_{1}t_{1}, σ_{1}^{−1}σ_{2}t_{2}, …, σ_{n−1}^{−1}σ_{n}t_{n}, σ_{n}^{−1}t_{n+1}), (1.2)

where t = (t_{1}, …, t_{n+1}) is the standard coordinate of A^{n+1}. (Here and in the following we will use superscripts to denote the result of the group action.) The convention throughout this paper is that for σ ∈ G[n] we will denote by σ_{i} its i-th component via isomorphism (1.1). For convenience, we introduce

σ_{i} = σ_{i}/σ_{i−1} with σ_{0} = σ_{n+1} = 1. (1.3)

In this way, (1.2) can be rewritten as t^{σ} = (σ_{1}t_{1}, …, σ_{n+1}t_{n+1}). Note that if we let

p : A^{n+1} → A^{1}, p(t) = t_{1} × ⋯ × t_{n+1},

then p is G[n]-equivariant with the trivial G[n]-action on A^{1}. We define

C[n] = C ×_{A^{1}} A^{n+1}

with the (unique) distinguished point 0 = 0 ×_{A^{1}} 0_{A^{n+1}} ∈ C[n]. We view C[n] as a C-scheme via the first projection. C[n] is a G[n]-scheme as well.

The standard model W [n] will be constructed as a desingularization of

W ×_{A^{1}} A^{n+1} = W ×_{C} C[n].

We begin with some more notation. For any integer n, we denote by [n] the set of integers between 1 and n. Let I ⊂ [n + 1] be any subset with cardinality |I| = m + 1. The subset I defines a unique increasing map [m + 1] → [n + 1] whose image is I. By abuse of notation, we will denote this map by I. Hence I(k) is the k-th element in I. Given any such I there is a standard embedding A^{m+1} → A^{n+1} that sends z ∈ A^{m+1} to w ∈ A^{n+1} via w_{I(k)} = z_{k} for k ∈ I and w_{l} = 1 for l ∉ I. It follows that it induces an immersion

\underline{γ_{I}} : C[m] → C[n]. (1.4)

We call this the standard embedding associated to I ⊂ [n + 1]. There is another embedding A^{m+1} → A^{n+1} via the rule w_{I(k)} = z_{k} for k ∈ I and w_{l} = 0 for l ∉ I. This induces an immersion A^{m+1} → C[n] whose image lies in a coordinate plane. We call this the coordinate plane associated to I and denoted it by C[n]_{I}. Lastly, when l is an integer in [n + 1], we denote by l the set of one element with l → [n + 1] the inclusion. Hence C[n]_{l} is the l-th coordinate axis of C[n]. Similarly, we let [l^{∨}] be the complement of l in [n]. Hence C[n]_{[l^{∨}]} is the coordinate hyperplane spanned by all coordinate axes except the l-th axis.

Now we construct W [n] by induction on n. For n = 0, W [0] = W. For n = 1, W_{1} = W ×_{C} C[1] is smooth except along the locus D ×_{C} 0, which is a smooth codimension 3 subscheme in W_{1} and the formal completions (the germs) of its normal slices in W_{1} is isomorphic to the formal completion of

X = {z_{1}z_{2} = t_{1}t_{2}} ⊂ A^{4} (1.5)

along its origin. Here we use (z_{1}, z_{2}, t_{1}, t_{2}) to denote the coordinate of A^{4}. After blowing up W_{1} along D ×_{C} 0 we obtain a smooth scheme W̃_{1} whose exceptional divisor is a P^{1} × P^{1} bundle over D. In the following, we will show how to contract one of these two P^{1}-factors from this divisor to obtain the desired scheme W [1].

We first work out the a desingularization of (1.5).

Lemma 1.1. There is a desingularization Z of (1.5) so that its exceptional locus Σ is a P^{1} and the proper transforms of the z_{1} and the t_{2}-axis (resp. the z_{2} and the t_{1}-axis) in Z intersect in Σ.

Proof. We first blow up the singular point (the origin) of the threefold X in (1.5). The resulting threefold, denoted by X̃, is isomorphic to the total space of the restriction of the tautological line bundle on P^{3} to the surface {w_{1}w_{2} = w_{3}w_{4}} ⊂ P^{3}. The exceptional divisor B ⊂ X̃ is isomorphic to the zero section of this line bundle, which is isomorphic to P^{1} × P^{1}. An isomorphism

P^{1} × P^{1} ⟶ B = {w_{1}w_{2} = w_{3}w_{4}} ⊂ P^{3}

is given by

([a_{0}, a_{1}], [b_{0}, b_{1}]) ↦ [a_{0}b_{1}, a_{1}b_{0}, a_{0}b_{0}, a_{1}b_{1}]. (1.6)

The normal bundle N_{B/X̃} is isomorphic to the restriction of the tautological line bundle to B, which is O_{P^{1}×P^{1}}(−1, −1), under the above isomorphism. Hence we can contract either of the P^{1}-factors in B to obtain a smooth threefold. In the following we will show that we can choose the contraction so that the proper transforms of the t_{1} and the z_{2}-axes intersect and the proper transforms of the t_{2} and the z_{1}-axes intersect.

Let p̃: X̃ → X be the obvious projection. Then under the isomorphism (1.6) the map p̃ is of the form

p̃:([a_{0}, a_{1}], [b_{0}, b_{1}], ζ) ↦ (z_{1}, z_{2}, t_{1}, t_{2}) = (a_{0}b_{1}ζ, a_{1}b_{0}ζ, a_{0}b_{0}ζ, a_{1}b_{1}ζ),

where ζ is the variable representing points on the total space of the tautological line bundle. Note that ζ has homogeneous degree (−1, −1), as a_{i} and b_{i} have homogeneous degree (1, 0) and (0, 1) respectively.^{3} Hence the proper transforms of the z_{1}, z_{2}, t_{1} and the t_{2}-axes intersect B at

([1, 0], [1, 0]), ([0, 1], [0, 1]), ([1, 0], [0, 1]) and ([0, 1], [1, 0]) ∈ P^{1} × P^{1},

respectively. Therefore if we contract the first P^{1}-factor the resulting threefold Z satisfies the required property. q.e.d.

For later application, we now work out an atlas of Z. By out construction, Z is the total space of the vector bundle associated to the locally free sheaf O_{P^{1}}(−1)^{⊕2}. Since Z is derived from X̃ by contracting the first P^{1} factor, we will use [b_{0}, b_{1}] as the homogeneous coordinate of the exceptional locus of p:Z → X. In this way, p:Z → X is of the form

3By this we mean ([a_{0}, a_{1}], [b_{0}, b_{1}], ζ) ∼ ([λa_{0}, λa_{1}], [μb_{0}, μb_{1}], λ^{−1}μ^{−1}ζ).

p : ([b_{0}, b_{1}], η_{1}, η_{2}) ↦ (b_{1}η_{1}, b_{0}η_{2}, b_{0}η_{1}, b_{1}η_{2}) (1.7)

and the contraction X̃ → Z is given by

([a_{0}, a_{1}], [b_{0}, b_{1}], ζ) ↦ ([b_{0}, b_{1}], a_{0}ζ, a_{1}ζ). (1.8)

Hence the projection Z → A^{2} is given by

φ : ([b_{0}, b_{1}], η_{1}, η_{2}) ↦ (t_{1}, t_{2}) = (b_{0}η_{1}, b_{1}η_{2}). (1.9)

From this we see that Z ×_{A^{2}} 0_{A^{2}} is a nodal curve obtained by adjoining an A^{1} (with η_{1} variable) to [0, 1] in P^{1} and an A^{1} (with η_{2} variable) to [1, 0] in P^{1}. Further if we let A_{1} (resp. A_{2}; resp. S) be the t_{1}-axis of A^{2} (resp. the t_{2}-axis; resp. a curve in A^{2} passing though the origin but not entirely contained in the two coordinate lines) then Z ×_{A^{2}} A_{1} (resp. Z ×_{A^{2}} A_{2}; resp. A ×_{A^{2}} S) is a smoothing of the nodal point [0, 1] (resp. the nodal point [1, 0]; resp. both nodal points) of Z ×_{A^{2}} 0_{A^{2}}. We also mention that near ([0, 1], 0, 0) (resp. ([1, 0], 0, 0)) the projection Z → A^{2} has the form (b_{0}, η_{1}, η_{2}) ↦ (b_{0}η_{1}, η_{2}) (resp. (b_{1}, η_{1}, η_{2}) ↦ (η_{1}, b_{1}η_{2})).

We continue our construction of W [1]. After blowing up the subscheme D×_{C} 0 in W ×_{C} C[1] we obtain the scheme W̃_{1} whose exceptional divisor is isomorphic to a P^{1} × P^{1}-bundle over D. By Lemma 1.1, we can contract either factor of this P^{1} × P^{1}-bundle to obtain a smooth scheme. We now specify the choice of the P^{1}-factor to be contracted. Let W [1] be the resulting scheme after contracting one such P^{1}-factor. Then W [1] has an induced projection π: W [1] → C[1] and the fiber of π over the origin 0 ∈ C[1], denoted by W [1]_{0}, is reduced with normal crossing singularity whose irreducible components are Y_{1}, Y_{2} and a ruled variety Δ̃ over D. The singular locus of W [1]_{0} consists of two disjoint divisors Y_{1}∩Δ̃ and Y_{2}∩Δ̃, both isomorphic to D. Following the proof of Lemma 1.1 and the description afterwards, we can contract one of the P^{1} factors so that the family W [1] ×_{C[1]} C[1]_{1} (resp. W [1] ×_{C[1]} C[1]_{2}) over C is a smoothing of the nodal divisor Y_{1}∩Δ̃ (resp. the nodal divisor Y_{2} ∩ Δ̃) of W [1] ×_{A^{2}} 0_{A^{2}}. Here C[1]_{i} ⊂ C[1] is the i-th coordinate line defined in the beginning of this subsection. It is direct to check that Δ̃ is the ruled variety P_{D}(1_{D} ⊕ N_{D_{1}/Y_{1}}) mentioned at the beginning of this section.

We let π_{2} : W [1] → A^{1} be the composite of W [1] → C[1] → A^{2} with the second projection A^{2} → A^{1}. It is easy to see that the singular locus of π_{2} is the proper transform of D × A^{1} ⊂ W ×_{A^{1}} A^{2} where D ×A^{1} → W ×_{A^{1}} A^{2} is the morphism induced by D → W and A^{1} → A^{2} as the t_{1}-axis. We denote the singular locus of π_{2} by D_{2}. Further, at each p ∈ D_{2}, there is a formal coordinate chart^{4} (z_{1}, …) of p so that π^{∗}:Γ(A^{2}) → Ô_{W [1],p} satisfies t_{1} ↦ z_{1} and t_{2} ↦ z_{2}z_{3}.

We now construct W [2]. We let W_{2} be the fiber product W [1]×_{A^{2}} A^{3} with A^{3} → A^{2} the morphism (t_{1}, t_{2}, t_{3}) ↦ (t_{1}, t_{2}t_{3}). Then W_{2} has singularity along D_{2}×_{A^{2}}A_{[23^{∨}]}^{3} ⊂ W [1]×_{A^{2}}A^{3}, where following our convention A_{[23^{∨}]}^{3} is the codimension two coordinate plane (line) spanned by all but the second and the third axes in A^{3}. Note that the singularity type of W_{2} along D_{2} ×_{A^{2}} A_{[23^{∨}]}^{3} is identical to the one of W_{2} along D ×_{A^{1}} 0_{A^{1}}. We now blow up W_{2} along its singular locus and then contract one copy of the P^{1}-bundle in the resulting exceptional divisor. We will contract the P^{1}-bundle so that the resulting scheme, denoted by W [2] → C[2], has the property that its fibers over the l-th axis C[2]_{l} ⊂ C[2] is a smoothing of the l-th nodal divisor of W [2]_{0} ≜ W [2] ×_{C[2]} 0. The proof of this is a word by word repetition of our earlier construction of W [1] from W ×_{A^{1}} A^{2}.

The model W [n]/C[n] is constructed by induction. Assume W [n − 1]/C[n − 1] was constructed, we then form W_{n} = W [n − 1] ×_{A^{n}} A^{n+1}, where A^{n+1} → A^{n} is defined by (…, t_{n+2}) ↦ (…, t_{n}, t_{n+1}t_{n+2}). We then resolve the singularity of W_{n} by a small resolution. We require that the resulting family W [n]/C[n] be so that W [n] ×_{C[n]} 0 ≅ W [n]_{0} and the fibers of W [n] over the l-th factor C[n]_{l} ⊂ C[n] is a smoothing of the l-th nodal divisor of W [n]_{0}. Again the details to work this out are exactly the same as in the construction of W [1].

Throughout this paper, we will use W [n]_{0} to denote the fiber of W [n] over 0 ∈ C[n] and denote by Δ_{1}, …, Δ_{n+2} the n + 2 irreducible components of W [n]_{0}, ordered as specified at the beginning of this section.

1.2 Construction of the stack of expanded degenerations

In this part, we will construct the stack 𝔚 of the expanded degenerations of W.

We begin with an investigation of the G[n]-group action on W [n].

4Namely they generate the maximal ideal of the formal ring Ô_{W [1],p}.

Clearly, since A^{n+1} → A^{1} is G[n]-equivariant, its action lifts to C[n] → C. To show that it lifts to W [n], we need to understand the corresponding G_{m}-action on the desingularization Z of X, constructed in Subsection 1.1. Following the notation there, G_{m} acts on X via

(z_{1}, z_{2}, t_{1}, t_{2})^{σ} = (z_{1}, z_{2}, σt_{1}, σ^{−1}t_{2}), σ ∈ G_{m}

and its lifting to Z is

([b_{0}, 1], η_{1}, η_{2})^{σ} = ([σb_{0}, 1], η_{1}, σ^{−1}η_{2}). (1.10)

Based on this, it is easy see that the G[n]-action on C[n] lifts uniquely to W [n].

For later application, we now give a precise description of the G[n]- action near the nodal divisor of W [n]_{0}. This can best be done by looking at the simplest case where W = A^{2}, C = A^{1} and W → C is given by t = u_{1}u_{2}, where (u_{1}, u_{2}) and t are the standard coordinates of A^{2} and A^{1}. With this choice of W → C, we can construct the corresponding W [n] → C[n]. To distinguish this from the general case, we will denote this special degeneration by Γ[n] → A^{n+1}.

Lemma 1.2. Let the notation be as before. Then Γ[n]_{0} is a chain of n + 2 curves of which the first and the last are isomorphic to A^{1} and the remainders are isomorphic to P^{1}. Further, we can cover Γ[n] by G[n]-invariant affine open subsets U_{1}, …, U_{n+1}, each isomorphic to A^{n+2}, so that if we denote by (u_{1}^{l}, …, u_{n+2}^{l}) the standard coordinates of U_{l} ≅ A^{n+2}, then:

(i) The restriction of π:Γ[n] → A^{n+1} to U_{l} is given by

π^{l} : (u_{1}^{l}, …, u_{n+2}^{l}) ↦ (…, u_{l−2}^{l}, u_{l−1}^{l}, u_{l}^{l}u_{l+1}^{l}, u_{l+2}^{l}, u_{l+3}^{l}, …).

(ii) The G[n]-action on U_{l} is given by

(u_{1}^{l}, …, u_{n+2}^{l})^{σ}

= (…, σ_{l−2}u_{l−2}^{l}, σ_{l−1}u_{l−1}^{l}, σ_{l−1}^{−1}u_{l}^{l}, σ_{l}u_{l+1}^{l}, σ_{l+1}u_{l+2}^{l}, σ_{l+2}u_{l+3}^{l}, …),

where σ_{i} = σ_{i}/σ_{i−1} with σ_{i} = 1 for i = 0, n + 1.

(iii) The transition function from (u_{·}^{l}) to (u_{·}^{l+1}) over U_{l} ∩ U_{l+1} is given by

(u_{1}^{l+1}, …, u_{n+2}^{l+1})

= (…, u_{l−2}^{l}, u_{l−1}^{l}, u_{l}^{l}u_{l+1}^{l}, 1/u_{l+1}^{l}, u_{l+2}^{l}u_{l+1}^{l}, u_{l+3}^{l}, …).

Proof. We prove this Lemma by induction on n. For n = 0, there is nothing to prove. We now assume that the Lemma holds for Γ[n − 1]. Namely, we have found coverings U_{1}, …, U_{n} of Γ[n − 1] that satisfy the required properties. Following the construction, Γ[n] is a small resolution of Γ[n − 1] ×_{A^{n}} A^{n+1}. Using the explicit description of π|_{U_{l}}: U_{l} → A^{n}, we see that

V_{l} ≜ U_{l} ×_{A^{n}} A^{n+1} ≅ A^{n+2}, where l < n,

are smooth G[n]-invariant open subsets of Γ[n]. Clearly, we can choose a unique coordinate (v_{·}^{l}) of V_{l} so that the projection V_{l} → U_{l} is given by

u_{i}^{l} = v_{i}^{l}, i ≤ n; u_{n+1}^{l} = v_{n+1}^{l}v_{n+2}^{l},

and the projection π^{l} : V_{i} → A^{n+1} is given by the formula (i) in the statement of the Lemma. A routine check shows that for l < n the G[n]-action on V_{l} and the transition function from V_{l−1} to V_{l} is exactly as shown in the statement of the Lemma.

It remains to check the case where l = n and n + 1. We now look at the chart U_{n}. Let φ : Γ[n] → Γ[n − 1] be the composite of the desingularization Γ[n] → Γ[n−1]×_{A^{n}} A^{n+1} with the obvious projection. Then

φ^{−1}(U_{n}) ≅ A^{n−1} × L^{⊕2},

where L is the tautological line bundle (degree −1) on P^{1} and L^{⊕2} is the total space of L^{⊕2}. We now cover P^{1} by B_{0} = {[b_{0}, 1]} and B_{1} = {[1, b_{1}]} ⊂ P^{1}. We choose trivializations L^{⊕2}|_{B_{0}} ≅ B_{0} × A^{2} and L^{⊕2}|_{B_{1}} ≅ B_{1}×A^{2} so that the transition function from B_{0}×A^{2} to B_{1}×A^{2} is given by

(b_{0}, ζ_{1}, ζ_{2}) ⟶ (b_{1}, ζ′_{1}, ζ′_{2}) = (1/b_{0}, ζ_{1}b_{0}, ζ_{2}b_{0}).

We let V_{n} be A^{n−1}×L^{⊕2}|_{B_{0}} and let V_{n+1} = A^{n−1}×L^{⊕2}|_{B_{1}} ⊂ A^{n−1}×L^{⊕2}. Then V_{1}, …, V_{n+1} form a covering of Γ[n]. Next for V_{n} we choose coordinates

(v_{1}^{n}, …, v_{n+2}^{n}) = (z_{1}, …, z_{n−1}, ζ_{1}, b_{0}, ζ_{2})

and for V_{n+1} we choose

(v_{1}^{n+1}, …, v_{n+2}^{n+1}) = (z_{1}, …, z_{n−1}, ζ′_{1}, b_{1}, ζ′_{2}).

We now check that this choice of coordinates satisfies the requirements of the Lemma.

We first check the transition functions. The transition function from V_{n} to V_{n+1} is determined by the transition function from L^{⊕2}|_{B_{0}} to L^{⊕2}|_{B_{1}}. Thus we have

(v_{1}^{n+1}, …, v_{n+2}^{n+1}) = (z_{1}, …, z_{n−1}, ζ′_{1}, b_{1}, ζ′_{2})
= (z_{1}, …, z_{n−1}, b_{0}ζ_{1}, 1/b_{0}, ζ_{2}b_{0})
= (v_{1}^{n}, …, v_{n−1}^{n}, v_{n}^{n}v_{n+1}^{n}, 1/v_{n+1}^{n}, v_{n+1}^{n}v_{n+2}^{n}).

This is the requirement in (iii) for l = n.

It remains to work out the transition function from V_{n−1} to V_{n}. We let v^{n−1} = (v_{1}^{n−1}, …, v_{n+2}^{n−1}) be a general point and let v^{n} = (v_{1}^{n}, …, v_{n+2}^{n}) be the point defined by the relation (iii) with u replaced by v and l replaced by n − 1. Then

v_{n+2}^{n} = v_{n+2}^{n−1}, v_{n+1}^{n} = v_{n+1}^{n−1}v_{n}^{n−1}, v_{n}^{n} = 1/v_{n}^{n−1}, v_{n−1}^{n} = v_{n−1}^{n−1}v_{n}^{n−1}.

Clearly, to prove (iii) we only need to check that the images of v^{n−1} and v^{n} in U_{n} coincide and their images in A^{n+1} also coincide. By definition, the image of v^{n−1} in A^{n+1} is

(v_{1}^{n−1}, …, v_{n−1}^{n−1}v_{n}^{n−1}, v_{n+1}^{n−1}, v_{n+2}^{n−1})

and the image of v^{n} in A^{n+1} is

(v_{1}^{n}, …, v_{n−1}^{n}, v_{n}^{n}v_{n+1}^{n}, v_{n+2}^{n}).

They coincide using the relations mentioned above. On the other hand the image of v^{n−1} in U_{n−1} is (…, v_{n}^{n−1}, v_{n+1}^{n−1}v_{n+2}^{n−1}) and its image in U_{n}, using (iii) and the induction hypothesis, is

(v_{1}^{n−1}, …, v_{n−1}^{n−1}v_{n}^{n−1}, 1/v_{n}^{n−1}, v_{n}^{n−1}v_{n+1}^{n−1}v_{n+2}^{n−1}).

