ON ABEL MAPS OF STABLE CURVES

LUCIA CAPORASO AND EDUARDO ESTEVES

Abstract. We construct Abel maps for a stable curve X. Namely, for each one-parameter deformation of X to a smooth curve, having regular total space, and each integer d ≥ 1, we construct by specialization a map α_{X}^{d} : Ẋ^{d} → \overline{P_{X}^{d}}, where Ẋ ⊆ X is the smooth locus, and \overline{P_{X}^{d}} is the moduli scheme for balanced line bundles on semistable curves over X. For d = 1, we show that α_{X}^{1} extends to a map \overline{α_{X}^{1}} : X → \overline{P_{X}^{1}}, and does not depend on the choice of the deformation. Finally, we give a precise description of when \overline{α_{X}^{1}} is injective.

Contents

1. Introduction 1 2. Néron models of Picard schemes 3 3. Abel maps to balanced Picard schemes 7 4. Modular interpretation of the first Abel map 16 5. Completing the first Abel map 22 References 32

1. Introduction

Let C be a smooth projective curve, and Pic^{d} C its degree-d Picard variety parametrizing line bundles of degree d on C. For each d > 0 there exists a remarkable morphism, often called the d-th Abel map:

\begin{matrix}C^{d} & ⟶ & Pic^{d} C \\ (p_{1}, …, p_{d}) & ↦ & O_{C}(∑ p_{i}). \end{matrix}

Such a map has been extensively studied and used in the literature. For example if d = 1, after a base point on C is chosen, it gives the Abel–Jacobi embedding C ↪ Pic^{1} C ≅ Pic^{0} C (unless C ≅ ℙ^{1}). For an interesting historic survey see [K04] or [K05].

What about Abel maps for singular curves? Not much is known for reducible curves, whereas the case of integral curves is better understood, as we will explain further down. In the present paper we construct Abel maps for stable curves (including reducible ones, of course).

As we see it, Abel maps should satisfy the following natural properties. First, they should have a geometric meaning. More explicitly, recall that for a smooth curve C the d-th Abel map is the moduli map of a natural line bundle on C^{d} × C; see 2.5. We want a similar property to hold for singular curves as well.

Second, Abel maps should vary continuously in families. In particular, given a one-parameter family of smooth curves specializing to a singular one, we expect the d-th Abel maps of the smooth fibers to specialize to the d-th Abel map of the singular fiber.

Both requirements turn out to be nontrivial. In order to address the second one we view stable curves as limits of smooth ones. More precisely, let X be a stable curve. Let f : X → B be a family of curves over a local one-dimensional regular base B, with regular total space X, smooth generic fiber and X as closed fiber. We observe that there exists a canonical way to partially extend the d-th Abel map of the generic fiber of f, by using the Néron model N_{f}^{d} of the degree-d Picard scheme of that fiber. The Néron mapping property yields a close relative of the d-th Abel map of X, defined on the nonsingular locus Ẋ^{d} ⊆ X^{d}, which we call the d-th Abel–Néron map of X; see 2.6. The target of this map is the closed fiber of N_{f}^{d}, rather than the Picard scheme.

The great advantage of Abel–Néron maps is their naturality, obtained directly from the universal property of Néron models. However, they have two major drawbacks. First, they do not have any a-priori modular interpretation. Second, they are not defined on the whole X^{d}.

To attack these problems we consider the modular compactified Picard variety introduced in [C94] and further studied in [C05]. If X is suitably general, more precisely “d-general” (Definition 3.6), there exists a natural proper B-scheme \overline{P_{f}^{d}} parametrizing equivalence classes of “balanced” line bundles on semistable curves; see 3.8. These are line bundles whose multidegree satisfies certain inequalities; see Definition 3.2. It is shown in [C05] that \overline{P_{f}^{d}} contains N_{f}^{d} as a dense open subscheme, hence it gives a geometrically meaningful description and completion of N_{f}^{d}.

So, assume for now that X is d-general. Let \overline{P_{X}^{d}} be the closed fiber of \overline{P_{f}^{d}}; it does not depend on the choice of f. Viewing N_{f}^{d} inside \overline{P_{f}^{d}}, it is thus

possible to exhibit the d-th Abel–Néron map as a map α_{X}^{d} : Ẋ^{d} → \overline{P_{X}^{d}} (see Proposition 3.10) and thus give it a geometric interpretation. Moreover, since \overline{P_{X}^{d}} is complete, we are able to study whether α_{X}^{d} extends.

Although α_{X}^{d} is modular, an explicit description for it is hard to exhibit in full generality; we do that only for curves with two components (see Proposition 3.12). Without such a description it is difficult to study whether and how α_{X}^{d} extends.

The case d = 1 turns out to be easier. By means of Theorem 4.6 we give an explicit description of the line bundle defining α_{X}^{1}. Using this description, we show in Corollary 4.10 that α_{X}^{1} does not depend on the choice of f, a remarkable property not to be expected in general for d > 1; see Remark 3.14.

Refining our modular description of α_{X}^{1}, we construct a completion for it as a regular map \overline{α_{X}^{1}} : X → \overline{P_{X}^{1}} (Theorem 5.5). Finally, we prove that \overline{α_{X}^{1}} is as close as it can be to an injection (see Proposition 5.9 for the precise statement).

Now suppose that X is not 1-general, so that X lies in a proper closed subset of \overline{M}_{g} for g even; see Proposition 3.15. Then \overline{P_{f}^{1}} fails to contain N_{f}^{1}; nevertheless our existence results do extend, suitably modified (see 5.10), whereas uniqueness and injectivity results (like Proposition 5.9) may fail. In this case the setup is significantly more complicated for standard technical reasons (presence of non-GIT-stable points, or of nonfine moduli spaces.) This is why we chose to first work under the assumption of 1-generality, and to later indicate, in 5.10 and 5.13, how to modify proofs and statements to include the special case.

Abel maps were constructed for all integral curves in [AK], and further studied in [EGK00], [EGK02] and [EK05]. In [AK], it is shown that the first Abel map of an integral singular curve is an embedding into its compactified Picard scheme.

Constructing Abel maps for reducible curves presents further difficulties, due to the lack of natural, separated target spaces. The use of Néron models as target spaces is not new in the literature: in [E98] Abel–Jacobi maps for nodal curves were studied by means of the Néron mapping property, similarly to what we do here with our Abel–Néron maps. However, Néron models are seldom proper and thus we cannot expect Abel maps to Néron models to be defined everywhere. In this framework, our contribution is that of bringing compactified Picard schemes into the picture. This enables us to compactify Néron models and hence to obtain a target space into which complete Abel maps could be defined. In fact, we prove that α_{X}^{1} extends; it is still an open problem how to extend α_{X}^{d} for d > 1.

It is our intention to construct the first Abel map of any Gorenstein singular curve, using the modular compactified Picard schemes introduced in [E01]. For such curves the connection with Néron models is not available. Nonetheless, prompted by the results of the present paper, we found significant evidence for the existence of a modular interpretation similar to the present case.

Our paper is organized as follows. Section 2 is devoted to preliminaries of various types. In Section 3 we describe degree-d Abel–Néron maps to the compactified Picard scheme. In Section 4 we establish the modular description of the Abel–Néron map in degree 1, and show that it is independent of the choice of the deformation. Finally, in Section 5 we construct the completed degree-1 Abel map, give it a modular description and study when it is injective. Finally, from 5.10 to the end of the paper, we explain how to handle the special case of non-1-general curves.

2. Néron models of Picard schemes

2.1. Setup. We work over a fixed algebraically closed field k. All schemes are assumed locally of finite type over k, unless stated otherwise

For us, a curve is a reduced and connected projective scheme of dimension 1. Mostly, we will deal with nodal curves, that is, curves whose only singularities are nodes.

A regular pencil (of curves) is a flat projective morphism f : X → B between connected, regular, schemes such that dim B = 1, every geometric fiber of f is a curve, and f is smooth over a dense open subscheme of B.

We call a regular pencil f : X → B local if B = Spec R, where R is a discrete valuation ring (having k as residue field). If X is the closed fiber, we will also say that f is a regular smoothing of X.