The image of v^{n} in U_{n} is

(v_{1}^{n}, …, v_{n−1}^{n}, v_{n}^{n}, v_{n+1}^{n}v_{n+2}^{n}).

They coincide as well. This proves the relation (iii). The part (ii) concerning the group action can be checked directly and will be omitted.

q.e.d.

To apply this Lemma to the general case W [n], we have the following useful observation. Let D̂ be the formal completion of W along D. For any open V ⊂ D, we let V̂ be V ⊂ D̂ endowed with the open subscheme structure. We choose V so that OV̂ ≅ O_{V} [[w_{1}, w_{2}]] and the induced morphism V̂ → A^{1} is given by t ↦ w_{1}w_{2}. We let Ṽ ⊂ W [n] be an open subset so that Ṽ ∩ (W [n] ×_{W} D) = W [n] ×_{W} V and let Ŵ [n] be the formal completion of Ṽ along W [n] ×_{W} V. Similarly we let U ⊂ Γ[n] be the preimage of 0_{A^{2}} ∈ A^{2} under Γ[n] → Γ[1] = A^{2} and let Γ̂[n] be the formal completion of Γ[n] along U. Then we have a G[n]-equivariant isomorphism

Ŵ [n] ≅ V × Γ̂[n]. (1.11)

We also denote by

\underline{Ψ} : G[n] × C[n] ⟶ C[n] and Ψ : G[n] × W [n] ⟶ W [n] (1.12)

the morphisms of G[n]-actions.

Corollary 1.3. Let G[n]_{l} be the subgroup of G[n] that is the lth copy of G_{m} in G[n], using the isomorphism (1.1). Then the G[n]_{l} action on Δ_{l+1} is induced by the linear action of G_{m} on 1_{D} and N_{D_{2}/Y_{2}} of weight 0 and 1, respectively. It acts trivially on all other Δ_{i}’s.

Corollary 1.4. Let h:W [n]_{0} → W [n]_{0} be an isomorphism commuting with W [n]_{0} → W_{0}. Then there is a σ ∈ G[n] so that h is identical to the automorphism of W [n]_{0} induced by the action of the σ. Namely h(w) = Ψ(σ, w) for and w ∈ W [n]_{0}.

Corollary 1.5. Let s ∈ C[n] be a closed point and let W [n]_{s} be the fiber of W [n] over s. Then the stabilizer of the W [n]_{s} (which is the subgroup of G[n] that fixes W [n]_{s}) is the trivial subgroup of G[n].

We now fix the notation for the group action. Let G be any group scheme acting on an S-scheme X via Ψ:G × X → X. Let f:T → X be any S-morphism of schemes and let λ: G′ → G be a homomorphism of groups. Then we define the induced group action morphism

Ψ_{f}^{λ} ≜ Ψ ∘ (λ × f) : G′ × T ⟶ X.

In case G = G′ and λ = 1_{G}, Ψ_{f}^{λ} will be shortened to Ψ_{f}; In case T = X and f = 1_{X}, then Ψ_{f}^{λ} will be shortened to Ψ^{λ}. Now let ρ: T → G and f:T → X be two morphisms. We define

f^{ρ} ≜ Ψ ∘ (ρ, f) : T ⟶ X, (1.13)

in case the group action Ψ is clear from the context. Later, we will encounter the situation where X is a scheme over a scheme Y and ρ is a morphism Y → G. By abuse of notation we will use ρ to denote the composite morphism X → Y → G as well and use f^{ρ} to denote the morphism induced by the group action, as in (1.13).

Let I ⊂ [n + 1] be a subset of l + 1 elements. It associates an embedding \underline{γ_{I}}:C[l] → C[n] (see (1.4)). From the construction of W [n], it is clear that there is a canonical W -isomorphism

γ_{I} : W [l] ≅ W [n] ×_{C[n]} C[l].

Now let J be the complement of I in [n+1] with entries j_{1} ≤ … ≤ j_{n−l}. We let λ_{J} : G[n − l] → G[n] be the homomorphism defined so that the j_{i}-th component of λ_{J}(σ) is σ_{i} (σ_{i} is the i-th component of σ) and all other components of λ_{J}(σ) are 1. Here the components of G[n] are defined via the isomorphism (1.1). Then by the explicit G[n]-action on A^{n+1}, we see immediately that

\underline{Ψ}λ_{J} ; \underline{γ_{I}} : G[n − l] × C[l] ⟶ C[n]

defined by the G[n]-action \underline{Ψ} is an open immersion. We denote the image open subscheme by C[ΣI] and denote W [n]×_{C[n]}C[ΣI] by W [ΣI]. Clearly, W [ΣI] is G[n − l]-invariant under the induced action

Ψ^{λ_{J}} : G[n − l] × W [n] ⟶ W [n].

Lemma 1.6. The scheme W [ΣI] (resp. C[ΣI]) is a principal fiber bundle over W [l] (resp. C[l]) with group G[n − l] via the action Ψ^{λ_{J}} (resp. \underline{Ψ}^{λ_{J}}) and the morphism γ_{I} (resp. \underline{γ}_{I}) is a section of this principal fiber bundle.

The proof is based on the parallel result concerning A^{l+1} → A^{n+1}, which is obvious. We shall omit the proof here.

The two principal fiber bundles are compatible in the sense that the following diagram is commutative:

G[n − l] × W [ΣI] ⟶^{Ψ^{λ_{J}}} W [ΣI]
↓  ↓
G[n − l] × C[ΣI] ⟶^{\underline{Ψ}^{λ_{J}}} C[ΣI].

Often, we have several morphisms r:S → C[n] and like to compare the corresponding fiber products W [n]×_{C[n]}S. Adopting the convention in groupoid, we shall view W [n] ×_{C[n]} S as the pullback of W [n] → C[n] via r: S → C[n]. Accordingly, we will denote such a fiber product by r^{∗}W [n].

Corollary 1.7. Let (s, S) be a pointed scheme, ι̃: S → C[n] be a morphism such that ι̃(s) ∈ C[ΣI] for some subset I ⊂ [n + 1] with |I| = l + 1. Let S_{0} = ι^{−1}(C[ΣI]), which is an open neighborhood of s in S. Then there is a morphism (ρ, ι) : S_{0} → G[n − l] × C[l] such that the restriction of ι̃ to S_{0} ⊂ S is identical to \underline{Ψ}_{\underline{γ}_{I}}^{λ_{J}} ∘ (ρ, ι). Further it induces a canonical isomorphism

ι^{∗}W [n] ×_{S} S_{0} ≅ ι^{∗}W [l],

compatible to the projections to W ×_{C} S_{0}, using the principal fiber bundle structure W [ΣI] → W [l] and the morphism γ_{I}.

We now introduce the notion of the expanded degenerations of W. Let S be any C-scheme. An effective degeneration over S is a Cmorphism ξ: S → C[n] for some n. Note that such a morphism comes with an S-family

W = W [n] ×_{C[n]} S ⟶ S

and the tautological projection W → W ×_{C} S. (For convenience, we will not distinguish the family W from the morphism ξ, and vice versa if no confusion arises. As a convention, we will call W the associated family of ξ and call ξ the associated morphism of W.) Let ξ_{1} :S → C[n_{1}] and ξ_{2} : S → C[n_{2}] be two effective degenerations. An effective arrow r : ξ_{1} → ξ_{2} consists of a standard embedding ι : C[n_{1}] → C[n_{2}] and a morphism ρ : S → G[n_{2}] so that (ι ∘ ξ_{1})^{ρ} = ξ_{2}. By Corollary 1.7, this identity defines a canonical S-isomorphisms ξ_{1}^{∗}W [n_{1}] ≅ ξ_{2}^{∗}W [n_{2}] compatible with their projections to W ×_{C} S. We call this the associated isomorphism of the arrow r : ξ_{1} → ξ_{2}. By abuse of notation, we will denote this isomorphism by r as well. Now let ξ_{1} and ξ_{2} be any two effective degenerations over S. We say ξ_{1} is equivariant to ξ_{2} via an effective arrow if there is either an effective arrow ξ_{1} → ξ_{2} or an effective arrow ξ_{2} → ξ_{1}. We say ξ_{1} and ξ_{2} are equivalent via a sequence of effective arrows if there is a sequence of effective degenerations η_{0}, …, η_{m} so that η_{0} = ξ_{1}, η_{m} = ξ_{2} and η_{i} is equivariant to η_{i+1} via an effective arrow for all i. Note that once such a sequence of effective arrows is given, then there is a unique induced S-isomorphism ξ_{1}^{∗}W [n_{1}] ≅ ξ_{2}^{∗}W [n_{2}] compatible with their projections to W ×_{C} S.

Now let W_{1} and W_{2} be two effective degenerations over S. We say W_{1} is isomorphic to W_{2} if W_{1} is S-isomorphic to W_{2} and the isomorphism is compatible to their tautological projections to W ×_{C} S. The following Lemma says that such isomorphisms are locally generated by a sequence of arrows.

Lemma 1.8. Let ξ_{1} and ξ_{2} be two effective degenerations over S so that their associated families W_{1} and W_{2} are isomorphic. Then to each p ∈ S there is an open neighborhood S_{0} of p ∈ S so that the (induced) isomorphism between W_{1} ×_{S} S_{0} and W_{2} ×_{S} S_{0} is induced by a sequence of effective arrows between ξ_{1} ×_{S} S_{0} and ξ_{2} ×_{S} S_{0}.

Proof. In case p lies over C − 0, then there is nothing to prove. Now assume p lies over 0 ∈ C. Let n be the integer so that W_{1} ×_{S} p has n + 2 irreducible components. Then by Corollary 1.7, there is an open neighborhood S_{0} of p ∈ S and two morphisms ξ′_{1}, ξ′_{2} :S_{0} → C[n] so that ξ_{i} ×_{S} S_{0} is equivalent to ξ′_{i} via an arrow. Thus to prove the Lemma it suffices to consider the case where n_{1} = n_{2} = n. Now we assume, ξ_{i} : S → C[n] and ξ_{i}(p) = 0 ∈ C[n]. Let φ : W_{1} ⟶ W_{2} be the W_{S}- isomorphism and let ϕ_{i} : W_{i} ⟶ W [n] be the associated morphisms. We define a functor ℑ𝔰𝔬𝔪[S] from the category of S-schemes to sets that associates to any S-scheme T the set of all morphisms ρ:T → G[n] such that with ι: W_{1} ×_{S} T → W_{1} the induced morphism (i.e., the first projection) then

(ϕ_{1} ∘ ι)^{ρ} = ϕ_{2} ∘ φ ∘ ι : W_{1} ×_{S} T ⟶ W [n].

(For the notation here see the formula (1.13).) By the existence theorem of Hilbert schemes developed by Grothendieck coupled with the trivialization of the stabilizers (see Corollary 1.5), ℑ𝔰𝔬𝔪[S] is represented by a subscheme R of S × G[n].

We now show that R is nonempty and dominates a neighborhood of p ∈ S. Since ξ_{1}(p) = ξ_{2}(p) = 0 ∈ C[n], the restriction of the isomorphism φ to fibers over p induces an isomorphism W [n]_{0} ≅ W [n]_{0} that commutes with W [n]_{0} → W_{0}. Then it is induced by an element g_{0} ∈ G[n] by Corollary 1.4. Hence (p, g_{0}) ∈ R. Let 𝔪 be the maximal ideal of the local ring O_{S,p}. By the previous discussion, we have ρ_{0} : p → G[n] so that (i_{0}, ρ_{0}) : p → S × G[n] factor through R ⊂ S × G[n], where i_{0} :p → S is the inclusion. Let B_{k} = Spec O_{S,p}/𝔪^{k} with i_{k}: B_{k} → S the inclusion morphism. Suppose for some k ≥ 1 we have

(i_{k−1}, ρ_{k−1}) : B_{k−1} ⟶ S × G[n],

extending (i_{0}, ρ_{0}) that factor through R ⊂ S × G[n]. We will show that ρ_{k−1} extends to ρ_{k} satisfying similar condition. Since G[n] is smooth, we can extend ρ_{k−1} to h_{k} : B_{k} → G[n]. Let ι_{k} : W_{1} ×_{S} B_{k} → W_{1} be the canonical immersion. We consider the problem of extending the W -morphism

(ϕ_{1} ∘ ι_{k−1})^{h_{k−1}} = ϕ_{2} ∘ φ ∘ ι_{k−1} : W_{1} ×_{S} B_{k−1} ⟶ W [n]

to a W -morphism

W_{1} ×_{S} B_{k} ⟶ W [n].

This is a typical extension problem. Since (ϕ_{1} ∘ ι_{k})^{h_{k}} and ϕ_{2} ∘ φ ∘ ι_{k} are two such extensions, by deformation theory they are related by an element in 𝔪^{k}/𝔪^{k+1} ⊗ K, where

K = ker { Hom_{W [n]_{0}}(Ω_{W [n]}, O_{W [n]_{0}}) ⟶

Hom_{W [n]_{0}}(Ω_{W} ⊗_{O_{W}} O_{W [n]_{0}}, O_{W [n]_{0}})}.

We claim that K is isomorphic to k^{⊕n} and is generated by the group action of G[n]. We let i : W [n]_{0} → W [n] be the immersion and j : W [n]_{0} → C[n] be the composite of i with the projection. Then we have the exact sequence

0 ⟶ j^{∗}Ω_{C[n]} ⟶ i^{∗}Ω_{W [n]} ⟶ Ω_{W [n]_{0}} ⟶ 0 (1.14)

and its induced exact sequence

Hom(i^{∗}Ω_{W [n]}, O_{W [n]_{0}}) ⟶^{β} Hom(j^{∗}Ω_{C[n]}, O_{W [n]_{0}}) ⟶^{δ}

Ext^{1}(Ω_{W [n]_{0}}, O_{W [n]_{0}}).

At the origin 0 ∈ C[n], we have a canonical isomorphism T_{0}C[n] ≅ T_{0}A^{n+1}. We let v_{l}: k → k^{⊕n+1} ≡ T_{0}A^{n+1} be the l-th component. The dual of v_{l}, which is v_{l}^{∨}: Ω_{C[n]} ⊗_{O_{C[n]}} k_{0} → k, defines a homomorphism v_{l}^{∨}:j^{∗}Ω_{C[n]} → O_{W [n]_{0}}. Now let

0 ≠ h ∈ Im(β) ⊂ Hom(j^{∗}Ω_{C[n]}, O_{W [n]_{0}}),

say given by a homomorphism j^{∗}Ω_{C[n]} → O_{W [n]_{0}}. Then it must be a linear combination of v_{l}^{∨}, say equal to a_{1}v_{1}^{∨} + ⋯ + a_{n+1}v_{n+1}^{∨}. Now let c be the extension class in Ext^{1}(Ω_{W [n]_{0}}, j^{∗}Ω_{C[n]}) of the exact sequence (1.14) and let

h^{∗} : Ext^{1}(Ω_{W [n]_{0}}, j^{∗}Ω_{C[n]}) ⟶ Ext^{1}(Ω_{W [n]_{0}}, O_{W [n]_{0}})

be the homomorphism induced by h. We claim that h^{∗}(c) ≠ 0. Indeed, let T_{2} = Spec k[t]/(t^{2}) → C[n] be the immersion representing the tangent vector v = a_{1}∂_{z_{1}} + ⋯ + a_{n+1}∂_{z_{n+1}} ∈ T_{0}C[n]. Then

W [n] ×_{C[n]} Spec k[t]/(t^{2}) ⟶ Spec k[t]/(t^{2})

is an infinitesimal smoothing of the l-th nodal divisor of W [n]_{0} whenever a_{l} ≠ 0, and consequently the exact sequence

0 ⟶ O_{W [n]_{0}} ⟶ Ω_{W [n]×_{C[n]}T_{2}} ⊗_{W [n]×_{C[n]}T_{2}} O_{W [n]_{0}} ⟶ Ω_{W [n]_{0}} ⟶ 0 (1.15)

does not split since h ≠ 0. On the other hand, this exact sequence is exactly defined by the extension class δ(h). In case h ∈ Im(β), then δ(h) = 0, violating the nonsplitness of (1.15). This proves β = 0. As a consequence

Hom(Ω_{W [n]_{0}}, O_{W [n]_{0}}) ≅ Hom(i^{∗}Ω_{W [n]}, O_{W [n]_{0}}).

Now let α be any element in K. Then α ∈ Hom(Ω_{W [n]_{0}}, O_{W [n]_{0}}) by the above identity. Hence the restriction of α to Δ_{l}, denoted by α_{l}, is a vector field of Δ_{l}. Since α lies in the kernel, we have α_{1} = α_{n+2} = 0. Furthermore, the restriction of α_{l} to the two (or one if l = 1 or n + 2) distinguished divisors is tangential to these divisors and must satisfy the compatibility condition α_{l}|_{D_{−}} = α_{l+1}|_{D_{+}}. Now let V_{l} be the subspace of H^{0}(T_{Δ_{l}}) consisting of the vector fields whose restrictions to the two distinguished divisors are tangential to the two divisors. Then V_{l} is canonically isomorphic to the direct sum H^{0}(T_{D}) ⊕ ℂ, where ℂ is generated by the vector field generated by the group action G[n]_{l}. Let e(α_{l}) be the component of α_{l} in H^{0}(T_{D}). Then the compatibility condition translates to e(α_{1}) = ⋯ = e(α_{n+2}). Adding the fact that α_{1} = 0, we conclude that e(α_{l}) = 0 for 2 ≤ l ≤ n + 1. Therefore, α_{l} must be a vector field tangential to the fibers of Δ_{l} → D and vanishes on the two distinguished divisors. By Corollary 1.4, such α is generated by the group action G[n].

If we apply this to the two extensions (ϕ_{1} ∘ ι_{k})^{h_{k}} and ϕ_{2} ∘ φ ∘ ι_{k}, we see that there is a morphism h_{k}^{+} :B_{k} → G[n] with h_{k}^{+}|_{B_{k−1}} :B_{k−1} → G[n] factor through {e} ⊂ G[n] such that with ρ_{k} = h_{k}^{+} · ι_{k},

(ϕ_{1} ∘ ι_{k})^{ρ_{k}} = ϕ_{2} ∘ φ ∘ ι_{k} : W_{1} ×_{S} B_{k} ⟶ W [n].

Since k is arbitrary, this implies that the projection R → S dominates a neighborhood of p ∈ S.

We let π: R → S be the projection. To complete the proof of the Lemma, we need to find a neighborhood R_{0} of (p, ρ_{0}) ∈ R, so that π|_{R_{0}} : R_{0} → S is an open neighborhood of p ∈ S. First, by Corollary 1.5, for any s ∈ S the set π^{−1}(s) has at most one point. Applying Corollary 1.5 again, we see that the choice of ρ_{k} in the proof is unique. Hence the formal completion of R along (p, g_{0}) is isomorphic to the formal completion of S along p, under π. Hence R → S is étale over a neighborhood of p ∈ S. Combining the one-to-one and étale property, we conclude that there is a neighborhood R_{0} ⊂ R of (p, g_{0}) so that π|_{R_{0}} :R_{0} → S is an open immersion. Let ρ:R_{0} → G[n] be the morphism given by the definition of the functor ℑ𝔰𝔬𝔪[S] and let η : R_{0} → S be the inclusion. Then ρ defines an effective arrow between ξ_{1} ×_{S} R_{0} and ξ_{2} ×_{S} R_{0}. This proves Lemma 1.8. q.e.d.

We now define the stack 𝔚 of the expanded degenerations of W.

Definition 1.9. Let S be any C-scheme. An expanded degeneration of W over S is a pair (W, ρ), where W is a family over S and ρ: W → W ×_{C} S is an S-projection, so that there is an open covering S_{α} of S such that for each S_{α} the restriction pair (W ×_{S} S_{α}, ρ|_{W×_{S}S_{α}}) is isomorphic to an effective expanded degeneration of W. Let (W, ρ) and (W′, ρ′) be two degenerations over S and S′ respectively. An arrow W → W′ consists of a C-morphism S → S′ and an S-isomorphism W → W′ ×_{S′} S compatible to their projections to W ×_{C} S.

We define the groupoid of the expanded degenerations of W to be the category 𝔚 whose objects are expanded degenerations of W over S, where S is a C-scheme. In the following, we will call expanded degenerations of W simply degenerations if its meaning is clear from the context. The functor 𝔭: 𝔚 → (Sch/C) is the one that sends degenerations over S to S. Arrows between two objects are those defined in the above Definition. Obviously, if ξ ∈ 𝔚 with 𝔭(ξ) = S and i: T → S is a morphism, then there is a unique pullback family i^{∗}ξ ∈ 𝔚 with an arrow i^{∗}ξ → ξ. In this way, (𝔚, 𝔭) forms a groupoid. For any S ∈ (Sch/C), we define 𝔚(S) to be the category of all degenerations over S.

Proposition 1.10. The groupoid 𝔚 is a stack.

Proof. This is straightforward and will be left to the readers. q.e.d.

2. Predeformable morphisms

Recall that a prestable curve is a connected complete curve with at most nodal singularity. (Since the marked points will not affect our discussion of predeformable morphisms, we will discuss curves without marked points in this section.) A morphism f whose domain is a prestable curve is called a prestable morphism. Let f: X → W [n] be a prestable morphism so that f(X) ⊂ W [n]_{0}. We say f is nondegenerate if no irreducible component of X is mapped entirely to the singular locus of W [n]_{0}. It is well-known that in general f may not be deformable to a prestable morphism to W [n] over a general point t ∈ C[n]. One set of conditions is purely local in nature. It reflects the fact that W [n]_{0} has nodal singularity. In this section, we will investigate such condition in details.

2.1 Morphisms of pure contact

We will investigate the following situation in this subsection. Let k[t] → k[w_{1}, w_{2}] be the homomorphism defined by t ↦ w_{1}w_{2} and let ϕ:k[[s]] → k[[z_{1}, z_{2}]] be the homomorphism defined by ϕ(s) = z_{1}z_{2} or ϕ(s) = z_{1}. Note that the former corresponds to a smoothing of the node and the latter corresponds to a family of smooth curves. We let A be a k[[s]]- algebra and let R = k[[z_{1}, z_{2}]] ⊗_{k[[s]]} A. We assume A is (s)-adically complete and let R̂ be the (z_{1}, z_{2})-adic completion of R. In this part, we will investigate the situation where there is a homomorphism ψ : k[t] → A with induced homomorphism ψ̃:k[t] → R̂ via ψ̃(t) = 1 ⊗ ψ(t) and a k[t]-homomorphism

φ : k[w_{1}, w_{2}] ⟶ R̂ (2.1)

satisfying the following nondegeneracy condition: The ideals (z_{1}, z_{2}) and (φ(w_{1}), φ(w_{2})) in R̂ satisfy

(z_{1}, z_{2})^{m} ⊂ (φ(w_{1}), φ(w_{2})) ⊂ (z_{1}, z_{2})

for some integer m.

Geometrically, this states that there are no irreducible components of the fibers of Spec R̂ → Spec A that are mapped entirely to the node in Spec k[w_{1}, w_{2}]/(w_{1}w_{2}).

We now state and prove our first technical result which explains the notion of curves of pure contact. We begin with the notion of normal forms of elements in R̂.

Lemma 2.1. Let R̂ be as above. Then any α ∈ R̂ has a unique expression, called its normal form, as

α = a_{0} + ∑_{i>0} a_{i}z_{1}^{i} + ∑_{i>0} b_{i}z_{2}^{i}, a_{i}, b_{i} ∈ A. (2.2)

We call such expansion the normal form of α.

Proof. Clearly, any α can be expressed as above. We now show that such expression is unique. Let I_{n} ⊂ R̂ be the A-submodule spanned by all z_{1}^{i}z_{2}^{j} satisfying |i − j| > n. Clearly, the quotient A-module R̂/I_{n} is a free A-module with basis z_{1}^{n}, …, z_{1}, 1, z_{2}, …, z_{2}^{n}. Let π_{n} : R̂ ⟶ R̂/I_{n} be the quotient homomorphism of A-modules. Assume α = 0 has an expression as in (2.2), then 0 = π_{n}(α) = a_{0} + ∑_{j≥1}^{n} a_{j}z_{1}^{j} + b_{j}z_{2}^{j} in R̂/I_{n}. Since R̂/I_{n} is a free A-module and z_{1}^{n} …, z_{2}^{n} is a basis, all a_{j} and b_{j} are zero for j ≤ n. Since n is arbitrary, this implies that all a_{j} and b_{j} are zero. This proves the Lemma. q.e.d.

Proposition 2.2. Let the notation be as above. We assume further that A is a local ring and is flat over k[t]. Then of the two choices of ϕ: k[[s]] → k[[z_{1}, z_{2}]] given before, only ϕ(s) = z_{1}z_{2} is possible. Further, possibly after exchanging z_{1} and z_{2}, the homomorphisms φ and ψ must be of the forms

φ(w_{1}) = z_{1}^{n}β_{1}, φ(w_{2}) = z_{2}^{n}β_{2} and ψ(t) = s^{n}ϵ

for some integer n, units β_{1} and β_{2} ∈ R̂ and a unit ϵ ∈ A such that β_{1}β_{2} = ϵ in R̂.

Proof. Let 𝔪 ⊂ A be the maximal idea and let

φ(w_{i}) = ∑_{j≥0} a_{i,j}z_{1}^{j} + ∑_{j≥0} b_{i,j}z_{2}^{j}, a_{i,0} = b_{i,0} (2.3)

be the normal form of φ(w_{i}). Since φ is nondegenerate, the homomorphism

φ_{0} : k[w_{1}, w_{2}] ⟶ k[[z_{1}, z_{2}]]/(z_{1}z_{2})

induced by φ and k[[z_{1}, z_{2}]]/(z_{1}, z_{2}) ≅ R̂/𝔪R̂ must be of the form (possibly after exchanging z_{1} and z_{2})

φ_{0}(w_{1}) = c_{1}z_{1}^{n_{1}} mod z_{1}^{n_{1}+1}  and
φ_{0}(w_{2}) = c_{2}z_{2}^{n_{2}} mod z_{2}^{n_{2}+1}, c_{1}, c_{2} ≠ 0,

for some positive integers n_{1} and n_{2}. Hence in (2.3), all a_{1,j} for j < n_{1} and b_{2,j} for j < n_{2} are in the maximal ideal 𝔪 ⊂ A. We let

I = ({a_{1,j}s^{j}, b_{1,j}, a_{2,j}, b_{2,j}s^{j}|j ≥ 0}) ⊂ A

be the ideal generated by the listed terms for all j ≥ 0.

We first show that I ⊂ 𝔪. After expressing φ(w_{1})φ(w_{2}) in its normal form φ(w_{1})φ(w_{2}) = c_{0} + ∑_{j>0} c_{j}z_{1}^{j} + d_{j}z_{2}^{j} we have

c_{k} = ∑_{i+j=k} a_{1,i}a_{2,j} + ∑_{i≥0} a_{1,k+i}b_{2,i}s^{i} + ∑_{i≥0} a_{2,k+i}b_{1,i}s^{i}, k > 0

and

d_{k} = ∑_{i+j=k} b_{1,i}b_{2,j} + ∑_{i≥0} b_{1,k+i}a_{2,i}s^{i} + ∑_{i≥0} b_{2,k+i}a_{1,i}s^{i}, k > 0.

Because φ(w_{1})φ(w_{2}) = φ(w_{1}w_{2}) is the image of t in A, all c_{k} and d_{k} for k > 0 are zero. Now let l be the largest integer so that b_{1,j} ∈ 𝔪 for j < l. Then by considering the relation d_{l+n_{2}} = 0, we obtain

∑_{i+j=l+n_{2}} b_{1,i}b_{2,j} + b_{1,n_{2}+l}a_{2,0} + b_{2,n_{2}+l}a_{1,0} ∈ 𝔪.

By assumption, all terms in the above expression except b_{1,l}b_{2,n_{2}} are in 𝔪. Hence b_{i,l}b_{2,n_{2}} ∈ 𝔪. Since b_{2,n_{2}} ∉ 𝔪, b_{1,l} ∈ 𝔪. This shows that there is no such l and hence all b_{1,j} are in 𝔪. Similarly, by considering the relation c_{k} = 0 we can show that all a_{2,j} ∈ 𝔪.

Now let n = min(n_{1}, n_{2}) and let I be the associated ideal of I in the quotient ring A/(s^{n}). Let 𝔪 be the maximal ideal of A/(s^{n}). We now show that I ⊂ 𝔪 · I. Note that in the expression of c_{k} and d_{k}, terms a_{2,k+i}b_{1,i}s^{i} and b_{1,k+i}a_{2,i}s^{i} are in 𝔪 · I for all k, i ≥ 0. We let J = 𝔪 · I + (s^{n}). Note that b_{2,k}s^{k} ∈ J for k ≥ n. Now let k be the smallest integer so that b_{2,k+i}s^{k+i} ∈ J for all i > 0. Hence k < n. Using c_{n_{1}−k} = 0, we obtain

∑_{n_{1}−k+i≥0} a_{1,n_{1}−k+i}b_{2,i}s^{i} + ∑_{i+j=n_{1}−k} a_{1,i}a_{2,j} ∈ 𝔪 · I,

which implies

a_{1,n_{1}}b_{2,k}s^{k} + ∑_{k−n_{1}<i<k} a_{1,n_{1}−k+i}b_{2,i}s^{i} + ∑_{k<i} a_{1,n_{1}−k+i}b_{2,i}s^{i}

≡ a_{1,n_{1}}b_{2,k}t^{k} mod J.

Hence b_{2,k}s^{k} ∈ J. By minimality of k we have b_{2,j}s^{j} ∈ J for all j ≥ 0. For similar reason, we can prove a_{1,k}s^{k} ∈ J for all k ≥ 0. To show a_{2,k} ∈ J, we use the relation c_{n_{1}+k} = 0, which combined with a_{2,k+i}b_{1,i} ∈ J and b_{2,j}s^{j} ∈ J implies ∑_{i+j=n_{1}+k} a_{1,i}a_{2,j} ∈ J. Now assume a_{2,j} ∈ J for j < k, then the above inclusion implies

a_{1,n_{1}}a_{2,k} + ∑_{i>0} a_{1,n_{1}−i}a_{2,k+i} + ∑_{i>0} a_{1,n_{1}+i}a_{2,k−i} ∈ J.

Since a_{1,n_{1}−i} ∈ 𝔪 and a_{2,k+i} ∈ I, we have a_{1,n_{1}}a_{2,k} ∈ J and hence a_{2,k} ∈ J. For similar reason, b_{1,k} ∈ J for all k > 0. Combined, this proves that I ⊂ 𝔪 · I. Since A is Noetherian, this is possible only if I = 0. This proves I ⊂ (s^{n}).

We next show n_{1} = n_{2}. Assume n = n_{1} < n_{2}. Then d_{n_{2}−n_{1}} = 0 implies that

∑_{i+j=n_{2}−n_{1}} b_{1,i}b_{2,j} + ∑_{i≥0} b_{1,n_{2}−n_{1}+i}a_{2,i}s^{i} + ∑_{i≥0} b_{2,n_{2}−n_{1}+i}a_{1,i}s^{i} = 0.

Note that all terms in this expression except b_{2,n_{2}}a_{1,n_{1}}s^{n_{1}} belong to 𝔪 · (s^{n}). This is impossible. Hence n_{1} = n_{2}.

We now look at the normal form of φ(w_{1}). Since b_{1,j} ∈ (s^{n}) for all j ≥ 0, there are b′_{1,j} ∈ A so that b_{1,j} = b′_{1,j}s^{n}. On the other hand, since a_{1,j}s^{j} ∈ (s^{n}), there are a′_{1,j} so that a_{1,j} = a′_{1,j}s^{n−j} for j < n. Using s = z_{1}z_{2}, we see that there are β_{1} and β_{2} in R̂ so that φ(w_{i}) = z_{i}^{n}β_{i}. Further by our choice of n both β_{1} and β_{2} are units in R̂. It remains to show that there is a unit ϵ ∈ A so that ψ(t) = s^{n}ϵ and β_{1}β_{2} = ϵ. First, since ψ(t) = φ(w_{1}w_{2}), which is in the ideal (s^{n}), there is an ϵ ∈ A so that ψ(t) = s^{n}ϵ. Hence s^{n}(β_{1}β_{2} − ϵ) = 0 in R̂. Because R̂ is flat over k[t], this implies that β_{1}β_{2} − ϵ = 0. Hence ϵ is a unit in A. This completes the proof of Proposition 2.2. q.e.d.

This Proposition inspires the notion of homomorphisms of pure contacts as follows.

Definition 2.3. Let the notation be as stated in the beginning of this subsection with A only assumed to be (s)-adically complete. We say φ has pure contact if, possibly after exchanging z_{1} and z_{2}, there is a unit β in R̂, a unit ϵ in A and an integer n such that φ(w_{1}) = βz_{1}^{n} and φ(w_{2}) = ϵβ^{−1}z_{2}^{n}.

The integer n is called the order of contact. Sometimes to emphasize this order we will say φ is of pure n-contact. The following is the main existence Lemma of this section.

Lemma 2.4. Let n be any positive integer. Suppose a_{1,n} and b_{2,n} in the normal forms of φ(w_{1}) and φ(w_{2}) in (2.3) respectively are units in A. Then there is an ideal I_{A} ⊂ A so that for any A → T the induced homomorphism

φ_{T} : k[w_{1}, w_{2}] ⟶ R̂ ⊗_{A} T

has pure n-contact if and only if A → T factors through A/I_{A} → T. The ideal I_{A} satisfies the following base change property: Let A → A′ be a homomorphism of rings and let φ′: k[w_{1}, w_{2}] → (k[[z_{1}, z_{2}]] ⊗_{k[[s]]} A′)ˆ , where (·)ˆ is the (z_{1}, z_{2})-adic completion of the respective ring, be the induced k[t]-homomorphism. Then I_{A′} = I_{A} · A′.

Proof. We let ζ, ϵ, ξ_{j}, η_{j}, where j runs through all positive integers, be indeterminants. We consider the ring B = A[[ζ, ζ^{−1}, ϵ, ϵ^{−1}, ξ, η]], where ξ and η mean (ξ_{1}, ξ_{2} ⋯) and (η_{1}, η_{2}, …) respectively. We let C be the (z_{1}, z_{2})-adic completion of k[[z_{1}, z_{2}]] ⊗_{k[[s]]} B. We let γ ∈ C and its inverse be

γ = ζ (1 + ∑_{j>0} ξ_{i}z_{1}^{i} + η_{j}z_{2}^{j}) and γ^{−1} = ζ^{−1} (1 + ∑_{j>0} ξ̃_{i}z_{1}^{i} + η̃_{j}z_{2}^{j}) ,

where ξ̃_{j} and η̃_{j} are elements in B. We consider the ideal J ⊂ C that is generated by all coefficients of z_{i}^{j} in the normal forms of

Φ_{1} = φ(w_{1}) − z_{1}^{n}ζ (1 + ∑_{j>0} ξ_{j}z_{1}^{j} + ∑_{j>0} η_{j}z_{2}^{j}) (2.4)

and

Φ_{2} = φ(w_{2}) − z_{2}^{n}ζ^{−1} (1 + ∑_{j>0} ξ̃_{j}z_{1}^{j} + ∑_{j>0} η̃_{j}z_{2}^{j}) ϵ. (2.5)

We now investigate these generators of J, or equivalently the relations in C/J. First note that the coefficient of z_{1}^{n+k} in Φ_{1} is a_{1,k+n}−ξ_{k}. Hence we obtain elements a_{1,k+n} − ξ_{k} in J for all k ≥ 1. Similarly, the coefficient of z_{1}^{n} in Φ_{1} is a_{1,n} − ζ, which by definition is in J. In the following we will call them the canonical relations of ξ_{k} and η, respectively. We let J_{1} be the ideal generated by a_{1,n} − ζ and all a_{1,k+n} − ζ_{k}. We next find relations of η_{j}. First, by expanding 1/γ in power series and using s = z_{1}z_{2}, we can express η̃_{k} canonically as η̃_{k} = −(η_{k} + ∑_{j≥1} c_{k,j}s^{j}), where c_{k,j} ∈ k[[ξ, η]]. Then the coefficient of z_{2}^{k+n} in Φ_{2} becomes

b_{2,n+k} + ζ^{−1}η_{k}ϵ + ∑_{j≥1} ζ^{−1}c_{k,j}s^{j}ϵ.

We let J_{2} be the ideal generated by J_{1} and this set of elements for all k > 0. Since c_{k,j} ∈ k[[ξ, η]], modulo J_{2} we can substitute ζ and all ξ_{l} by elements in A using relations in J_{1} and then substitute all η_{l} in c_{k,j} by −ζϵ^{−1}b_{2,n+k} − ∑ c_{k,j}s^{j}. This way we obtain

η_{k} − (d_{1,k} + ∑_{j≥2} c̃_{k,j}s^{j}) ∈ J_{2},

where d_{k,1} ∈ A and c̃_{k,1} ∈ A[[η, ϵ^{−1}]]. Note that here we have used the fact that A is complete with respect to the ideal (s). Repeating this procedure, we eventually obtain an element η_{k} − f_{k}(ϵ^{−1}) ∈ J_{2}, where f_{k}(ϵ^{−1}) ∈ A[[ϵ^{−1}]]. It remains to find relation in ϵ. By using the coefficient of z_{2}^{n} in Φ_{2}, we obtain an element

b_{2,n} − ϵζ^{−1}(1 + ∑_{j≥1} h_{j}s^{j}) ∈ J,

where h_{j} ∈ k[[ξ, η]]. We then let J_{0} ⊂ C be the ideal generated by J_{2} and this element. Then again modulo J_{0}, which amounts to repeatedly replacing ζ, ξ_{k} and η_{k} by relations in J_{0}, we obtain an element −1 + ζϵ^{−1}b_{2,n}+sϕ(ϵ^{−1}) ∈ J_{0}, where ϕ(ϵ^{−1}) ∈ A[[ϵ^{−1}]]. By iteration and taking the limit, we obtain a canonical element ϵ − h ∈ J for some h ∈ A. It is clear from this construction that h is a unit since b_{2,n} is a unit. This way we have found elements c_{1,k} and c_{2,k} ∈ A for k ≥ 1 and a unit h ∈ A so that

J_{0} = ({ξ_{k} − c_{1,k}, η_{k} − c_{2,k} | k > 0}; ζ − a_{1,n}, ϵ − h) ⊂ C. (2.6)

From this set of generators, we see that the canonical homomorphism A → C/J_{0} is an isomorphism. Since J_{0} is a subideal of J, we have quotient homomorphism C/J_{0} → C/J. We let I_{A} ⊂ A be the kernel of A → C/J_{0} → C/J. We will show that the ideal I_{A} satisfies the property specified in the statement of the Lemma. First, from the construction it is direct to check that I_{A} satisfies the base change property. Now let A → T and φ_{T} be as in the Lemma so that φ_{T} has pure n-contact.

Then it follows from our construction that the ideal I_{T} = {0}. Therefore A → T factors through A/I_{A} → T since I_{T} = I_{A} · T. This proves one direction of the Lemma. Now assume A → T is a homomorphism so that it factors through A/I_{A} → T. Then by the base change property we have I_{T} = {0}. It follows from the construction of I_{T} that φ_{T} does have pure n-contact. This completes the proof of Lemma 2.4. q.e.d.

2.2 Families of predeformable morphisms

In this subsection, we will introduce the notion of predeformable morphisms. We will then prove that for any family of nondegenerate stable morphisms f:X → W [n] over S, the locus S_{pd} ⊂ S of all predeformable morphisms admits a natural closed subscheme structure.

We begin with some notation that will be used throughout this section. We let l be any integer in [n + 1]. Following our convention in Section 1, we let l^{∨} be the complement of l in [n + 1] and C[n]_{[l^{∨}]} the smooth divisor C ×_{A^{1}} H_{l}^{n+1} ⊂ C[n], where H_{l}^{n+1} ⊂ A^{n+1} is the l-th coordinate hyperplane. We define

C_{0}[n] ≜ C[n] ×_{C} 0 = ∪_{l=1}^{n+1}C[n]_{[l^{∨}]}.

We denote by D_{l} the nodal divisor of W [n]×_{C[n]} C[n]_{[l^{∨}]} and by Δ_{l}^{−} and Δ_{l}^{+} its two smooth irreducible components. According to our ordering of the irreducible components of W [n]_{0}, Δ_{l}^{−} is on the left of Δ_{l}^{+}. Then W [n] ×_{C[n]} C[n]_{[l^{∨}]} is the union of Δ_{l}^{−} and Δ_{l}^{+} intersecting transversely along D_{l}. Let π_{l}:C[n] → A^{1} be induced by the l-th projection A^{n+1} → A^{1}. For any point q ∈ D_{l} there is a neighborhood W of q ∈ W [n] with two regular functions w_{1} and w_{2} so that w_{1} = 0 (resp. w_{2} = 0) defines the divisor W ∩ Δ_{l}^{+} (resp. W ∩ Δ_{l}^{−}). By adjusting w_{2} if necessary, we can assume w_{1}w_{2} is the image of t_{l} via Γ(A^{n+1}) → Γ(W). Hence by shrinking W if necessary, we obtain a smooth morphism

(2.7) ψ : W ⟶ Θ_{l} ≜ Spec k[w_{1}, w_{2}] ⊗_{k[t_{l}]} k[t].

Here k[t_{l}]→k[w_{1}, w_{2}] is defined by t_{l} ↦ w_{1}w_{2} and k[t] = k[t_{1}, …, t_{n+1}]. Later we will call W with ψ implicitly understood an admissible chart of W [n] at q ∈ D_{l}.

Let f : X → W [n] be a prestable morphism. We say f is nondegenerate if no irreducible component of X is mapped entirely to any of D_{1}, …, D_{n+1}. Now assume f is nondegenerate and let p ∈ f^{−1}(D_{l}). We define the notion of upper and lower contact order of f along p.

Let X̂ be the formal completion of X along p and let Δ̂_{l}^{−} (resp. Δ̂_{l}^{+}; resp. D̂_{l}) be the formal completion of Δ_{l}^{−} (resp. Δ_{l}^{+}; resp. D_{l}) along f(p). For any irreducible component V̂ of X̂, we can define the contact order of f along V̂ as follows. Assume f(V̂) ⊂ Δ̂_{l}^{−}. We let z be a generator of the maximal ideal of OV̂ and let w be a generator of the ideal I_{D̂_{l}⊂Δ̂_{l}^{−}} ⊂ OΔ̂_{l}^{−}. Let f^{∗}: OΔ̂_{l}^{−} → OV̂ be the induced homomorphism. Then f^{∗}(w) = z^{δ}β for a positive integer δ and a unit β ∈ OV̂. The integer δ is the contact order of f along V̂. The contact order of f along V̂ when f(V̂) ⊂ Δ̂_{l}^{+} is defined similarly. We define the left contact order (resp. right contact order) of f at p, denoted δ(f, p)^{−} (resp. δ(f, p)^{+}), to be the sum of the contact orders of f along all irreducible components of X̂ that are mapped to Δ̂_{l}^{−} (resp. Δ̂_{l}^{+}).