For each regular pencil f : X → B we shall let K := k(B), the field of rational functions of B, and denote by X_{K} the generic fiber of f. Notice that X_{K} is a smooth curve over K.

Given any morphism f : X → B, and any integer d ≥ 1, let f_{d} : X_{B}^{d} → B denote the d-th fibered power of X over B. If f is a regular pencil, we denote the open subset of X_{B}^{d} where f_{d} is smooth by Ẋ_{B}^{d}; so

Ẋ_{B}^{d} := X_{B}^{d} ∖ Sing(f_{d})

If f is a local regular pencil, let Ẋ^{d} denote the closed fiber of Ẋ_{B}^{d} → B; so

Ẋ^{d} = {(p_{1}, …, p_{d}) : p_{i} ∈ X ∖ X_{sing}}.

Given any morphism f : X → B and any B-scheme T, the base change of f to T is denoted f_{T} : X_{T} → T.

2.2. The relative Picard scheme. Let f : X → B be a regular pencil, and d an integer. The closed fibers of f are geometric, by our general assumption, and the general fiber is smooth. Thus the irreducible components of the fibers of f are geometrically irreducible. By a theorem of Mumford’s, [BLR], Thm. 2, p. 210, the (relative) Picard scheme Pic_{f} of f exists, and is locally of finite type over S. Furthermore, Pic_{f} is formally smooth over B by [BLR], Prop. 2, p. 232, whence smooth over B by [BLR], Prop. 6, p. 37.

Let Pic_{f}^{d} be the degree-d Picard scheme of f, the open subscheme of Pic_{f} parametrizing line bundles of relative degree d. Given any B-scheme T and any line bundle L on X_{T} of f_{T} -relative degree d, there is a moduli map associated to L,

(1)  \begin{aligned}μ_{L} : & T & ⟶ & Pic_{f}^{d} \\ & t & ↦ & L_{t}, \end{aligned}

where L_{t} ∈ Pic^{d} X_{t} is the restriction of L to the fiber X_{t} := f_{T}^{−1}(t). The map μ_{L} determines L up to tensoring with pullbacks of line bundles from T. Notice that to a map T → Pic_{f}^{d} there does not necessarily correspond a line bundle on X_{T}, though the line bundle will exist, for instance, if f admits a section; see [BLR], Prop. 4, p. 204.

2.3. Néron models of Picard schemes. Let f : X → B be a regular pencil, and d an integer. Recall that a basic characteristic (and a drawback for various applications) of the Picard scheme Pic_{f}^{d} is that it is not separated over B, if f has reducible special fibers. One way to fix this is to introduce the Néron model:

N_{f}^{d} := N(Pic^{d} X_{K}).

The Néron model is a smooth, separated (possibly not proper) scheme of finite type over B with generic fiber equal to Pic^{d} X_{K}, which satisfies a fundamental mapping property that uniquely determines it. Namely, for every smooth B-scheme Z each map Z_{K} → Pic^{d} X_{K} extends uniquely to a map Z → N_{f}^{d}; see [BLR], Def. 1, p. 12.

The existence of N_{f}^{d} for any regular pencil f is likely well known. Since this result is fundamental for our work, but we could not find the precise statement to refer to, we sketch a proof of it using results in [BLR]. First, assume that f is local, that is, B is the spectrum of a discrete valuation ring R. Then there is a Néron model of Pic^{d} X_{K} over B, which is equal to Pic_{f}^{d} if f is smooth. Indeed, since Pic^{d} X_{K} is a Pic^{0} X_{K}-torsor, by descent theory we may assume that R is a strictly Henselian ring; see [BLR], Cor. 3, p. 158. In this case, f admits a section through its smooth locus, by [BLR], Prop. 5, p. 47. This section can be used to produce a B-isomorphism Pic_{f}^{d} → Pic_{f}^{0}. We may thus assume that d = 0. In this case, there is a Néron model of Pic^{0} X_{K} over B because Pic^{0} X_{K} is an Abelian variety over K; see [BLR], Cor. 2, p. 16. Furthermore, if f is smooth, Pic_{f}^{0} is an Abelian B-scheme, whence is the Néron model of Pic^{0} X_{K} over B by [BLR], Prop. 8, p. 15.