Definition 2.5. Let f : X → W [n] be a nondegenerate prestable morphism. We say f is predeformable along D_{l} if δ(f, p)^{−} = δ(f, p)^{+} for all p ∈ f^{−1}(D_{l}). We say f is predeformable if f is predeformable along all D_{l}.

Note that if f^{−1}(D_{l}) = ∅, then f is automatically predeformable along D_{l}.

In the remainder part of this section, we fix a scheme S over C and a family

f:X → W [n] (2.8)

of nondegenerate prestable morphisms over S. Our goal is to construct a maximal closed subscheme S_{pd} ⊂ S so that it parameterizes all predeformable morphisms in S.

We begin our investigation with the decomposition of f along D_{l}. We let S_{l} be S×_{C[n]}C[n]_{[l^{∨}]} endowed with the reduced scheme structure. We have the following Decomposition Lemma whose proof follows from the standard deformation theory of nodal singularity of curves.

Lemma 2.6. Let the notation be as above. We let X_{l} = X ×_{S} S_{l} and let f_{l} be the restriction of f to X_{l}. Then there are two flat families of curves (not necessarily connected) Y_{−} and Y_{+} over S_{l}, a pair of S_{l}- morphisms g_{±}:Y_{±} → Δ_{l}^{±}, a scheme Σ finite and étale over S_{l} and a pair of S_{l}-immersions ι_{±}:Σ → Y_{±} of which the followings hold: The original family X_{l} is the gluing of Y_{−} and Y_{+} along ι_{−}(Σ) ⊂ Y_{−} and ι_{+}(Σ) ⊂ Y_{+}, and the restriction of f_{l} to Y_{±} ⊂ X_{l} is g_{±}:Y_{±} → Δ_{l}^{±} ⊂ W [n].

Later, we will write X = Y_{−} ⊔ Y_{+} and f = g_{−} ⊔ g_{+} to indicate that (Y_{±}, g_{±}) are the decomposition of (f, X).

Lemma 2.7. Let S_{pd,l} ⊂ S be the set of all closed points ξ ∈ S so that f_{ξ} is predeformable along D_{l}. Then S_{pd,l} is closed in S.

Proof. It suffices to prove the case where S is reduced and irreducible, which we assume now. Let S → A^{1} be induced by S → C[n] ⟶^{π_{l}} A_{l}^{n+1} ≅ A^{1}. In case S is dominant over A^{1}, then Proposition 2.2 implies that S_{pd,l} = S as topological spaces and hence the Lemma holds. Now assume S → A^{1} factor through 0 ∈ A^{1}. Since S is reduced, S → C[n] factors though C[n]_{[l^{∨}]} ⊂ C[n] and then S = S_{l} as schemes. Let ι_{±}: Σ → Y_{±} and g_{±}: Y_{±} → Δ_{l}^{±} be the decomposition of f provided by Lemma 2.6. Let π: Σ → S be the projection. We let ι:Σ → X be the composite ι_{±}:Σ → Y_{±} → X. As before, for any ξ ∈ S we denote by f_{ξ} the restriction of f to the fiber over ξ and denote by Σ_{ξ} the fiber of Σ over ξ. Then the left and the right contact order of f along ι(Σ) define two functions

δ(f_{π(·)}, ι(·))^{∓} : Σ ⟶ ℤ. (2.9)

It follows from the definition that ξ ∈ S_{pd,l} if the following two conditions hold:

(i) For any x ∈ Σ_{ξ}, δ(f_{ξ}, ι(x))^{+} = δ(f_{ξ}, ι(x))^{−}.

(ii) The preimage f_{ξ}^{−1}(D_{l}) is identical to ι(Σ_{ξ}).

Applying the usual semi-continuity theorem, we see that both δ(f_{π(·)}, ι(·))^{±} are upper semicontinuous. Hence (i) and (ii) combined implies that S_{pd,l} is a constructible set. Therefore, to show that S_{pd,l} is closed it suffices to show that if R is a discrete valuation domain with residue field k and quotient field K, and if Spec K → S_{pd,l} extends to Spec R → S, then the extension Spec R → S factor through S_{pd,l} ⊂ S.

Now we let Spec R → S be a morphism so that Spec K factors through S_{pd,l} ⊂ S. We let f_{R}:X_{R} → W [n] be the pull back family over Spec R via Spec R → S. By abuse of notation, we still denote by Y_{±}, g_{±} and (ι_{±}, Σ) the decomposition data of the R-curve (f_{R}, X_{R}). Let η = Spec K be the generic point of Spec R. Then by assumption, Conditions (i) and (ii) hold for ξ = η. Now let ξ ∈ Spec R be any element. As before, for α = − or + we let g_{α,ξ} and Y_{α,ξ} be the restrictions of the respective families to the fiber over ξ. We let g_{α,ξ}^{∗}(D_{l}) be the pullback divisor where D_{l} is viewed as a divisor in Δ_{l}^{α}. Since f_{R} is nondegenerate,

(2.10) deg(g_{α,ξ}^{∗}(D_{l}), Y_{α,ξ}) ≥ ∑_{x∈ι(Σ)∩X_{ξ}} δ(f_{ξ}, ι(x))^{α}, α = − and + .

Clearly, the equality in (2.10) holds for α if g_{α,ξ}^{∗}(D_{l}) = ι_{α}(Σ)∩X_{ξ}. Hence Condition (ii) is equivalent to the assumption that the equalities in the above two inequalities hold. Because the function of contact order is upper semicontinuous, for the closed point η_{0} ∈ Spec R,

∑_{x∈ι(Σ)∩X_{η_{0}}} δ(f_{η_{0}}, ι(x))^{−} ≥ ∑_{x∈ι(Σ)∩X_{η}} δ(f_{η}, ι(x))^{−}.

On the other hand, since f is a flat family of nondegenerate morphisms, we have

deg(g_{−,η_{0}}^{∗}(D_{l}), Y_{−,η_{0}}) = deg(g_{−,η}^{∗}(D_{l}), Y_{−,η}).

Combined with the inequality in (2.10) for α = − and ξ = η_{0} and the equality in (2.10) for α = − and ξ = η, we see that the equality in (2.10) for α = − and ξ = η_{0} must also hold. For similar reasons, the equality in (2.10) also holds for α = + and ξ = η_{0}. Hence the two contact order functions in (2.9) must be constant functions. Therefore, (i) and (ii) hold at η_{0} and hence Spec R → S factors through S_{pd,l} ⊂ S. This shows that S_{pd,l} is closed in S. q.e.d.

Let S_{pd} be the subset of S consisting of ξ ∈ S so that f_{ξ} is predeformable. Clearly, S_{pd} = ∩_{l=1}^{n+1}S_{pd,l}. Hence S_{pd} is closed in S. In the remainder of this section, we will give S_{pd} a canonical closed scheme structure.

We begin with the notion of parameterizations of the formal neighborhood of nodes of prestable curves. Let π : X → S be a family of prestable curves and let X_{node} ⊂ X be the locus of all nodes of the fibers of X over S, endowed with the reduced scheme structure. Let p ∈ X_{node} be any closed point and let V ⊂ X_{node} be an affine neighborhood of p ∈ X_{node}. Then by shrinking V if necessary we can find a scheme U containing V as its closed subscheme so that there is an étale r:U → S extending π|_{V} :V → S.^{5} We let V̂ be the formal completion of U along V. We now argue that V̂ is independent of the choice of U. Indeed, in case r′: U′ → S is another such U and V̂′ is the formal completion of U′ along V. Then since both Û → S and Û′ → S are étale, by the topological invariance of étale morphisms [31, p. 30]^{6} , the scheme V̂ is canonically isomorphic to V̂′. Thus V̂ is independent of the choice of U. Now let XV̂ be X ×_{S} V̂. Clearly, V ⊂ X_{node} is a closed subscheme of XV̂. We let X̂V̂ be the formal completion of XV̂ along V ⊂ XV̂.

5Such (U, r) can be constructed by elementary method. For instance, this follows from the proof of Lemma 1.5 in [23].

6In [31] this theorem was stated for the case V ⊂ V̂ is defined by a nilpotent ideal.

However, the proof can be adopted without change to cover our case.

Lemma 2.8. Let π: X → S be a family of prestable morphisms as before and let p ∈ X_{node} be any closed point. Then there is an affine neighborhood V ⊂ X_{node} of p so that with V̂ and X̂V̂ constructed as before, we have a morphism \underline{ϕ}:V̂ → Spec k[[s]] and a V̂ -isomorphism

ϕ : X̂V̂ ⟶ Spec(k[[z_{1}, z_{2}]] ⊗_{k[[s]]} Γ(OV̂))ˆ ,

where (·)ˆ is the (z_{1}, z_{2})-adic completion of the respective ring. We will call ϕ a parameterization of the formal neighborhood of the nodes of X along V. Furthermore, all such parameterizations satisfy the following uniqueness property. Let ϕ′ be another parameterization of X along V. Then there is a unit u in (k[[z_{1}, z_{2}]]⊗_{k[[s]]} Γ(OV̂))ˆ and a unit ϵ in Γ(OV̂) such that \underline{ϕ′}^{∗}(s) = ϵ\underline{ϕ}^{∗}(s) and that the Γ(OV̂)-automorphism (ϕ′ ∘ ϕ^{−1})^{∗} of R̂ is defined by

(ϕ′ ∘ ϕ^{−1})^{∗}(z_{1}) = z_{1}u and (ϕ′ ∘ ϕ^{−1})^{∗}(z_{2}) = z_{2}u^{−1}ϵ.

Finally, the choice of such (u, ϵ) is unique.

Proof. The existence of such a parameterization follows immediately from the deformation of nodes [4]. This also follows from the local parameterization constructed in [23, Section 1]. Now let ϕ and ϕ′ be two such parameterizations of X along V. Because \underline{ϕ′}^{∗}(s) and \underline{ϕ}^{∗}(s) define the same divisor in Û corresponding to the subscheme where the nodes are not smoothed, there is a unit ϵ ∈ Γ(OV̂) such that \underline{ϕ′}^{∗}(s) = ϵ\underline{ϕ}^{∗}(s). Hence ϕ^{′∗}(z_{1}z_{2}) = ϵϕ^{∗}(z_{1}z_{2}) and hence

(ϕ′ ∘ ϕ^{−1})^{∗}(z_{1}z_{2}) = (ϕ^{∗−1} ∘ ϕ^{′∗})(z_{1}z_{2}) = ϵz_{1}z_{2}. (2.11)

It follows from the deformation of nodes that there are units u_{1} and u_{2} in R̂ such that

(ϕ′ ∘ ϕ^{−1})^{∗}(z_{1}) = z_{1}u_{1} and (ϕ′ ∘ ϕ^{−1})^{∗}(z_{2}) = z_{2}u_{2}. (2.12)

Using the above relation, we obtain su_{1}u_{2} = sϵ. Hence u_{2} = ϵ(u_{1}^{−1} + e) for some e ∈ R̂ satisfying se = 0. We now demonstrate that if we choose u_{1}, u_{2} and ϵ appropriately, then e will be zero. We first write e in its normal form

e = α_{0} + ∑_{i≥1} z_{1}^{i}α_{i} + z_{2}^{i}β_{i}, α_{i}, β_{i} ∈ Γ(Û).

Since se = 0, by the uniqueness of the normal form we have z_{1}β_{i} = 0 and z_{2}α_{i} = 0 for i ≥ 1 while sα_{0} = 0. We let u_{1} = γ_{0} + ∑ z_{1}^{i}γ_{i} + z_{2}^{i}δ_{i} be its normal form. Since u_{1} is a unit, γ_{0} must be a unit. We now let ϵ′ = ϵ + γ_{0}α_{0}. Then since sα_{0} = 0, we can replace ϵ in (2.11) by ϵ′. Then the new error term e′ = u_{2} − u_{1}^{−1}ϵ′, which obeys u_{2} = ϵ′(u_{1}^{−1} + e′), belongs to the ideal (z_{1}, z_{2}). Hence without loss of generality we can assume the ϵ, u_{1} and u_{2} are chosen a priori so that the α_{0} in the normal form of e is zero. On the other hand, since z_{2}α_{i} = 0 for i > 0, we can replace u_{2} by u′_{2} = u_{2} − ∑_{i>0} z_{1}^{i}α_{i} so that (2.11) and (2.12) still hold. Hence we can assume a priori that all α_{i} = 0. Lastly, we let u′_{1} = u_{1}(1 + eu_{1}ϵ^{−1})^{−1}. Clearly, z_{1}u_{1} = z_{1}u′_{1} and u_{2} = ϵu_{1}^{′−1} since u_{2} = u_{1}^{−1}ϵ + e. Thus (ϵ, u) with u = u′_{1} is what we want.

The uniqueness of (u, ϵ) can be proved following same strategy. Since a more general version will be proved in [23, Section 1], we will omit the proof here. q.e.d.

We keep the integer l ∈ [n] and continue to use the notation developed so far. Let π : Σ → S_{l} be the projection and let p ∈ Σ be any closed point so that π(p) ∈ S_{pd,l}. We pick a connected neighborhood V of p ∈ Σ and a local parameterization ϕ of the formal neighborhood of the nodes of X along V. Let A = Γ(OV̂) and let V̂ = Spec A. Let R̂ be the (z_{1}, z_{2})-adic completion of k[[z_{1}, z_{2}]] ⊗_{k[[s]]} A. Without loss of generality, we can assume that the image f(V) is contained in an admissible chart (W, ψ) of W [n] constructed in the beginning of this subsection. Let

φ : k[w_{1}, w_{2}] ⟶ R̂ (2.13)

be the morphism induced by f, the parameterization ϕ and ψ, and the projection Θ_{l} → Spec k[w_{1}, w_{2}]. Let 𝔪 be the ideal of the point ξ ∈ V, where ξ = π(p). Since f_{ξ} is a predeformable morphism, the left and the right contact order of f_{ξ} at p coincide, which we denote by n. Hence possibly after exchanging z_{1} and z_{2}, near p we have f_{ξ}^{∗}(w_{i}) = c_{i}z_{i}^{n} for some units c_{i} in k[[z_{1}, z_{2}]]. Hence in the normal form φ(w_{i}) = ∑ a_{i,j}z_{1}^{j} + b_{i,j}z_{2}^{j}, as in (2.3), both a_{1,n} and b_{2,n} are not in 𝔪. Hence for sufficiently small neighborhood V_{0} of p ∈ V both a_{1,n} and b_{2,n} are units in B = Γ(O_{V_{0}}). Therefore we can apply Lemma 2.4 to obtain an ideal I_{B} ⊂ B for such B.

Definition 2.9. Let the notation be as before. Let V ⊂ Σ be an affine neighborhood of p ∈ Σ so that f(V) is contained in an admissible neighborhood (W, ψ) of f(p) ∈ W [n]. We say f is predeformable near p if for some neighborhood V_{0} of p ∈ V both a_{1,n} and b_{2,n} are units in B = Γ(O_{V_{0}}) and further the ideal I_{B} ⊂ B just constructed is the zero ideal in B. The family f is said to be predeformable along D_{l} if it is predeformable at every p ∈ f^{−1}(D_{l}). We say f is predeformable if it is predeformable along all divisors D_{l}.

A remark about this definition is in order. First, in defining the notion of predeformable at p we need to choose a parameterization of the formal neighborhood of the nodal singularity of X along V. Because of the second part of Lemma 2.8, this definition does not depend on this choice. For the same reason, this definition is also independent of the choice of (W, ψ).

We still denote by f a family of nondegenerate morphisms to W [n] over S. We will conclude this section by showing that there is a canonical closed subscheme structure on S_{pd} ⊂ S making it the closed subscheme parameterizing all predeformable morphisms over S.

Definition 2.10. Let f be a family of nondegenerate morphisms to W [n] over S. We define a functor 𝔉_{pd,l}[f, S] (resp. 𝔉_{pd}[f, S]) that associates to any scheme T over C the set of morphisms T → S so that the pullback family f_{T} : X_{T} → W [n] is predeformable along the divisor D_{l} (resp. all divisors D_{l}).

When the choice of f is apparent from the context, we will omit the reference f and abbreviate 𝔉_{pd,l}[f, S] to 𝔉_{pd,l}[S].

Theorem 2.11. Let f : X → W [n] be a family of nondegenerate stable morphisms. Then the functor 𝔉_{pd,l}[S] and 𝔉_{pd}[S] are represented by closed subschemes of S.

Proof. Clearly, should 𝔉_{pd,l}[S] be represented by a subscheme in S, then set-theoretically it must be S_{pd,l}. Hence it suffices to give S_{pd,l} a closed scheme structure so that it represents the functor 𝔉_{pd,l}[S]. As before, we let S_{l} be S ×_{A^{n+1}} H_{l}^{n+1} endowed with the reduced scheme structure and Ŝ_{l} be the formal completion of S along S_{l}. We let V̂ be any connected component of Ŝ_{l} endowed with the open scheme structure and let V be V̂ endowed with the reduced scheme structure. Our first objective is to show that S_{pd,l} ∩ V̂ admits a canonical scheme structure (as closed scheme of V̂) so that it represents the functor 𝔉_{pd,l}[V̂]. We continue to use the notation developed so far. For instance (Y_{±}, g_{±}) and (ι_{±}, Σ) are the decomposition data of the restriction of f to V ⊂ S along the divisor D_{l}. We let Σ̂ → V̂ be the étale morphism so that Σ̂ ×V̂ V ≅ Σ. We let π:Σ → V and π̂:Σ̂ → V̂ be the projections.

Let ξ ∈ S_{pd,l} ∩ V be any closed point and let p_{1}, …, p_{k} be the set of points in π^{−1}(ξ). Note that p_{1}, …, p_{k} are exactly the preimage set f_{ξ}^{−1}(D_{l}). Hence each p_{α} is associated with an integer n_{α} that is the contact order of f_{ξ} at p_{α}. For each α between 1 and k, we pick an affine open neighborhood Û_{α} of p_{α} ∈ Σ̂. We let X_{α} = X ×_{S} Û_{α} and let f_{α}: X_{α} → W [n] be the pullback of f to X_{α}. We let U_{α} be Û_{α} ∩ Σ. Then the multi-section ι:Σ → X defines a section ι_{α}:U_{α} → X ×Û_{α} U_{α}. Without loss of generality, we can assume X_{α} admits a parameterization of the formal neighborhood of its nodes along ι_{α}(U_{α}). By shrinking U_{α} if necessary, we can assume that there is an admissible chart (W_{α}, ψ_{α}) of W [n] near f_{ξ}(p_{α}) so that f_{α}(ι_{α}(U_{α})) is entirely contained in W_{α}. Therefore, following the construction associated to Definition 2.9, by shrinking U_{α} if necessary we have an ideal sheaf I_{α} ⊂ OÛ_{α} so that it defines the subscheme of Û_{α} that parameterizes all morphisms in the family f_{α}: X_{α} → W [n] that are predeformable and have contact order n_{α} along the nodes ι_{α}(U_{α}) ⊂ X_{α}.

To obtain the ideal sheaf of OV̂ that defines the desired closed subscheme structure of S_{pd,l} ∩ V̂ we need to descend the “intersection” of I_{α} since S_{pd,l} is defined by the requirement that all nodes are predeformable. As before, we denote by k the number of elements in π^{−1}(ξ) for ξ ∈ V. Since V is connected and Σ → V is étale and finite, k is independent of the choice of ξ ∈ V. We consider the fiber product, denoted Σ̂^{×k}, of k-copies of Σ̂ over V̂. We denote by Σ^{×k} accordingly the fiber product of k copies of Σ over V. Note that points in Σ^{×k} are (a_{1}, …, a_{k}) with not necessarily distinct a_{1}, …, a_{k} ∈ π^{−1}(ξ) for some ξ ∈ V. We let Γ ⊂ Σ^{×k} be the subset of those points (a_{1}, …, a_{k}) ∈ Σ^{×k} so that all a_{1}, …, a_{k} are distinct. Since Σ → V is finite and étale, and that each fiber of π:Σ → V has cardinal k, Γ is an open subset of Σ^{×k}. Then since Σ^{×k} is homeomorphic to Σ̂^{×k}, Γ is an open subset of Σ̂^{×k}. We let Γ̂ be Γ ⊂ Σ̂^{×k} endowed with the open scheme structure. Then the obvious projection Γ̂ → V̂ is étale and finite. Now let p̂r_{α}: Γ̂ → Σ̂ be the morphism induced by the α-th projection of Σ̂^{×k}, which is étale.

Now consider any point η = (p_{1}, …, p_{k}) ∈ Γ̂. To each α between 1 and k we let Û_{α} ⊂ Σ̂ be the neighborhood of p_{α} ∈ Σ̂ and I_{α} ⊂ OÛ_{α} the ideal sheaf just constructed. We let Û ⊂ Γ̂ be an open neighborhood of η so that p̂r_{α}(Û) ⊂ Û_{α} for each α. We then let Ĩ_{α} be the ideal sheaf of OÛ that is the pullback of I_{α} ⊂ OÛ_{α} under p̂r_{α}|Û: Û → Û_{α}, and let J ⊂ OÛ be the intersection J = Ĩ_{1} ∩ ⋯ ∩ Ĩ_{l}. This ideal sheaf defines a close subscheme of Û, denoted by Z(J). Now let Φ : Γ̂ → Ŝ be the composite Γ̂ → V̂ → Ŝ. It may happens that Z(J) and Φ^{−1}(S_{pd,l}) are different as subsets of Γ̂. However, by the proof of the closedness of S_{pd,l}, the connected component of Z(J) containing η coincides with the connected component of Φ^{−1}(S_{pd,l}) containing η. Hence by shrinking Û if necessary, we can assume Z(J) is connected. Now let fÛ be the family over Û that is the pullback of the family f:X → W [n] via Γ̂ ⟶^{Φ} Ŝ → S. Since for ξ ∈ Û the preimage f_{ξ}^{−1}(D_{l}) has exactly k points, it is direct to check that the subscheme Z(J) represents the functor 𝔉_{pd,l}[Û].

Now by repeating this procedure, we can cover Γ̂ by open subsets Û_{β} with ideal sheaves J_{β} ⊂ OÛ_{β}, for β = 1, …, L, such that the closed subscheme Z(J_{β}) defined by the ideal sheaf J_{β} represents the functor 𝔉_{pd,l}[Û_{β}]. By the universal property of these schemes, whenever Û_{α} ∩ Û_{β} ≠ ∅ the ideal sheaves J_{α} and J_{β} coincide over Û_{α} ∩ Û_{β}. Thus they patch together to form an ideal sheaf J ⊂ OΓ̂. Again, by the universal property, this ideal sheaf forms a descent data for the faithfully flat étale morphism Γ̂ → V̂. Hence it descends to an ideal sheaf I ⊂ OV̂ which defines a closed subscheme Z(I). It follows from the previous construction that Z(I) represents the functor 𝔉_{pd,l}[V̂].

We now show that the functor 𝔉_{pd,l}[S] is represented by a closed subscheme. Since each connected component V̂ of Ŝ_{l} has a closed scheme structure that represents the functor V̂, the functor 𝔉_{pd,l}[Ŝ_{l}] is represented by a closed subscheme of Ŝ_{l}. We denote this subscheme by Ŝ_{pd,l}. It remains to show that there is a closed scheme structure on S_{pd,l} so that it represents the functor 𝔉_{pd,l}[S]. As before, we let t_{l} be the l-th standard parameter of A^{n}. We let K ⊂ O_{S} be the subsheaf of elements annihilated by some power of t_{l}. Since S is noetherian, there is an integer m_{0} so that t_{l}^{m_{0}}K ⊂ O_{S} is zero. Similarly, we let K̂ be the subsheaf of OŜ consisting of elements annihilated by some power of t_{l}. Then K̂ is canonically isomorphic to K. As an ideal sheaf, K̂ defines a closed subscheme Ŝ_{fl} ⊂ Ŝ_{l}. Clearly, Ŝ_{fl} is flat over Spec k[t_{l}] near 0. Hence by Proposition 2.2, the restriction of the family f to Ŝ_{fl} is predeformable along D_{l}. Consequently, the ideal sheaf of Ŝ_{pd,l} ⊂ Ŝ_{l} is contained in K̂. Using the natural isomorphism K̂ ≅ K, this ideal sheaf defines an ideal sheaf J of O_{S}. By our construction the support of the subscheme defined by J coincides with S_{pd,l}. We define the closed scheme structure of S_{pd,l} to be the one defined by J. It is straightforward to check that with this closed scheme structure, S_{pd,l} represents the functor 𝔉_{pd,l}[S].

As to 𝔉_{pd}[S], it is clear that it is represented by the intersection of the closed subschemes ∩_{l=1}^{n}S_{pd,l}. This completes the proof of Theorem 2.11.

q.e.d.

3. Moduli stack of stable morphisms

In this section, we will introduce the notion of stable morphisms to the stack 𝔚. We will then show that the moduli of such stable morphisms form an algebraic stack (i.e., a Deligne-Mumford stack).

3.1 Stable morphisms

We first define the notion of stable morphisms in our setting. Let S be any C-scheme. A prestable morphism to 𝔚(S) consists of a triple (f, X , W) where X is a family^{7} of marked connected, prestable curves over S, W is a member in 𝔚(S) and f is an S-morphism f : X → W that is stable as a family of morphisms to W. When there is no confusion, we will abbreviate the triple (f, X , W) to f. Let (f′, X′, W′) be another prestable morphism over S. An isomorphism from f to f′ consists of an S-isomorphism r_{1} : X → X′ (as pointed curves) and an arrow r_{2} : W → W′ so that r_{2} ∘ f = f′ ∘ r_{1}. When f ≡ f′, then such isomorphisms are called the automorphisms of f. To define the group scheme of the automorphisms of f, we follow the usual procedure. Let 𝔄𝔲𝔱_{𝔚}(f) be the functor that associates any S-scheme T the set of all automorphisms of f ×_{S} T. 𝔄𝔲𝔱_{𝔚}(f) is represented by a group scheme over S, denoted Aut_{𝔚}(f) and called the automorphism group of f.

Let f be a map f : X → W [n] with X proper and reduced. We fix an ample line bundle H on W. Using W [n] → W, H pulls back to a line bundle over W [n], which we denote by H if there is no confusion. We define the degree of f to be the degree of f^{∗}H over X. We fix a triple of integers Γ = (b, g, k) where k will be the number of marked points of the domain curve, g will be the arithmetic genus of the domain curve and d will be the degree of the map. We say f:X → W [n] is a prestable map of topological type Γ if f : X → W [n] is an ordinary stable morphism whose domain is a connected, k-pointed arithmetic genus g nodal curve and f has degree b. We fix such an H and Γ once and for all in this section.

Definition 3.1. Let S be any C-scheme and f : X → W be a family of prestable morphisms to 𝔚(S). Suppose W is an effective degeneration, say given by a C-morphism r: S → C[n]. Then we say f is a family of predeformable morphisms to 𝔚(S) if the associated morphism f̃: X → W [n] is a family of predeformable morphisms; We say f is a family of stable morphisms to 𝔚(S) if in addition to f being a family of predeformable morphisms we have that for every point ξ ∈ S the automorphism group Aut_{𝔚}(f_{ξ}) of f_{ξ}: X_{ξ} → W [n] is finite. We say f is a family of topological type Γ if for every ξ ∈ S the restriction of f to the fiber of X over ξ is of topological type Γ. In general, given a family of prestable morphisms f to 𝔚(S), we say that f is a family of predeformable (resp. stable, resp. of topological type Γ) morphisms to 𝔚(S) if there is an open covering S_{α} of S such that to each α the family W ×_{S} S_{α} is an effective family over S_{α} and the restriction of f to X ×_{S} S_{α} is a family of predeformable (resp. stable, resp. of topological type Γ) morphisms to 𝔚(S_{α}).

7All families are flat families unless otherwise is mentioned.

Clearly, the definition of stability does not depend on the choice of the local representatives of the family W. The goal of this section is to construct the moduli stack of all marked stable morphisms to 𝔚 of given topological type Γ. In the sequel of this paper, we will use this moduli stack to study the degeneration of GW-invariants.

We begin with the definition of the groupoid of stable morphisms to 𝔚. We fix the relative ample line bundle H on W and the triple Γ. We define 𝔐(𝔚, Γ) to be the category whose objects are all families of stable morphisms to 𝔚 of topological type Γ. We define 𝔭:𝔐(𝔚, Γ) → (Sch/C) to be the morphism that sends families over S to S. Let ξ = {f : X → W} ∈ 𝔐(𝔚, Γ)(S), ξ′ = {f′ : X′ → W′} ∈ 𝔐(𝔚, Γ)(S′) and let h : S′ → S be an arrow (a morphism). An arrow r : ξ′ → ξ with 𝔭(r) = h consists of an arrow W′ → W covering h (i.e., a W_{S′}- isomorphism W ×_{S′} S′ ≅ W′) and an S′-isomorphism X ×_{S} S′ ≅ X′ (as marked curves) so that

X′  ⟶^{f′} W′
↓≅  ↓≅
X ×_{S} S′ ⟶^{f×_{S}S′} W ×_{S} S′

is commutative. In such case, we call ξ′ the pullback of ξ via h:S′ → S. Obviously, this makes (𝔐(𝔚, Γ), 𝔭) a groupoid.

Before proceeding, we need to study the structure of stable morphisms to 𝔚. Let f : X → W [n] be a predeformable morphism with B ⊂ X the divisor of the marked points. We assume that f factor through W [n]_{0} ⊂ W [n]. By the decomposition Lemma, f splits X, according to the nodal divisors of W [n]_{0}, into components X_{1}, …, X_{n+2} (some of them may be empty), marked points ι_{i}^{+} : Σ_{i} → X_{i+1} and ι_{i}^{−} : Σ_{i} → X_{i}, where Σ_{i} is a finite set, so that f(X_{i}) ⊂ Δ_{i} ⊂ W [n]_{0} and that X is the result of successively gluing X_{i} and X_{i+1} along ι_{i}^{+} :Σ_{i} → X_{i+1} and ι_{i}^{−}:Σ_{i} → X_{i}. Namely,

X = X_{1} ⊔ ⋯ ⊔ X_{n+2} and f = f_{1} ⊔ ⋯ ⊔ f_{n+2}.

Now let A be any purely one-dimensional closed subset of X. We define the weight of f along A to be

ω(f, A) = deg(A, f^{∗}H) + 2g(A) − 2 + #(B ∩ A) + #τ(A),

where τ(A) is the set of smooth points of A that are nodes of X. (We agree ω(f, A) = 0 if A = ∅.) Note that if A_{1} and A_{2} are two such purely one-dimensional closed subsets with no common irreducible components, then

ω(f, A_{1} ∪ A_{2}) = ω(f, A_{1}) + ω(f, A_{2}).

Lemma 3.2. The predeformable morphism f : X → W [n]_{0} is a stable morphism in 𝔐(𝔚, Γ)(k) if and only if ω(f, X_{i}) > 0 for all i = 2, …, n + 1. ^{8}

Proof. Let i be any integer between 2 and n + 1. We let

Ψ_{i}:G_{m} × Δ_{i} ⟶ Δ_{i}

be the group action given in Corollary 1.3. For any closed purely onedimensional subset A of X_{i} we define the relative automorphism group Aut_{𝔚}(f, A)_{rel} to be the set of pairs (a, b) where a: A → A is an automorphism leaving B ∩ A and τ(A) fixed and b is an element in G_{m} such that Ψ_{i}(b, f_{i}(w)) = f_{i}(a(w)) for all w ∈ A. Let σ ∈ Aut_{𝔚}(f) be any automorphism. Then σ induces a permutation of Σ_{i} for each i. As a result, it defines a homomorphism

Aut_{𝔚}(f) ⟶ S_{Σ_{1}} × ⋯ × S_{Σ_{n+1}},

where S_{Σ_{i}} is the permutation group of Σ_{i}. Clearly, its kernel is

Aut_{𝔚}(f, X_{1})_{rel} × ⋯ × Aut_{𝔚}(f, X_{n+2})_{rel}.

8For simplicity we agree 𝔐(𝔚, Γ)(Spec k) = 𝔐(𝔚, Γ)(k).

Since f is an ordinary stable morphism, Aut_{𝔚}(f, X_{1})_{rel} and Aut_{𝔚}(f, X_{n+2})_{rel} are finite. Hence Aut_{𝔚}(f) is finite if and only if Aut_{𝔚}(f, X_{i})_{rel} are finite for i = 2, …, n + 1.

Now assume A is a connected component of X_{i} so that f(A) is not a single point set and that Aut_{𝔚}(f, A)_{rel} is infinite. Then the following must hold: f(A) must be contained in a fiber of Δ_{i} → D; A must be a smooth rational curve; A contains no marked points and τ(A) consists of exactly two nodal points of X, one lies in ι_{i−1}^{+}(Σ_{i−1}) and the other lies in ι_{i}^{−}(Σ_{i}). Further if we let P^{1} ⊂ Δ_{i} be the image of A then f|_{A} : A → P^{1} is a branched covering ramified at τ(A). We call such connected components trivial components. Clearly, if A is not a trivial component, then Aut_{𝔚}(f, A)_{rel} is finite. On the other hand, if A is a trivial component then Aut_{𝔚}(f, A)_{rel} → G_{m} is surjective. Hence Aut_{𝔚}(f, X_{i})_{rel} is finite if and only if not all connected components of X_{i} are trivial.

To complete the proof of the Lemma, it suffices to show that ω(f, X_{i}) = 0 if and only if all connected components of X_{i} are trivial. Let A be any connected component of X_{i}. In case deg(A, f^{∗}H) > 0, then since H is very ample we can assume without loss of generality that deg(A, f^{∗}H) > 3, and hence ω(f, A) > 0. In case deg(A, f^{∗}H) = 0, then f(A) must be contained in a fiber of Δ_{i} → D. Since deg(X, f^{∗}H) > 0 and X is connected, f(A) can not be a single point. Hence f(A) is a fiber of Δ_{i} → D. Because f is predeformable, A contains at least two nodal points of X, one lies in f^{−1}(D_{i−1}) and the other lies in f^{−1}(D_{i}). Hence ω(f, A) ≥ 0. Moreover, the equality holds if and only if g(A) = 0, A contains no marked points and contains exactly two nodal points of X. Again since f is predeformable, it is easy to see that this is possible only if A is a trivial component. This shows that any connected component A of X_{i} has ω(f, A) ≥ 0 and the equality holds if and only if A is a trivial component.

Now we assume f is stable. Then to each i between 2 and n + 1 at least one connected component of X_{i} is nontrivial. Then combined with the fact that the weight of any connected components of X_{i} are nonnegative, we have ω(f, X_{i}) > 0. Conversely, if for some i between 2 and n + 1 the weight ω(f, X_{i}) = 0, then all connected components of X_{i} are trivial. Since to each component A of X_{i} the automorphism group Aut_{𝔚}(f, A)_{rel} surjects onto G_{m}, Aut_{𝔚}(f, X_{i})_{rel} will surjects onto G_{m} as well. Thus Aut_{𝔚}(f, X)_{rel}, and hence Aut_{𝔚}(f) is infinite. This proves Lemma 3.2. q.e.d.

For convenience we define the norm of the triple Γ = (b, g, k) to be |Γ| = b + 2g − 2 + k.

Lemma 3.3. Let f : X → W [n]_{0} be a stable morphism in 𝔐(𝔚, Γ)(k) and let X_{1}, …, X_{n+2} be the splitting components of X. Then

|Γ| = ω(f, X_{1}) + ⋯ + ω(f, X_{n+2}).

We need a more general form of this Lemma. Let S = Spec R be a C-scheme where R is a discrete valuation domain. Let f : X → W be a family of stable morphisms in 𝔐(𝔚, Γ)(S) and ι: S → C[n] be the morphism associated to W. Let η_{0} ∈ S be the closed point and η ∈ S be the generic point. Let Δ_{1}^{0}, …, Δ_{n_{0}+2}^{0} be the (n_{0} + 2)-irreducible components of W [n] ×_{C[n]} ι(η_{0}) and Δ_{1}, …, Δ_{n_{1}+2} be the (n_{1} + 2)- irreducible components of W [n] ×_{C[n]} ι(η). We say Δ_{i}^{0} ⪯ Δ_{j} if Δ_{i}^{0} is contained in the closure of Δ_{j}.

Lemma 3.4. Let the notation be as before and let X_{i}^{0} (resp. X_{j}) be the splitting components of X ×_{X} η_{0} (resp. X ×_{S} η). Then for each j ∈ [1, n_{1} + 2],

ω(f_{η}, X_{j}) = ∑_{Δ_{i}^{0}⪯Δ_{j}} ω(f_{η_{0}}, X_{i}^{0}).

Proof. The proofs of these two Lemmas are standard and will be omitted. q.e.d.

We next introduce the standard choice of ample line bundle on the domain and the target of a stable morphism in 𝔐(𝔚, Γ). We begin with line bundles on W [n]. As before, we fix the relative ample line bundle H on W/C and its pullback on W [n], still denoted by H. For l ∈ [n], the divisor W [n] ×_{C[n]} C[n]_{[l^{∨}]} ⊂ W [n] is the union of two smooth irreducible components, one left and one right according to our convention of ordering. We denote the right component by Ξ_{l}. (Note that Ξ_{l} ×_{C[n]} 0 contains the last irreducible component of W [n]_{0}.) Then associating to any sequence of integers [a] = (a_{1}, …, a_{n}) we define a line bundle

H_{[a]} = H(a_{1}Ξ_{1} + ⋯ + a_{n+1}Ξ_{n+1})

on W [n]. Since W [n]×_{C} (C −0) → W ×_{C} (C −0) is G[n]-equivariant and since restricting to W [n] ×_{C} (C − 0) the line bundle H_{[a]} is the pullback of H from W ×_{C} (C − 0), the trivial G[n]-linearization of H over W pullback to a unique G[n]-linearization of H_{[a]} over W [n] ×_{C} (C − 0).

Further, this linearization extends to a G[n]-linearization of H_{[a]} over W [n]. We call this the canonical G[n]-linearization of H_{[a]}.

Let Δ → D be the ruled variety that is one of the middle components of W [n]_{0}, where n > 1. We denote by H_{Δ} the pullback of the ample line bundle H via Δ → D → W. Without loss of generality, we can assume H on W is sufficiently ample so that for any integer 0 ≤ a < b ≤ |Γ| the divisor H_{Δ}(aD_{+} + bD_{−}) is ample on Δ, where D_{±} are the two distinguished divisors of Δ. We now choose a standard choice of ample line bundle on the target of f : X → W [n]_{0}. Let X_{1}, …, X_{n+2} be the splitting components of X. We define

a_{i}(f) = ∑_{j=1}^{i} ω(f, X_{j}),

and define the standard line bundle on W [n]_{0} (associated to f) to be

H_{[f]} = H(a_{1}(f)Ξ_{1} + ⋯ + a_{n+1}(f)Ξ_{n+1}).

Because {a_{i}(f)} is a strictly increasing sequence, H_{[f]} is ample on W [n]_{0}.

Let S be any scheme and f : X → W be a family in 𝔐(𝔚, Γ)(S). We now extend this construction to the family W/S. Let ξ ∈ S be any closed point and let U be an open neighborhood of ξ ∈ S so that W ×_{S} U is isomorphic to the pullback ρ^{∗}W [n] for a ρ : U → C[n]. Without loss of generality, we can assume ρ(ξ) = 0 ∈ C[n]. Let Ξ_{1}, …, Ξ_{n} be the divisors of W [n] mentioned before. Let f_{ξ} : X_{ξ} → W [n]_{0} be the restriction of f to the fiber over ξ. We then form the line bundle

H_{[f_{ξ}]} = H(a_{1}(f_{ξ})Ξ_{1} + ⋯ + a_{n+1}(f_{ξ})Ξ_{n+1})

on W [n]. Because of Lemma 3.4, after shrinking U if necessary, for any η ∈ U the restriction of H_{[f_{ξ}]} to W_{η} = W [n] ×_{C[n]} ρ(η) is the standard choice of ample line bundle associated to f_{η}. We define the standard choice of ample line bundle on W ×_{S} U to be ρ^{∗}H_{[f_{ξ}]} and denote it by H_{[f,U]}. Repeating this procedure, we can find an open covering U_{α} of S so that each W×_{S}U_{α} is isomorphic to ρ_{α}^{∗}W [n_{α}] for some ρ_{α}:U_{α} → C[n_{α}] and over W ×_{S} U_{α} we have the standard choice of ample line bundle H_{[f,U_{α}]}. By Lemma 1.8 we can assume that whenever U_{αβ} = U_{α}∩U_{β} ≠ ∅ then there is an arrow φ_{αβ} making the diagram

ρ_{α}^{∗}W [n_{α}] ×_{U_{α}} U_{αβ} ⟶^{φ_{βα}} ρ_{β}^{∗}W [n_{β}] ×_{U_{β}} U_{αβ}
↓  ↓
X ×_{S} U_{αβ}  X ×_{S} U_{αβ}

commute. Using the G[n]-linearization, such an arrow defines a canonical isomorphism

ψ_{βα} : φ_{βα}^{∗}H_{[f,U_{β}]} ≅ H_{[f,U_{α}]}.

Since ψ_{βα} is canonical, the collection {H_{[f,U_{α}]}} forms a line bundle on W. In the following, we will call the collection H_{[f,U_{α}]} the standard choice of relative ample line bundle on W and denote it by H_{[f,W]}.

The standard choice of the ample line bundle on the domain of f is

H_{[f,X]} = f^{∗}H_{[f,W]}(5B) ⊗ ω_{X /S}^{⊗5},

where ω_{X /S} is the relative dualizing sheaf and B ⊂ X is the divisor of the marked points.

3.2 Basic Properties of 𝔐(𝔚, Γ)

In this subsection, among other things, we will show that 𝔐(𝔚, Γ) is a separated and proper algebraic stack over C.

Proposition 3.5. The groupoid 𝔐(𝔚, Γ) over C is a stack.

Proof. Following the definition of stacks (cf. [40]), it suffices to show the following:

(i) For any scheme S over C and two stable morphisms ξ_{1} and ξ_{2} ∈ 𝔐(𝔚, Γ)(S), the functor

ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) : (Sch/S) ⟶ (Sets)

which associates to any morphism φ: T → S the set of isomorphisms in 𝔐(𝔚, Γ)(T) between φ^{∗}ξ_{1} and φ^{∗}ξ_{2} is a sheaf in the étale topology.

(ii) Let {S_{i} → S} be a covering of S (over C) in the étale topology. Let ξ_{i} ∈ 𝔐(𝔚, Γ)(S_{i}) and let φ_{ij} : ξ_{i}|_{S_{i}×_{S}S_{j}} → ξ_{j}|_{S_{i}×_{S}S_{j}} be isomorphisms in 𝔐(𝔚, Γ)(S_{i} ×_{S} S_{j}) satisfying the cocycle condition. Then there is a ξ ∈ 𝔐(𝔚, Γ)(S) with isomorphism ψ_{i}: ξ|_{S_{i}} → ξ_{i} so that

φ_{ij} = (ψ_{i}|_{S_{i}×_{S}S_{j}}) ∘ (ψ_{j}|_{S_{i}×_{S}S_{j}})^{−1}.