Now, consider the general case. Let U ⊆ B be the largest open subscheme over which f is smooth, and set h := f|_{f^{−1}(U)}. As we saw above, Pic_{h}^{d} restricts to the Néron model of Pic^{d} X_{K} locally around each point of U. Then Pic_{h}^{d} is the Néron model of Pic^{d} X_{K} over U by [BLR], Prop. 4, p. 13. Finally, the local existence of Néron models of Pic^{d} X_{K} around each point of B, and the existence of the Néron model over the dense open subscheme U ⊆ B imply the (global) existence of the Néron model over B, by [BLR], Prop. 1, p. 18.

Since Pic_{f}^{d} is smooth over B, a first consequence of the mapping property of the Néron model N_{f}^{d} is the existence of a canonical B-morphism

(2)  q_{f} : Pic_{f}^{d} ⟶ N_{f}^{d}

which is the identity on the generic fiber.

Assume now that the geometric fibers of f are nodal. Let X be a closed fiber of f. In the description of the Néron model, and also in our paper, the following set Tw_{f} X of (isomorphism classes of) distinguished line bundles plays an important role:

Tw_{f} X := \frac{\{O_{X} (D)|_{X} : D ∈ Div X with Supp D ⊂ X\}}{≅} ⊂ Pic^{0} X.

The divisors D appearing above are simply sums with integer coefficients of the components of X, which are Cartier divisors of X because X is regular. Line bundles in Tw_{f} X are called twisters. Here is a useful observation:

(3)  ∀ T, T′ ∈ Tw_{f} X,  T = T′ ⇔ \underline{de}g T = \underline{de}g T′.

Since twisters are specializations of the trivial line bundle of the generic fiber, O_{X_{K}}, all of them must be identified in any separated quotient of Pic_{f}^{0}. In particular, q_{f}(T) = q_{f}(O_{X}) for each T ∈ Tw_{f} X.

We shall now identify multidegrees that differ by multidegrees of twisters. Let γ be the number of irreducible components of X, and set

Λ_{X} := \{\underline{de}g T : T ∈ Tw_{f} X\} ⊆ ℤ^{γ}.

Define now an equivalence relation “≡” on multidegrees by setting

\underline{d} ≡ \underline{d}′ ⇔ \underline{d} − \underline{d}′ ∈ Λ_{X}.

The set of multidegree classes \underline{d} + Λ_{X} with fixed total degree |\underline{d}| equal to d is denoted by Δ_{X}^{d}. Thus

(4)  Δ_{X}^{d} := \frac{\{\underline{d} ∈ ℤ^{γ} : |\underline{d}| = d\}}{≡}.

It is well known that Δ_{X}^{0} is a finite group, a purely combinatorial invariant of X often called the degree class group of X, or the group of connected components of N_{f}^{d}. In addition, for each d there is a (nonunique) bijection Δ_{X}^{0} → Δ_{X}^{d}, obtained by summing with any multidegree \underline{d} with |\underline{d}| = d.

For each δ ∈ Δ_{X}^{d}, let \underline{d} be any multidegree representing δ, and set

(5)  Pic_{f}^{δ} := Pic\underline{d} ; f = \{L ∈ Pic_{f}^{d} : \underline{de}g L|_{X} = \underline{d}\} ⊂ Pic_{f}^{d} .

This definition is easily seen not to depend on the choice of the representative \underline{d} (cf. [C05], 3.9).

Assume now that f is a regular smoothing of X. At this point we are able to describe the Néron model of Pic^{d} X_{K}:

(6)  N_{f}^{d} ≅ \frac{∐_{δ∈Δ_{X}^{d}} Pic_{f}^{δ}}{∼_{K}} ,

where ∼_{K}” denotes the gluing along the generic fiber, equal to Pic^{d} X_{K}; see [C05], Lemma 3.10.