We first prove (i). Let ξ_{i}, i = 1 and 2, be represented by f_{i}:X_{i} → W_{i}. To each morphism ρ: U → S we define ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U) to be the set of all arrows from ρ^{∗}ξ_{1} to ρ^{∗}ξ_{2}. This defines a functor

ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) : (étale opens in S) ⟶ (Sets).

Clearly it is a presheaf. To show that this is a sheaf it suffices to show that if U = ∪_{j∈J}U_{j} is an étale open covering of U, then

ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U) ⟶^{e} ∏_{j∈J} ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U_{j})

⟶^{p_{1},p_{2}} ∏_{i,j∈J} ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U_{i} ×_{S} U_{j})

is an equalizer diagram. This means that if r = {r_{j}|j ∈ J} is an element in the middle term above, then

r_{i}|_{U_{i}×_{S}U_{j}} ≡ r_{j}|_{U_{i}×_{S}U_{j}} (3.1)

for all i, j ∈ J if and only if there is an r ∈ ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U) so that its restriction to U_{i} is r_{i}. By definition, each r_{i} consists of two isomorphisms r_{i,1} and r_{i,2} and a commutative diagram

X_{1}|_{U_{i}} ⟶^{f_{1}|_{U_{i}}} W_{1}|_{U_{i}}
r_{i,1}↓  ^{r_{i,2}}↓
X_{2}|_{U_{i}} ⟶^{f_{2}|_{U_{i}}} W_{2}|_{U_{i}}.

The condition (3.1) makes {r_{i,1}|i ∈ J} a descent data for an isomorphism between X_{1} and X_{2} and makes {r_{i,2}|i ∈ J} a descent data for an isomorphism between W_{1} and W_{2}. Since ∐ U_{i} → U is an étale covering, by Descent Lemma there is an isomorphism r_{1} : X_{1} → X_{2} and r_{2} : W_{1} → W_{2} restricting to r_{i,1} and r_{i,2} respectively. Hence r = (r_{1}, r_{2}) ∈ ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2})(U) is the element so that e(r) = r. This proves (i).

The proof of (ii) is similar. It amounts to construct a family X over S, a family W in 𝔚(S) and an S-morphism f : X → W, which form an element ξ ∈ 𝔐(𝔚, Γ)(S), so that ξ_{i} = ξ|_{S_{i}} and the identity map is φ_{ij}. This is again a standard application of Descent Lemma and will be omitted. q.e.d.

Lemma 3.6. Let S be any C-scheme and let ξ_{1}, ξ_{2} ∈ 𝔐(𝔚, Γ)(S) be any two families over S. Then the functor ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) is represented by a scheme quasiprojective over S.

Proof. We let ξ_{i} be represented by f_{i} : X_{i} → W_{i}. Let H_{[f_{i},W_{i}]} and H_{[f_{i},X_{i}]} be the standard choices of relative ample line bundles on W_{i} and on X_{i}. It follows from the uniqueness of these line bundles that whenever T → S is a morphism and (r_{1}, r_{2}) is a pair of isomorphisms shown below that makes the diagram

X_{1} ×_{S} T ⟶^{r_{1}} X_{2} ×_{S} T
f_{1}↓  ^{f_{2}}↓
W_{1} ×_{S} T ⟶^{r_{2}} W_{2} ×_{S} T.

commutative then

r_{1}^{∗}H_{[f_{2},X_{2}]} ≅ H_{[f_{1},X_{1}]} and r_{2}^{∗}H_{[f_{2},W_{2}]} ≅ H_{[f_{1},W_{1}]}.

Hence ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) is isomorphic to the functor associating to T → S the set of pairs of isomorphisms of polarized projective schemes over T satisfying the obvious compatibility condition. It is a routine application of Grothendieck’s results on the representability of the Hilbert scheme and related functor [33], see also [4, p. 84], that the functor ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) is represented by a scheme quasiprojective over S. This completes the proof of Lemma 3.6. q.e.d.

In the following, we will denote the scheme representing the functor ℑ𝔰𝔬𝔪_{S}(ξ_{1}, ξ_{2}) by Isom_{S}(ξ_{1}, ξ_{2}).

Lemma 3.7. The scheme Isom_{S}(ξ_{1}, ξ_{2}) is quasifinite and unramified over S.

Proof. To prove that Isom_{S}(ξ_{1}, ξ_{2}) is unramified over S, we only need to check the case where S is itself a closed point. Namely S = Spec k, in which case Isom_{S}(ξ_{1}, ξ_{2}) is either empty or is isomorphic to Aut_{𝔚}(ξ_{1}). Since char k = 0, it is well-known that Aut_{𝔚}(ξ_{1}) is unramified over k if there are no vector fields v_{1} of X_{1} and vector fields v_{2} of W_{1} (whose pushforward to W is trivial) so that they are compatible via f_{1} : X_{1} → W_{1}. Such vector field does not exist because ξ_{1} is stable. Hence Isom_{S}(ξ_{1}, ξ_{2}) is unramified over S. Further, since Isom_{S}(ξ_{1}, ξ_{2}) is of finite type over S, it is quasifinite over S. This proves the Lemma. q.e.d.

Lemma 3.8. Let S = Spec R be a C-scheme where R is a discrete valuation domain with residue field k and quotient field K. We denote by η_{0} and η the closed and the generic point of S. Let ξ_{1}, ξ_{2} ∈ 𝔐(𝔚, Γ)(S) be two families over S so that there is an arrow r_{K} from ξ_{1} ×_{S} η to ξ_{2} ×_{S} η. Then possibly after a base change r_{K} extends to an arrow r_{R} from ξ_{1} to ξ_{2}.

Proof. Without loss of generality we can assume that ξ_{i} is represented by f_{i}: X_{i} → W [n_{i}] with ι_{i}: S → C[n_{i}] the associated morphism so that ι_{i}(η_{0}) = 0 ∈ C[n_{i}]. We will prove the case where S → C does not factor through 0 ∈ C. The other case is similar and will be left to readers.

We let η and η_{0} be the generic and the closed point of S, as stated in the Lemma. Let X_{K} = X_{1} ×_{S} η, which is isomorphic to X_{2} ×_{S} η by assumption. Let f_{K}: X_{K} → W be the composite X_{1} ×_{S} η → W [n_{1}] → W. Since S → C does not factor through 0 ∈ C, f_{K} is an ordinary stable morphism. By the property of stable morphisms, possibly after a base change S̃ → S the morphism f_{K} ×_{S} S̃ extends to a family of stable morphisms f:X → W over S̃. For simplicity by replacing S with S̃ we can assume S̃ = S. We consider the induced morphism

p_{i} ∘ f_{i} : X_{i} ⟶ W,

where p_{i}: W [n_{i}] → W is the canonical projection. By our assumption, the restriction of p_{i} ∘ f_{i} to X_{i} ×_{S} η is isomorphic to f_{K}. Therefore by the property of stable morphisms f : X → W is the stabilization of p_{i} ∘ f_{i}:X_{i} → W for i = 1, 2. In particular, there is a unique contraction morphism q_{i}:X_{i} → X so that

p_{i} ∘ f_{i} = f ∘ q_{i} : X_{i} ⟶ W.

Now let X_{i} = X_{i} ×_{S} η_{0} and let A ⊂ X_{1} be any irreducible component contracted under q_{1}. Following the same argument in the proof of Lemma 3.2, one can show that deg(f_{1}^{∗}H, A) = 0, f_{1}(A) is contained in Δ_{l} ⊂ W [n_{1}]_{0} for some 2 ≤ l ≤ n_{1} and f_{1}(A) is contained in a fiber of Δ_{l} → D. There are two cases we need to distinguish. One is when f_{1}(A) is a point. Then that A is contracted by q_{1} implies A contains at least two nodal points of X_{1}. On the other hand, when f_{1}(A) is not a point, then f_{1}(A) is a fiber of Δ → D and hence because f_{1} ×_{S} η_{0} is stable and hence predeformable, the component A contains at least two nodal points of X_{1} as well. Because this is true for every irreducible components in X_{1} contracted under q_{1}, by the property of stable contraction, an irreducible component A′ of X_{1} is contracted under q_{1} if and only if it is isomorphic to P^{1}, it contains exactly two nodal points but no marked points and deg(f_{1}^{∗}H, A′) = 0. Then since f_{1} ×_{S} η_{0} is stable, such A′ must be a trivial component of X_{1}. Conversely, any trivial component of X_{1} obviously will be contracted under q_{1}. This shows that the morphism q_{i} : X_{i} → X contracts exactly all trivial components of X_{i}.

Consequently, if we let E_{i} ⊂ X_{i} be the union of all trivial components in X_{i}, then the contractions q_{1} and q_{2} define an isomorphism

φ : X_{1} − E_{1} ⟶^{≅} X_{2} − E_{2} (3.2)

extending the isomorphism X_{1} ×_{S} η ≅ X_{2} ×_{S} η.

To proceed, we assume n_{1} ≥ n_{2} without loss of generality. Note that W [n_{1}] is a resolution of W [n_{2}] ×_{A^{n_{2}+1}} A^{n_{1}+1}, ^{9} which induces a morphism q:W [n_{1}] → W [n_{2}]. We consider

q ∘ f_{1} : X_{1} ⟶ W [n_{2}].

Clearly no irreducible components of X_{1} are mapped entirely to the nodal divisor of W [n_{2}]_{0} unless n_{1} > n_{2} and in which case some nontrivial components of X_{1} will be mapped to and only to the n_{2}-th nodal divisor of W [n_{2}]_{0}. Let ι̃_{1} :S → C[n_{2}] be the morphism induced by q∘f_{1}. Because S → C does not factor through 0 ∈ C, there are isomorphisms r_{1,η} and r_{2,η} fitting into the commutative diagram

\begin{matrix}X_{1} ×_{S} η & ⟶^{q∘f_{1}} & W [n_{2}] ×_{C[n_{2}]} ι̃_{1}(η) \\ ↓^{r_{1,η}} & & ↓^{r_{2,η}} \\ X_{2} ×_{S} η & ⟶^{f_{2}} & W [n_{2}] ×_{C[n_{2}]} ι_{2}(η) \end{matrix} (3.3)

By Lemma 1.8 and Corollary 1.7, r_{2,η} is induced by the G[n_{2}]-action on W [n_{2}] via a morphism ρ: Spec K → G[n_{2}]. Let v be the uniformizing parameter of R and let ρ be defined by

ρ^{∗}(σ_{1}) = c_{1}v^{a_{1}}, …, ρ^{∗}(σ_{n_{2}}) = c_{n_{2}}v^{a_{n_{2}}} ,

where all c_{i} are units in R and a_{i} are integers. Here we follow the convention

G[n_{2}] = (G_{m})^{×n_{2}} and Γ(G[n_{2}]) = k[σ_{1}, σ_{1}^{−1}, …, σ_{n_{2}}, σ_{n_{2}}^{−1}].

Now let A be an irreducible component of X_{1} whose image under f_{1} is contained in the first irreducible component of W [n_{1}]_{0}. Then p_{1} ∘ f_{1}(A) is not contained in D ⊂ W, A is a nontrivial component and f_{2} ∘ φ(A) is also contained in the first component of W [n_{2}]_{0}. Now let A be a nontrivial irreducible component of X_{1} whose image under f_{1} is contained in the second irreducible component of W [n_{1}]_{0}. (Here we assume n_{1} ≥ 1 since otherwise there is nothing to prove.) We let h : (η̃_{0}, T) → X_{1} − E_{1}, where E_{1} is defined in (3.2), be a curve so that h(η̃_{0}) is a closed point in A in general position and the composite T ⟶^{h} X_{1} → S is a branched covering ramified at η̃_{0}. We pick an affine open V ⊂ W so that the image of h(η̃_{0}) under p_{1} ∘ f_{1} is contained in V ∩ D. Let T̂ be the formal completion of T along η̃_{0}, let V̂ be the formal completion of V along D ∩ V and let ĥ:T̂ → X_{1} be the induced morphism. Then the morphism

9A^{n_{1}+1} → A^{n_{2}+1} is defined by (t_{1}, …, t_{n_{1}+1}) ↦ (t_{1}, …, t_{n_{2}}, Π_{j=n_{2}+1}^{n_{1}+1}t_{j}).

q ∘ f_{1} ∘ ĥ : T̂ → W [n_{2}]

factor through W [n_{2}] ×_{W} V̂ → W [n_{2}], which we denote by

ĥ_{1} : T̂ ⟶ W [n_{2}] ×_{W} V̂ ≅ (V ∩ D) × Γ̂[n_{2}].

Here Γ̂[n_{2}] is the scheme defined before (1.11). We let

ĥ_{2} : T̂ ⟶ W [n_{2}] ×_{W} V̂ ≅ (V ∩ D) × Γ̂[n_{2}]

be the morphism induced by f_{2} ∘ φ ∘ h:T → W [n_{2}]. Because of (3.3), if we let pr_{2} be the second projection of (V ∩ D) × Γ̂[n_{2}] and let

Φ_{Γ} :G[n_{2}] × Γ̂[n_{2}] → Γ̂[n_{2}]

be the group action morphism, then

(pr_{2} ∘ ĥ_{1})^{ρ} = pr_{2} ∘ ĥ_{2} : T̂ ⟶ Γ̂[n_{2}].

To utilize this identity, we use the open subset U_{1} of Γ[n_{2}] constructed in Lemma 1.2. Since pr_{2} ∘ ĥ_{1}(η̃_{0}) is contained in the second irreducible component of Γ̂[n_{2}], if we let (u_{i}^{1}) be the coordinate chart of U_{1} introduced in Lemma 1.2, then

(pr_{2} ∘ ĥ_{1})^{∗}(u_{j}^{1}) = α_{j} ∈ OT̂, j = 1, …, n_{2}.

Because pr_{2} ∘ ĥ_{1}(η̃_{0}) is in the smooth locus of Γ[n_{2}]_{0}, all α_{j} are units in OT̂ except α_{1}, which is in the maximal ideal 𝔪T̂ of OT̂. On the other hand, because T → S is flat, there is a 0 ≠ c ∈ 𝔪T̂ so that R → OT̂ is given by v ↦ c. Therefore using the formula in Lemma 1.2,

((pr_{2} ∘ ĥ_{1})^{ρ})^{∗}(u_{i}^{1}) = (α_{1}, c_{1}c^{a_{1}}α_{2}, w_{3}, …, w_{n_{2}+2}) (3.4) for some w_{3}, …, w_{n_{2}+2} in the quotient field of OT̂. Hence if we let π:Γ(n_{2}) → A^{n_{2}+2} be the projection, then ((π ∘ (pr_{2} ∘ ĥ_{1})^{ρ})^{∗} maps

t_{1} ↦ c_{1}c^{a_{1}}α_{1}α_{2}, t_{2} ↦ w_{3}, …, t_{n_{2}+1} ↦ w_{n_{2}+2}.

Because this composition specializes to 0_{A^{n_{2}+1}} ∈ A^{n_{2}+1} by assumption, all w_{i} are regular and are in the maximal ideal 𝔪T̂. Furthermore, we know pr_{2} ∘ ĥ_{2} does not specialize to the first nodal point of Γ[n_{2}]_{0}, therefore c_{1}c^{a_{1}}α_{2} is not in 𝔪T̂. Since α_{2} is a unit in OT̂, c_{1} is a unit in R and c is in the maximal ideal 𝔪T̂ ⊂ OT̂, this implies a_{1} ≤ 0. Similarly, we can pick a nontrivial irreducible component A of X_{2} so that its image under f_{2} is contained in the second component of W [n_{2}]_{0}. Then because f_{1}∘φ^{−1}(A) is not contained in the first nodal divisor of W [n_{2}]_{0}, a parallel argument shows that a_{1} ≥ 0. Therefore, a_{1} = 0.

Repeating this argument, using the fact that no irreducible components of X_{1} are mapped entirely to the first n_{2} nodal divisors of W [n_{2}]_{0} under q ∘ f_{1}, we can show that all a_{1}, …, a_{n_{2}} are zero. Therefore, ρ:Spec K → G[n_{2}] extends to ρ_{R}:S → G[n_{2}].

Now we prove that the arrow r_{K} extends to an arrow r_{R} from ξ_{1} to ξ_{2}. Let f̃_{1} :X̃_{1} → W [n_{2}] be the stable contraction of q ∘ f_{1} :X_{1} → W [n_{2}] as a morphism to W [n_{2}]. Then X̃_{1} ×_{S} η is X_{1} ×_{S} η, which is isomorphic to X_{2} ×_{S} η via the arrow r_{K}. We consider the morphism

(f̃_{1})^{ρ_{R}} : X̃_{1} ⟶ W [n_{2}].

By assumption, its restriction to X̃_{1} ×_{S} η is identical to the restriction of f_{2} to X_{2} ×_{S} η via X̃_{1} ×_{S} η ≅ X_{2} ×_{S} η. On the other hand, both (f̃_{1})^{ρ_{R}} and f_{2} are stable extensions of a stable morphism from X_{1} ×_{S} η ≅ X_{2} ×_{S} η to W [n_{2}]. Hence by the uniqueness of the extensions of ordinary stable morphisms, the isomorphism X_{1} ×_{S} η ≅ X_{2} ×_{S} η extends to an isomorphism X̃_{1} ≅ X_{2} and the morphism (f̃_{1})^{ρ_{R}} is identical to f_{2}. This shows that (f̃_{1})^{ρ_{R}} maps no irreducible components of X̃_{1} ×_{S} η_{0} to the nodal divisors of W [n_{2}]_{0}. We claim that n_{1} must be equal to n_{2}. Otherwise, all irreducible components of X_{1} that are mapped to Δ_{n_{2}+1} ⊂ W [n_{1}]_{0} (under f_{1}) are trivial components, contradicting to Lemma 3.2. Once we have n_{1} = n_{2}, then f̃_{1} = f and hence X̃_{1} = X_{1} and f_{2} = f_{1}^{ρ_{R}} follows immediately. This defines an arrow r_{R} that is an extension of r_{K}. This proves Lemma 3.8. q.e.d.

Lemma 3.9. Let S = Spec R be a C-scheme where R is a discrete valuation domain as before and let η be the generic point of S. Let ξ_{K} ∈ 𝔐(𝔚, Γ)(η) be a family of stable morphisms over Spec K. Then possibly after a base change S̃ → S, ξ_{K} ×_{S} S̃ extends to a family ξ ∈ 𝔐(𝔚, Γ)(S̃).

Proof. Let ξ_{K} be represented by f_{K} : X_{K} → W [n]. We will prove the case where Spec K → C does not factor though 0 ∈ C and leave the other case to the readers.

Since Spec K → C does not factor through 0 ∈ C, we can assume without loss of generality that f_{K} is represented by f_{K} : X_{K} → W. Then f_{K} is an ordinary stable morphism over Spec K. By the property of stable morphism, possibly after a base change S̃ → S, the morphism f_{K} ×_{S} S̃ extends to an S̃-family of ordinary stable morphism f_{1} :X_{1} → W. Again by replacing S with S̃ we can assume S = S̃. Let η_{0} and η be the closed and the generic points of S. In case S → C sends η_{0} to C − 0, then f_{1} is already a family of stable morphisms in 𝔐(𝔚, Γ)(S). Now assume η_{0} is mapped to 0 ∈ C. Let ι : S → C[n] be any Cmorphism. Then since W [n]×_{C[n]}η is canonically isomorphic to W ×_{C} η, f_{K}:X_{K} → W induces a morphism f̃_{K}:X_{K} → W [n]. We say f_{K} admits a partial extension to W [n] if after a base change S̃ → S the morphism f̃_{K} ×_{S} S̃ extends to a family of quasistable morphisms over S̃

f_{n} : X_{n} ⟶ W [n] (3.5)

so that the associated ι:S̃ → C[n] maps the closed point of S̃ to 0. Here we say f_{n} is quasistable if Aut_{𝔚}(f_{n,η_{0}}) is finite, where the automorphism group is the set of pairs (a, b) such that a is an automorphism of the domain of f_{n,η_{0}} and b ∈ G[n] such that f_{n} ∘ a = (f_{n,η_{0}})^{b}. (As before, we let f_{n,η_{0}} be the restriction of f_{n} to the fiber X_{n,η_{0}} = X_{n} ×_{S} η_{0}.) It is clear that the extension f_{1} is a quasistable extension to W = W [1]. On the other hand, by the proof of Lemma 3.2 any quasistable extension f_{n} satisfies n ≤ |δ|.

Now we let n be the largest possible integer so that there is a quasistable extension f_{n}. We show that f_{n} defines an extension of f_{K}. Let f_{n} be a quasistable extension as in (3.5) via ι:S → C[n]. If

f_{n,η_{0}} : X_{n,η_{0}} ⟶ W [n]_{0}

is nondegenerate, then Proposition 2.2 implies that f_{n} is predeformable. Hence f_{n} ∈ 𝔐(𝔚, Γ)(S) as desired. Now assume f_{n,η_{0}} is degenerate. Let A be an irreducible component of X_{n,η_{0}} so that f_{n,η_{0}}(A) is contained, say, in the l-th nodal divisor of W [n]_{0}. We distinguish the case where l ≤ n from the case where l = n + 1. We first consider the case l ≤ n. We let v be a uniformizing parameter of R and let ρ:Spec K → G_{m} be defined by ρ^{∗}(σ) = v^{α} for an integer α. Let λ_{l}: G_{m} ≅ G[n]_{l} → G[n] be the l-th one-parameter subgroup of G[n]. We consider

(f_{n,η})^{λ_{l}∘ρ} : X_{n,η} ⟶ W [n]. (3.6)

As in the proof of Lemma 3.8, we can pick a curve h : (η̃_{0}, T) → X_{n} covering \underline{h}:(η̃_{0}, T) → (0, C) so that \underline{h} is flat, h(η̃_{0}) ∈ A and is in general position of A. Further we can find an integer α so that

(f_{n} ∘ h)^{λ_{l}∘ρ∘\underline{h}} : T ⟶ W [n]

specializes to a point in the (l + 1)-th irreducible component of W [n]_{0} and is away from the nodal divisors of W [n]_{0}. A simple analysis using the covering U_{l} of Γ[n] shows that if we let

f̃_{n} : X̃_{n} → W [n]

be the extension of (3.6) to a family of stable morphisms, then f̃_{n} is still quasistable. Further, some irreducible component of X̃_{n,η_{0}} is mapped entirely to the (l +1)-th nodal divisor of W [n]_{0}. This shows that we can assume without loss of generality that f_{n} maps an irreducible component A of X_{n,η_{0}} to D_{n+1} ⊂ W [n]_{0}.

We now show that there is a quasistable extension of f_{K} to W [n+1]. Let f_{n} and A ⊂ X_{n,η_{0}} be as before so that f_{n}(A) ⊂ D_{n}. We let C[n] ⊂ C[n+1] and W [n] → W [n+1] be the standard embedding associated to [n] ⊂ [n+1]. Namely, they are induced by the embedding A^{n+1} → A^{n+2} that keep the last coordinate 1. Let f̃_{n}:X_{n} → W [n+1] be the composite of f_{n} with W [n] → W [n + 1]. Then the same technique as before shows that we can find an integer α so that if we let ρ : Spec K → G_{m} be defined by ρ^{∗}(σ) = v^{α}, then the extension (possibly after a base change S̃ → S) of

(f̃_{n})^{λ_{n}∘ρ} : X_{n,η} ⟶ W [n + 1]

to a family of ordinary stable morphisms

f_{n+1} : X_{n+1} ⟶ W [n + 1]

is quasistable. This contradicts to the maximal assumption of n. Hence f_{n} must already be a family of nondegenerate morphism. Since S̃ → C is flat, it must be predeformable and thus in 𝔐(𝔚, Γ)(S̃), extending ξ_{K} ×_{S} S̃ ∈ 𝔐(𝔚, Γ)(η). This proves Lemma 3.9. q.e.d.

Theorem 3.10. The moduli stack 𝔐(𝔚, Γ) of stable morphisms to 𝔚 of topological type Γ is separated and proper over C. Furthermore, it is a Deligne-Mumford stack.

Proof. The fact that the moduli stack 𝔐(𝔚, Γ) is separate over C follows from Lemma 3.8 and that it is proper over C follows from Lemma 3.9. It remains to show that it is algebraic. Namely, 𝔐(𝔚, Γ) admits an étale cover by a scheme of finite type. We now show that it admits an étale covering by a quasiprojective scheme.

Let 𝔐(W [n], Γ) be the moduli stack of stable morphisms from pointed curves to W [n] of type Γ, and let 𝔐(W [n], Γ)^{st} be the substack of 𝔐(W [n], Γ) consisting of all predeformable morphisms that are stable in the sense of Definition 3.1. By Theorem 2.11, 𝔐(W [n], Γ)^{st} is a locally closed substack of 𝔐(W [n], Γ). Since members of 𝔐(W [n], Γ)^{st} are stable morphisms to 𝔚, there is a natural morphism 𝔐(W [n], Γ)^{st} → 𝔐(𝔚, Γ). Because of Lemma 3.3, there is an integer n so that 𝔐(W [n], Γ)^{st} ⟶ 𝔐(𝔚, Γ) is surjective. Further, since W [n] is a G[n]-variety, 𝔐(W [n], Γ) admits a natural G[n]-action. By the definition of stability (cf. Definition 3.1), 𝔐(W [n], Γ)^{st} is G[n]-invariant and the G[n] action on 𝔐(W [n], Γ)^{st} has only finite stabilizer. Hence 𝔐(W [n], Γ)^{st}/G[n] is a DM-stack. Finally, it is clear that the induced morphism

𝔐(W [n], Γ)^{st}/G[n] ⟶ 𝔐(𝔚, Γ)

is finite and étale. This proves that 𝔐(𝔚, Γ) is an algebraic (i.e., Deligne-Mumford) stack. q.e.d.

4. Stacks of relative stable morphisms

Let D ⊂ Z be a smooth connected divisor (called the distinguished divisor) in a smooth projective variety. In this section, we will construct the moduli stack of relative stable morphisms with prescribed contact with D. We will prove some basic property of this moduli stack and describe its relation with the stack of stable morphisms constructed in the previous section. This construction can easily be generalized to the case where D is smooth but not necessarily connected.

4.1 Standard models

We first construct the standard models (Z[n], D[n]) which are the building blocks of the stack of expanded relative pairs of (Z, D). For n = 0, we let (Z[0], D[0]) = (Z, D). For n = 1, Z[1] is the blowing-up of Z × A^{1} along D × {0} ⊂ Z × A^{1}. Its distinguished divisor D[1] is the proper transform of D × A^{1} ⊂ Z × A^{1} in Z[1]. We now let W → A^{1} be Z[1] → A^{1}. We shall view the ruled variety Δ = P(1_{D} ⊕ N_{D/Z}) and Z in W_{0}(= W ×_{A^{1}} 0) as its top (left) and its bottom (right) components. With this understanding, we can construct a sequence of varieties W [n] over A^{n+1} for all n ≥ 1 as in Section 1 with the projection π: W [n] → Z × A^{n+1}. We define Z[n] = W [n − 1] and define its distinguished divisor D[n] to be the proper transform of D × A^{n} ⊂ Z × A^{n} in Z[n]. We call (Z[n], D[n]) with the tautological projection Z[n] → Z × A^{n} an expanded relative pair of (Z, D). Note that under this projection D[n] is isomorphic to D×A^{n} ⊂ Z ×A^{n}. We denote the projection Z[n] → A^{n} by π.

Clearly, the fiber of Z[n] over 0 ∈ A^{n}, denoted Z[n]_{0}, is reduced with normal crossing singularity. It consists of a chain of smooth varieties, one Z and n ruled varieties Δ. We give these n components an ordering so that the first component of Z[n]_{0} contains the distinguished divisor D[n]_{0} and the remainder components are ordered according to their intersection chain structure. Namely, the k-th component will intersect the (k + 1)-th component. We call this the standard ordering of Z[n]_{0}. We continue to denote by G[n] the product of n copies of G_{m}. Since the G_{m}-action on Z × A^{1} defined via (z, t)^{σ} = (z, σ^{−1}t) leaves D × {0} ⊂ Z × A^{1} fixed, it lifts to an action on Z[1] → A^{1}. In general, we let G[n] acts on A^{n} via

(t_{1}, …, t_{n})^{σ} = (σ_{2}t_{1}, σ_{3}t_{2}, …, σ_{n+1}t_{n}). (4.1)

(Recall σ_{i} = σ_{i}/σ_{i−1}, see (1.3).) Note that if we embed A^{n} in A^{n+1} in the standard way as the n + 1-th coordinate hyperplane H_{n+1}^{n+1} and let G[n] acts on A^{n+1} as defined in Section 1, then the embedding A^{n} ⊂ A^{n+1} is G[n]-equivariant. It is direct to check that there is a unique G[n]-action on Z[n] that makes Z[n] → Z × A^{n−1} G[n]-equivariant, where the G[n] action on Z × A^{n−1} is (z, t)^{σ} = (z, t^{σ}) defined in (4.1).

For later application, we need to consider Z[n] with the reversed ordering of its components. In the following for any n we define 1_{A^{n}}^{o} : A^{n} → A^{n} be the morphism (t_{1}, …, t_{n}) ↦ (t_{n}, …, t_{1}). We let Z[n]^{o} be Z[n] with the tautological projection π^{o} ≡ 1_{A^{n}}^{o} ∘ π: Z[n] → A^{n}. At the same time we reverse the order of the components of Z[n]_{0} so that the first irreducible component of Z[n]_{0}^{o} is Z while its last component contains the distinguished divisor. Note that the restriction of Z[n]_{0}^{o} to the l-th coordinate line is still a smoothing of the l-th nodal divisor of Z[n]_{0}^{o}. Accordingly, we let the G[n]-action on A^{n} be

(t_{1}, …, t_{n})^{σ^{o}} = (σ_{1}t_{1}, …, σ_{n}t_{n}). (4.2)

We next describe the connection between the standard model constructed in Section 1 with the relative standard models so constructed. Let W → C be the pair in Section 1 with D_{1} ⊂ Y_{1} and D_{2} ⊂ Y_{2} the two irreducible components of W_{0} with their distinguished divisors. Using the pair (Y_{i}, D_{i}) we can construct the relative pair (Y_{i}[n], D_{i}[n]) with the associated projection Y_{i}[n] → Y_{i} × A^{n}. Now let l be any integer in [n]. We consider the pairs

D_{1}[l] × A^{n−l} ⊂ Y_{1}[l]^{o} × A^{n−l} and A^{l} × D_{2}[n − l] ⊂ A^{l} × Y_{2}[n − l].

We have the canonical isomorphisms

D_{1}[l] × A^{n−l} ⟶^{≅} (D_{1} × A^{l}) × A^{n−l} ⟶^{≅} D_{1} × A^{l} × A^{n−l}

and canonical isomorphisms

A^{l} × D_{2}[n − l] ⟶^{≅} A^{l} × (D_{2} × A^{n−l}) ⟶^{≅} D_{2} × A^{l} × A^{n−l},

where the second arrow is induced by exchanging the factor A^{l−1} and D_{2}. Combined with the canonical isomorphism D_{1} ≅ D_{2}, we have

D_{1}[l] × A^{n−l} ≅ A^{l} × D_{2}[n − l]. (4.3)

Hence we can glue Y_{1}[l]^{o} × A^{n−l} with A^{l} × Y_{2}[n − l] along

D_{1}[l] × A^{n−l} ⊂ Y_{1}[l]^{o} × A^{n−l} and A^{l} × D_{2}[n − l] ⊂ A^{l} × Y_{2}[n − l]

according to the isomorphism (4.3) to obtain a new scheme, denoted by

Y_{1}[l]^{o} × A^{n−l} ⊔ A^{l} × Y_{2}[n − l]. (4.4)

The group G[n] acts on (4.4) as follows: We let A^{l}×A^{n−l} → A^{n+1} be the embedding as coordinate plane defined by (r, s) ↦ (r, 0, s). Then the G[n]-action on A^{n+1} (see (1.2)) lifts to an action on A^{l} × A^{n−l}. Namely for σ = (σ_{1}, …, σ_{n−1}) ∈ G[n],

(r_{1}, …, r_{l})^{σ} = (σ_{1}r_{1}, …, σ_{l}r_{l}) and
(s_{1}, …, s_{n−l})^{σ} = (σ_{l+2}s_{1}, …, σ_{n+1}s_{n−l}).

Let ρ_{−}:G[n] → G[l] and ρ_{+} :G[n] → G[n − l] be the homomorphism

ρ_{−}(σ) = (σ_{1}, …, σ_{l}) and ρ_{+}(σ) = (σ_{l+1}, …, σ_{n}).

Thus r^{σ} = r^{ρ_{−}(σ)^{o}} and s^{σ} = s^{ρ_{+}(σ)}, where r^{σ} and s^{σ} are the actions defined above and r^{ρ_{−}(σ)^{o}} and s^{ρ_{+}(σ)} are the actions defined in (4.1) and (4.2). We define the G[n]-action on Y_{1}[l]^{o} × A^{n−l} and on A^{l} × Y_{2}[n − l] be

(x, s)^{σ} = (x^{ρ_{−}(σ)^{o}}, s^{ρ_{+}(σ)}) and (r, y)^{σ} = (r^{ρ_{−}(σ)^{o}}, y^{ρ_{+}(σ)}).

It is direct to check that the isomorphism (4.3) is G[n]-equivariant, and hence (4.4) is a G[n]-scheme.

Proposition 4.1. Let π:W [n] → A^{n+1} be the associated morphism and π_{l} : W [n] → A^{1} be the composite of π with the l-th projection of A^{n+1}. Then we have a canonical isomorphism

Y_{1}[l]^{o} × A^{n−l} ⊔ A^{l} × Y_{2}[n − l] ≅ W [n] ×_{π_{l}} 0_{A^{1}} ⊂ W [n].^{10}

Further the above inclusion and isomorphism are G[n]-equivariant.

Proof. The proof is straightforward and will be omitted. q.e.d.

Now let (Y_{i}, D_{i}), i = 1 and 2, be effective relative pairs over S associated to τ_{i}:S → A^{l_{i}} defined by

(Y_{i}, D_{i}) = (Y_{i}[l_{i}], D_{i}[l_{i}]) ×_{A^{l_{i}}} S.

As before, we define τ_{1}^{o} = 1_{A^{l_{1}}}^{o} ∘ τ_{1} :S → A^{l_{1}} and let Y_{1}^{o} be Y_{1}[l_{1}]^{o} ×_{τ_{1}^{o}} S. Obviously Y_{1}^{o} = Y_{1} with only the ordering of the components of its fibers reversed. Since D_{1} (resp D_{2}) is canonically isomorphic to D_{1} × S (resp. D_{2} × S), we can construct a new scheme over S by gluing Y_{1}^{o} and Y_{2} along D_{1} and D_{2} using D_{1} ≅ D_{2}. Since both D_{1} and D_{2} are smooth over S, the resulting scheme Y_{1}^{o} ⊔ Y_{2} is a flat family of schemes with normal crossing singularity relative to S. Let A^{n} ≅ A^{l_{1}} × A^{1} × A^{l_{2}}, n = l_{1} + l_{2} + 1, be the canonical isomorphism keeping the order of each copies of A^{1} in A^{n+1}. Then we have an embedding A^{l_{1}} × A^{l_{2}} → A^{n+1} sending (r, s) ∈ A^{l_{1}} × A^{l_{2}} to (r, 0, s) ∈ A^{n+1}. Let τ : S → A^{n+1} be the composite

τ : S ⟶^{(τ_{1}^{o},τ_{2})} A_{l_{1}} × A_{l_{2}} ⟶ A_{n+1}. (4.5)

10By W [n] ×_{π_{l}} 0_{A^{1}} we mean W [n] ×_{A^{1}} 0_{A^{1}} with W [n] → A^{1} given by π_{l}.

Corollary 4.2. Let the notation be as above and let τ : S → A^{n} be defined in (4.5). Then the associated family W = τ^{∗}W [n] is canonically isomorphic to Y_{1}^{o} ⊔ Y_{2}.

Proof. Clearly, the composite of τ:S → A^{n+1} with pr_{l_{1}} :A^{n+1} → A^{1} is trivial. Hence by Proposition 4.1

W [n] ×_{A^{n+1}} S ≅ (Y_{1}[l_{1}]^{o} × A^{l_{2}} ⊔ A^{l_{1}} × Y_{2}[l_{2}]) ×_{A^{l_{1}}×A^{l_{2}}} S
≅ Y_{1}[l_{1}]^{o} ×_{A^{l_{1}}} S ⊔_{D_{1}[l_{1}]×_{A^{l_{1}}} S=D_{2}[l_{2}]×_{A^{l_{2}}} S} Y_{2}[l_{2}] ×_{A^{l_{2}}} S,

which is isomorphic to Y_{1}^{o} ⊔ Y_{2}. This proves the Corollary. q.e.d.

We now define the stack of expanded relative pairs of (Z, D). Let S be any scheme. An effective family of expanded relative pair of (Z, D) (in short relative pair) is the associated family (Z, D) of a morphism τ:S → A^{n} for some n defined by

Z = τ^{∗}Z[n] = Z[n] ×_{A^{n}} S and D = τ^{∗}D[n] = D[n] ×_{A^{n}} S.

We call D the distinguished divisor of Z. Now let

\underline{Φ} : G[n] × A^{n} ⟶ A^{n} and Φ : G[n] × Z[n] ⟶ Z[n]

be the G[n]-action on A^{n} and on Z[n] defined earlier in this section. Then for any τ : S → A^{n} and ρ : S → G[n], the group action \underline{Φ} defines a new morphism τ^{ρ} : S → A^{n} and thus a new family of relative pair (τ^{ρ}^{∗}Z[n], τ^{ρ}^{∗}D[n]). Using the group action of G[n] on Z[n], there is a canonical isomorphism between the pair (τ^{∗}Z[n], τ^{∗}D[n]) and (τ^{ρ}^{∗}Z[n], τ^{ρ}^{∗}D[n]). As in the case for W [n], given two effective relative pairs ξ_{1} = (Z_{1}, D_{1}) and ξ_{2} = (Z_{2}, D_{2}) over S associated to morphisms τ_{1} : S → A^{n_{1}} and τ_{2} : S → A^{n_{2}} respectively, an effective arrow from ξ_{1} to ξ_{2} consists of a standard embedding ι: A^{n_{1}} → A^{n_{2}} (associated to an increasing map [n_{1}] → [n_{2}], see discussion at the beginning of Subsection 1.1) and a morphism ρ:S → G[n_{2}] such that (ι ∘ τ_{1})^{ρ} = τ_{2}. Similar to the discussion before Lemma 1.8, this identity defines a canonical S-isomorphism of pairs

(τ_{1}^{∗}Z[n_{1}], τ_{1}^{∗}D[n_{1}]) ≅ (τ_{2}^{∗}Z[n_{2}], τ_{2}^{∗}D[n_{2}]),

compatible to their projections to Z × S. Along the same line, we can define an effective arrow between ξ_{1} and ξ_{2} to be either an effective arrow from ξ_{1} to ξ_{2} or an effective arrow from ξ_{2} to ξ_{1}. We say ξ_{1} and ξ_{2} are equivalent via a sequence of effective arrows if there is a sequence of effective families η_{0}, …, η_{m} so that ξ_{1} = η_{0}, ξ_{2} = η_{m} and that η_{i} is equivalent to η_{i+1} via an effective arrow. Note that this sequence of effective arrows induces a canonical isomorphism of (Z_{1}, D_{1}) and (Z_{2}, D_{2}). We have a partial inverse to this, similar to Lemma 1.8.

Lemma 4.3. Let ξ_{i} = (Z_{i}, D_{i}), i = 1 and 2, be two effective relative pairs over S associated to τ_{i}:S → A^{n_{i}}. Suppose there is an isomorphism (Z_{1}, D_{1}) ≅ (Z_{2}, D_{2}) compatible to their projections to Z × S. Then to each p ∈ S there is an open neighborhood S_{0} of p ∈ S such that over S_{0} the induced isomorphism Z_{1} ×_{S} S_{0} ≅ Z_{2} ×_{S} S_{0} is induced by a sequence of effective arrows between ξ_{1} ×_{S} S_{0} and ξ_{2} ×_{S} S_{0}.

Proof. The proof is similar to that of Lemma 1.8 and will be omitted. q.e.d.

We define a family of relative pairs over S to be a pair (Z, D), where Z is an S-family with an S-projection Z → Z × S and D is a Cartier divisor of Z, such that there is an open covering S_{α} of S so that each (Z ×_{S} S_{α}, D ×_{S} S_{α}) is isomorphic to an effective relative pair associated to some τ_{α} : S_{α} → A^{n_{α}} and the isomorphism Z ×_{S} S_{α} ≅ τ_{α}^{∗}Z[n_{α}] is compatible to their projections to Z × S_{α}. Note that then D is isomorphic to D×S ⊂ Z ×S under the projection Z → Z ×S. We call Z ×_{S} S_{α} ≅ Z[n_{α}]×_{A^{nα}} S_{α} with its associated data a local representative of (Z, D). We say two such families (Z_{1}, D_{1}) and (Z_{2}, D_{2}) are isomorphic if there is an S-isomorphism (Z_{1}, D_{1}) → (Z_{2}, D_{2}) compatible to the projections Z_{1} → Z × S and Z_{2} → Z × S. Clearly, if (Z, D) is a relative pair over S and ρ : S′ → S is a morphism, then (Z ×_{S} S′, D ×_{S} S′) coupled with the obvious projection Z ×_{S} S′ → Z × S′ is a relative pair over S′, called the pullback relative pair.

Definition 4.4. We define ℨ^{rel} to be the category whose objects are families of expanded relative pairs of (Z, D) over S for some scheme S. Let S_{i}, i = 1 and 2, be any two schemes and ξ_{i} ∈ ℨ^{rel}(S_{i}) be two objects in ℨ^{rel}. An arrow from ξ_{1} to ξ_{2} consists of an arrow (a morphism) ρ: S_{1} → S_{2} and an isomorphism of relative pairs ξ_{1} ≅ ρ^{∗}ξ_{2}. We define 𝔭: ℨ^{rel} → (Sch) to be the functor that send families over S to S. The pair (ℨ^{rel}, 𝔭) forms a groupoid.

Proposition 4.5. The groupoid (ℨ^{rel}, 𝔭) is a stack.

Proof. The proof is straightforward and will be omitted. q.e.d.

In the following, we will call ℨ^{rel} the stack of expanded relative pairs of (Z, D).

Now let (Y_{1}, D_{1}) and (Y_{2}, D_{2}) be the two relative pairs associated to the family W/C and let 𝔜_{1}^{rel} and 𝔜_{2}^{rel} be the associated stacks of expanded relative pairs of (Y_{1}, D_{1}) and (Y_{2}, D_{2}), respectively. Then the correspondence (Y_{1}, Y_{2}) ↦ Y_{1}^{o} ⊔ Y_{2} defined in Corollary 4.2 induces a canonical morphism of stacks

𝔜_{1}^{rel} × 𝔜_{2}^{rel} ⟶ 𝔚 ×_{C} 0 ⊂ 𝔚. (4.6)

4.2 Relative stable morphisms

Let (Z, D) be the pair and ℨ^{rel} the stack of expanded relative pairs of (Z, D). In this part we define the notion of relative stable morphisms to ℨ^{rel}. We fix an ample line bundle H on Z, called a polarization of (Z, D). We first introduce the notion of admissible weighted graphs (in short admissible graphs).

In this paper, by a graph Γ we mean a finite collection of vertices, edges, legs and roots. Here an edge is as usual a line segment with both ends attached to vertices of Γ. A leg or a root is a line segment with only one end attached to a vertex of Γ. We will denote by V (Γ) the set of vertices of Γ.

Definition 4.6. An admissible weighted graph Γ is a graph without edges coupled with the following additional data:

1. An ordered collection of legs, an ordered collection of weighted roots and two weight functions g : V(Γ) → ℤ_{≥0} and b : V (Γ) → A_{2}Z/ ∼_{alg}.

2. The graph is relatively connected in the sense that either V(Γ) consists of a single element or each vertex in V(Γ) has at least one root attached to it.

In this section, we will reserve the integer k to denote the number of legs and the integer r to denote the number of roots of Γ. We will denote by μ_{1}, …, μ_{r} the weights of the r roots.

Given two admissible graphs Γ_{1} and Γ_{2}, an isomorphism Γ_{1} ≅ Γ_{2} is an isomorphism between their respective vertices, legs and roots (order preserving in the later two cases) so that it preserves the weights of the roots and the two sets of weights of the vertices.

We now define the relative stable morphisms to (Z[n], D[n]) of type Γ.

Definition 4.7. Let S be an A^{n}-scheme and Γ be an admissible graph with r roots, k legs and l vertices v_{1}, …, v_{l}. An S-family of relative stable morphisms to (Z[n], D[n]) of type Γ is a quadruple (f, X , q_{i}, p_{j}) as follows:

1. X is a disjoint union of X_{1}, …, X_{l} such that each X_{i} is a flat family of prestable curves^{11} over S of arithmetic genus g(v_{i}).

2. q_{i}: S → X, i = 1, …, r and p_{j}: S → X, j = 1, …, k, are disjoint sections away from the singular locus of the fibers of X /S so that q_{i}(S) ⊂ X_{j} (resp. p_{i}(S) ⊂ X_{j}) if the i-th root (resp. i-th leg) is attached to the j-th vertex of Γ.

3. f : X → Z[n] is an A^{n}-morphism (i.e., an S-morphism X → Z[n]×_{A^{n}}S) so that as Cartier divisors f^{−1}(D[n]) = ∑_{i=1}^{r} μ(x)q_{i}(S), and that to each closed s ∈ S the class φ_{∗}(f(X_{s})) is algebraically equivalent to b(v). Here φ: Z[n] → Z be the tautological projection.

4. Finally, each morphism f|_{X_{i}}, considered as a morphism whose domain is X_{i} with all marked sections in X_{i}, is a family of stable morphisms to Z[n].

Note that a necessary condition for the existence of relative stable morphisms of type Γ is that to each v ∈ V (Γ), ∑_{i≺v} μ_{i} = b(v)·D, where i ≺ v means that the i-th root is attached to the vertex v. We will call q_{i} the distinguished marked sections and call p_{j} the ordinary marked sections.

We now fix an admissible graph Γ. Let f : X → Z[n] be a relative stable morphism to (Z[n], D[n]) of type Γ. Let l be any integer in [n]. Then Z[n] ×_{A^{n}} H_{l}^{n} has normal crossing singularity whose singular locus is isomorphic to D × A^{n−1}. We denote this nodal divisor by B_{l}. As in Section 2, we call f nondegenerate if to each s ∈ S there are no irreducible components of X_{s} mapped entirely to the union of B_{1}, …, B_{n} under f. We call f predeformable if f is predeformable along B_{1}, …, B_{n} as defined in Section 2.

We now change our view point of f. Since S is an A^{n}-scheme,

(Z[n] ×_{A^{n}} S, D[n] ×_{A^{n}} S) ∈ Ob(ℨ^{rel}(S)),

11Recall by prestable we mean that X /S is a family of connected, complete and nodal curves.

and hence f can be viewed as an S-family of morphisms to ℨ^{rel}. We say f, considered as an S-family of morphisms to ℨ^{rel}, is relative prestable (resp. predeformable; resp. of type Γ) if f, considered as an S-family of morphisms to Z[n], is relative stable (resp. predeformable; resp. or type Γ). As before, we define Aut_{ℨ}(f) to be the functor associating to each S-scheme T the set of all (a, b), where a is an automorphism of X ×_{S} T as pointed nodal curve over T, namely a leaves all marked (distinguished and ordinary) sections of X ×_{S} T fixed, and b is a morphism T → G[n] such that if we let f_{T} be the morphism X ×_{S} T → Z[n] induced by f then

(f_{T})^{b} = f_{T} ∘ a : X ×_{S} T → Z[n].

Here (f_{T})^{b} is the morphism resulting from the G[n]-action on Z[n] associated to b:T → G[n] (see (1.13)). Obviously, this functor is represented by a group scheme over S, called the automorphism group of f and denoted by Aut_{ℨ}(f).

Let S be any scheme and (Z, D) ∈ Ob(ℨ^{rel}(S)) be any object. Then S has an open covering by {S_{α}} so that over each S_{α} we have isomorphism

(Z, D) ×_{S} S_{α} ≅ (Z[n_{α}] ×_{A^{nα}} S_{α}, D[n_{α}] ×_{A^{nα}} S_{α})

for some S_{α} → A^{n_{α}}. For any S-family of morphisms f:X → Z, we will call the induced morphisms f_{α} : X ×_{S} S_{α} ⟶ Z[n_{α}] local representatives of f.

Definition 4.8. An S-family of relative prestable morphisms to ℨ^{rel} of type Γ is a family f : X → Z, where (Z, D) ∈ Ob(ℨ^{rel}(S)), so that all its local representative f_{α} are relative stable (as morphisms to Z_{α}) of type Γ. We say f is stable if in addition to it being prestable, all its local representatives are predeformable and that for all closed s ∈ S the automorphism groups Aut_{ℨ}(f_{s}) are finite.

It is routine to check that the definition of Aut_{ℨ}(f_{s}) and the definition of relative stable morphisms to ℨ^{rel} is independent of the choices of local representatives.

Definition 4.9. Let (Z, D) be as before and let Γ be an admissible graph. We define 𝔐(ℨ^{rel}, Γ) to be the category whose objects are families of relative stable morphisms to ℨ^{rel} of type Γ. As usual, 𝔐(ℨ^{rel}, Γ)(S) is the subcategory of families over S. Let ξ_{1} and ξ_{2} be objects in 𝔐(ℨ^{rel}, Γ)(S) represented by relative stable morphisms f_{1} : X_{1} → Z_{1} and f_{2} : X_{2} → Z_{2}. An arrow ξ_{1} → ξ_{2} in 𝔐(ℨ^{rel}, Γ)(S) covering 1_{S} : S → S consists of a pair (φ_{1}, φ_{2}), where φ_{1} is an Sisomorphism X_{1} → X_{2} preserving all ordinary marked points and φ_{2} is an arrow Z_{1} → Z_{2} in ℨ^{rel}(S) covering 1_{S} so that f_{2}∘φ_{1} = φ_{2}∘f_{1}. In case ξ_{1} ∈ 𝔐(ℨ^{rel}, Γ)(S) and ξ_{2} ∈ 𝔐(ℨ^{rel}, Γ)(T), then an arrow ξ_{1} → ξ_{2} consists of a morphism σ:S → T and an arrow ξ_{1} → σ^{∗}ξ_{2} in 𝔐(ℨ^{rel}, Γ)(S) covering 1_{S}. Let 𝔭: 𝔐(ℨ^{rel}, Γ) → (Sch) be the functor that sends families over S to S. The category 𝔐(ℨ^{rel}, Γ) coupled with the functor 𝔭 form a groupoid, called the groupoid of the relative stable morphisms to ℨ^{rel} of type Γ.

Theorem 4.10. The groupoid 𝔐(ℨ^{rel}, Γ) is an algebraic stack. It is separated and proper over k.

Proof. The proof is parallel to that of 𝔐(𝔚, Γ) and will be omitted. q.e.d.

We now move back to the family W defined before. We fix an ample line bundle H on W as before and let H_{Y_{i}} be the restriction of H to Y_{i}. We let (Y_{1}, D_{1}) and (Y_{2}, D_{2}) be the two relative pairs from decomposing W_{0}. We let 𝔜_{1}^{rel} and 𝔜_{2}^{rel} be the stack of expanded relative pairs of (Y_{1}, D_{1}) and (Y_{2}, D_{2}) respectively. In the remainder part of this section, we will investigate the relation of the stacks 𝔐(𝔜_{1}^{rel}, Γ_{1}), 𝔐(𝔜_{2}^{rel}, Γ_{2}) and 𝔐(𝔚, Γ). We begin with a discussion of gluing a pair of relative stable morphisms

(f_{1}, f_{2}) ∈ 𝔐(𝔜_{1}^{rel}, Γ_{1}) × 𝔐(𝔜_{2}^{rel}, Γ_{2})

to form a stable morphism in 𝔐(𝔚, Γ). Here as usual the domain of the new morphism will be the gluing of the domains of f_{1} and f_{2}. In terms of their associated graphs, this amounts to connecting the roots in the graph Γ_{1} to the roots in the graph Γ_{2} to form a new graph with edges.^{12}

Definition 4.11. Let Γ_{1} and Γ_{2} be two admissible graphs with identical numbers of roots and k_{1} and k_{2} legs respectively. We let r be the number of roots of Γ_{1}, let k = k_{1} + k_{2} and let I ⊂ [k] be a subset of k_{1} elements. We say that the triple (Γ_{1}, Γ_{2}, I) is an admissible triple if the following two conditions hold:

1. The weight assignments μ_{1,i} and μ_{2,i} of the roots of Γ_{1} and Γ_{2} coincide in the sense that μ_{1,i} = μ_{2,i} for all i = 1, …, r.

12Namely we identify the free end of a root with the free end of another root to form an edge.

2. After connecting the i-th root in Γ_{1} with the i-th root in Γ_{2} for all i, we obtain a new graph with k-edges (but without roots). We demand that this graph is connected.

Let η = (Γ_{1}, Γ_{2}, I) be an admissible triple. Using the subset I ⊂ [k] we obtain a unique bijection [k_{1}] ∪ [k_{2}] → [k] so that it preserves the orders of [k_{1}] and [k_{2}] and the image of [k_{1}] is I ⊂ [k]. Using this bijection, we get a unique ordering of the legs of this new graph. The genus of this graph is

g(η) = r + 1 − |V (Γ)| + ∑_{v∈V(Γ_{1})∪V(Γ_{2})} g(v)

and the degree of this graph is

b(η) = ∑_{i=1}^{k_{1}} b_{Γ_{1}}(i) · c_{1}(H_{Y_{1}}) + ∑_{i=1}^{k_{2}} b_{Γ_{2}}(i) · c_{1}(H_{Y_{2}}).

For simplicity, we denote by |η| the triple

|η| = (g(η), b(η), k).

Now let f_{1} :X_{1} →Y_{1} be a relative stable morphism in 𝔐(𝔜_{1}^{rel}, Γ_{1})(S), where (Y_{1}, D_{1}) ∈ 𝔜_{1}^{rel}(S), and let q_{1,i}:S → X_{1} be the i-th distinguished section. Then each f_{1} ∘ q_{1,i} : S → Y_{1} factor through D_{1} ⊂ Y_{1}, which composed with the first projection pr_{D} of D ≅ D×S defines a morphism S → D. This way we obtain an evaluation morphism

q(f_{1}) = (pr_{D} ∘ f_{1} ∘ q_{1,1}, …, pr_{D} ∘ f_{1} ∘ q_{1,r}) : S → D^{r}. (4.7)

Similarly, let f_{2} : X_{2} → Y_{2} be in 𝔐(𝔜_{2}^{rel}, Γ_{2})(S) with distinguished sections q_{2,i}. Then we have a similarly defined evaluation morphism q(f_{2}) : S → D^{r}. Now we construct a new family of morphisms over S. We glue the i-th sections q_{1,i}(S) ⊂ X_{1} with the i-th section q_{2,i}(S) ⊂ X_{2} for all i = 1, …, r to obtain an S-family of nodal curves X_{1} ⊔ X_{2}. Since η = (Γ_{1}, Γ_{2}, I) is admissible, the new family X_{1} ⊔ X_{2}/S has connected fibers. Also the data I ⊂ [k] defines a natural ordering of the k ordinary marked sections of X_{1}⊔X_{2}. Now let Y_{1}^{o}⊔Y_{2} be the S-scheme constructed in Corollary 4.2, then the pair of morphisms (f_{1}, f_{2}) defines a morphism

f_{1} ⊔ f_{2} : X_{1} ⊔ X_{2} ⟶ Y_{1}^{o} ⊔ Y_{2} (4.8)

if and only if

q(f_{1}) ≡ q(f_{2}) : S → D^{r}.

We state it as a Proposition.

Proposition 4.12. Let η = (Γ_{1}, Γ_{2}, I) be an admissible triple and let f_{i} be relative stable morphisms in 𝔐(𝔜_{i}^{rel}, Γ_{i})(S), i = 1, 2. Suppose q(f_{1}) ≡ q(f_{2}), then the morphism f_{1} ⊔ f_{2} is an S-family of stable morphisms in 𝔐(𝔚, Γ)(S) with Γ = |η|.

Proof. The proof is straightforward and will be omitted. q.e.d.

Let f ∈ 𝔐(𝔚, Γ)(S) be an S-family of stable morphisms to W. In case there is an admissible triple η = (Γ_{1}, Γ_{2}, I) and families f_{i} ∈ 𝔐(𝔜_{i}^{rel}, Γ_{i})(S) so that f = f_{1} ⊔ f_{2}, then we say f is η-decomposable or f is η-decomposed into families f_{1} and f_{2}.

We let 𝔐(𝔜_{i}^{rel}, Γ_{i}) → D^{r} be the morphisms induced by q in (4.7). Then this Proposition implies that we have a natural morphism of stacks

(4.9) 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔚, Γ).

Note that if r = 0, then η is admissible implies that one of Γ_{i} is empty. Then the fiber product is understood to be either 𝔐(𝔜_{2}^{rel}, Γ_{2}) in case Γ_{1} = ∅ or the otherwise.

Clearly, the above morphism factor through the substack 𝔐(𝔚, Γ) ×_{C}0, and is finite and representable. We let 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η) be the image stack in 𝔐(𝔚, Γ). We denote the induced morphism by

(4.10) Φ_{η} : 𝔐(𝔜_{1}^{rel}, Γ_{1}) ×_{D^{r}} 𝔐(𝔜_{2}^{rel}, Γ_{2}) ⟶ 𝔐(𝔜_{1}^{rel} ⊔ 𝔜_{2}^{rel}, η).

We now investigate the degree of Φ_{η}. Let f:X → W [n]_{0} be a stable morphism that lies in the image of Φ_{η}. By definition, there is an integer l ∈ [n + 1] so that we can η-decompose f along the divisor D_{l} ∩ W [n]_{0}. Namely, we have a pair of relative stable morphisms f_{i} : X_{i} → Y_{i}[n_{i}], where n_{1} = l − 1 and n_{2} = n + 1 − l, an ordered finite set Σ and embeddings Σ ⊂ X_{i} so that X is the result of gluing X_{1} and X_{2} along Σ ⊂ X_{1} and Σ ⊂ X_{2}, and the restriction of f to X_{i} is f_{i} using the gluing Y_{1}[n_{1}]^{o} ⊔ Y_{2}[n_{2}] = W [n]_{0}. Further, if we view Σ ⊂ X_{i} as its ordered distinguished marked points, then the topological type of f_{i} with the induced ordering on its ordinary marked points is exactly the Γ_{i} in η = (Γ_{1}, Γ_{2}, I). Here I ⊂ [k] is the subset of those marked points that are in X_{1}. We fix such a ordering on Σ.

Now let σ ⊂ S_{r} be any permutation. Then σ induces a new ordering on Σ, which we denote by Σ^{σ}. Accordingly, for the Γ_{i} in η = (Γ_{1}, Γ_{2}, I), we define Γ_{i}^{σ} to be the graph Γ_{i} with its roots reordered according to σ. Namely, the j-th root of Γ_{i}^{σ} is the σ^{−1}(j)-root of Γ_{i}. We define η^{σ} = (Γ_{1}^{σ}, Γ_{2}^{σ}, I). For any two admissible triples η_{1} and η_{2} we say η_{1} ∼ η_{2} (called equivalent) if there is a σ ∈ S_{r} so that η_{1} is isomorphic to η_{2}^{σ}. We define Eq(η) to be the subgroup of σ ∈ S_{r} so that η ∼ η^{σ}. Note that η ∼ η^{σ} implies that whenever σ(j_{1}) = j_{2} then the j_{1}-th and the j_{2}-th roots of Γ_{i} have identical weights μ_{j_{1}} = μ_{j_{2}} and are attached to the same vertex of Γ_{i}.

Proposition 4.13. The morphism Φ_{η} is finite and étale. It has pure degree^{13} | Eq(η)|.

Proof. The proof that the morphism Φ_{η} is finite and étale is straightforward, and will be left to readers. We now check that the degree of Φ_{η} is as stated. Let ξ ∈ Im(Φ_{η}) be any element. It follows from the discussion preceding to the statement of this Proposition that there is an l ∈ [n+1] so that the decomposition of f along the divisor D_{l} ∩W [n]_{0} is the η-decomposition of f. Because f is stable, for all other l′ ∈ [n+1]−l the decomposition types of f along D_{l′} ∩ W [n]_{0} will be different from η (i.e., ≁). We now let ξ_{i} be represented by f_{i} : X_{i} → Y_{i}[n_{i}], after fixing an ordering on Σ = f^{−1}(D_{l}) so that the topological type of ξ is Γ_{i}. Thus f_{i} ∈ 𝔐(𝔜_{i}^{rel}, Γ_{i}) and f = f_{1} ⊔ f_{2}. Now let σ ∈ S_{r} be any permutation. We let f_{i}^{σ} be the same morphism f_{i}: X_{i} → Y_{i}[n_{i}] except that the distinguished marked points of X_{i} are reordered according to σ. This way

(f_{i}^{σ}, f_{2}^{σ}) ∈ 𝔐(𝔜_{1}^{rel}, Γ_{1}^{σ}) ×_{D^{r}} 𝔐(𝔜_{1}^{rel}, Γ_{1}^{σ}).

Clearly, it is in the domain of Φ_{η} if and only if η ∼ η^{σ}.

To derive the degree formula, we need to investigate the automorphism group Aut(ξ) of ξ ∈ 𝔐(𝔚, Γ) and the automorphism group Aut(ξ_{i}) of 𝔐(𝔜_{i}^{rel}, Γ_{i}). First, the automorphism group Aut((ξ_{1}, ξ_{2})) is naturally isomorphic to Aut(ξ_{1}) × Aut(ξ_{2}). Because f is derived from gluing f_{1} and f_{2} along Σ ⊂ X_{1} and Σ ⊂ X_{2}, elements in Eq(η) induce permutations of Σ. Hence we obtain a natural homomorphism of groups h:Aut(ξ) → S_{r} whose kernel is isomorphic to Aut(ξ_{1})×Aut(ξ_{2}). Further the image of h lies in the subgroup Eq(η) ⊂ S_{r} and the coset Eq(η)/ Im(η) is exactly the set Φ_{η}^{−1}(ξ). This shows that Φ_{η}^{−1}(ξ) consists of exactly | Eq(η)|/| Im(h)| distinct elements. Hence the degree of Φ_{η} is

| Eq(η)|/| Im(η)| · | Aut(ξ)| · 1/| Aut(ξ_{1}) × Aut(ξ_{2})| = | Eq(η)|.

This proves Proposition 4.13. q.e.d.

13Here we say f:X → Y has pure degree d if for any integral A ⊂ Y the restriction X ×_{Y} A → A has degree d.

5. Some Comments

It is interesting to work out the analogous construction for the moduli spaces of stable sheaves over schemes. Before the discovery of SWinvariants, there were a lot of research work on using degeneration methods to study the Donaldson invariants. The differential geometric approach to this is well understood. But the algebro-geometric approach to this is still missing. It is clear that one should consider the moduli of stable sheaves over the stack 𝔚 and the stack ℨ^{rel}. It is interesting to work this out in detail. One technical issue is that after using the expanded degenerations one should be able to avoid non-locally freeness (of sheaves) along the nodal divisor. This is the case for curves, following the pioneering work of [13]. It is true for surfaces, which is essentially proved by Gieseker and the author in [12].

A more challenging degeneration problem is to workout a degeneration of GW-invariants for the families W → C whose singular fibers are only assumed to have normal crossing singularities. Recall that the family W/C studied in this paper only deal with the case where W_{0} has two irreducible components that intersect along a connected smooth divisor. It is easy to see that this construction can be generalized to the case where W_{0} is only assumed to have normal crossing singularity and its singular locus is smooth. When the singular locus of W_{0} is not smooth, it is not clear what constitutes the class of expanded degenerations of W/C. The progress along this line is important to the study of degeneration of moduli spaces.

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Stanford University, Stanford, CA 94305