Let N_{X}^{d} denote the closed fiber of N_{f}^{d}. Observe that N_{X}^{d} is a disjoint union of finitely many copies of the generalized Jacobian of X: picking a representative \underline{d}^{δ} for each class δ ∈ Δ_{X}^{d}, we have

N_{X}^{d} ≅ ∐_{δ∈Δ_{X}^{d}} Pic^{\underline{d}^{δ}} X.

Although the above isomorphism is not canonical, we see that the scheme structure of N_{X}^{d} does not depend on f. The closed points of N_{X}^{d} are in 1-1 correspondence with the degree-d line bundles on X modulo twisters. In particular, for d = 0, we have q_{f}^{−1}(q_{f}(O_{X})) = Tw_{f}X.

2.4. Néron maps. Let f : X → B be a regular pencil. Let T be a B-scheme, and L a line bundle on X_{T} of relative degree d over T. Let μ_{L} : T → Pic_{f}^{d} be the moduli map of L, defined in 2.2. Consider the composition:

\overline{μ_{L}} : T ⟶^{μ_{L}} Pic_{f}^{d} ⟶^{q_{f}} N_{f}^{d}.

We call \overline{μ_{L}} the Néron map of L. Notice that L is certainly not determined by its Néron map, not even modulo pullbacks of line bundles on T. In fact, if D ⊂ X is a Cartier divisor entirely supported on a closed fiber of f, then L ⊗ O_{X_{T}} (D_{T}) has the same Néron map as L, because N_{f}^{d} → B is separated.

2.5. Abel–Néron maps. Let us recall the precise definition of the Abel map of a smooth curve, using the same set up of [GIT], Section 6 p.118, 119. Let h : C → S be a smooth curve over a scheme S, so that h is a smooth morphism whose fibers are curves. For each integer d ≥ 1, let C_{S}^{d} be the d-th fibered power of C over S. There is a canonical S-morphism

(7)  \begin{matrix}C_{S}^{d} & ⟶ & Pic_{h}^{d} \\ C_{s}^{d} ∋ (p_{1}, …, p_{d}) & ↦ & O_{C_{s}}(p_{1} + ⋯ + p_{d}), \end{matrix}

defined over each s ∈ S by taking a d-tuple of points of the fiber C_{s} to the line bundle associated to their sum, which we shall call the d-th Abel map of h. Recall that the above map is the moduli map of a natural line bundle on C_{S}^{d} ×_{S} C, namely the one associated to the Cartier divisor ∑_{1}^{d} S_{i}, where each S_{i} is the image of the i-th natural section σ_{i} of the first projection C_{S}^{d} ×_{S} C → C_{S}^{d}, given by

(8)  σ_{i}(p_{1}, …, p_{d}) = ((p_{1}, …, p_{d}), p_{i}).

We may apply this construction to a regular pencil f : X → B. First of all, since X_{K} is smooth over K, we may consider the d-th Abel map of X_{K}:

(9)  α_{K}^{d} : X_{K}^{d} ⟶ Pic^{d} X_{K}.

The above map extends to a map Ẋ_{B}^{d} → Pic_{f}^{d}. Indeed, the extension is the moduli map of the line bundle associated to the Cartier divisor E^{d} of Ẋ_{B}^{d} ×_{B} X, where E^{d} is the sum of the images of the d natural sections σ_{1}, …, σ_{d} given by (8) of the first projection Ẋ_{B}^{d} ×_{B} X → Ẋ_{B}^{d}. Composing with the map q_{f} of (2), we obtain the Néron map of O_{Ẋ_{B}^{d}×_{B}X} (E^{d}) (cf. 2.4), which is also an extension of α_{K}^{d}.

The first simple but crucial observation is the following (well-known) fact:

Lemma - Definition 2.6. Let f : X → B be a regular pencil. For each integer d ≥ 1 there exists a unique morphism, which we call the d-th Abel– Néron map of f,

N(α_{K}^{d}) : Ẋ_{B}^{d} ⟶ N(Pic^{d} X_{K}) = N_{f}^{d}

whose restriction to the generic fiber is α_{K}^{d}. The map N(α_{K}^{d}) is the Néron map of O_{Ẋ_{B}^{d}×_{B}X} (E^{d} +D) for every Cartier divisor D of Ẋ_{B}^{d} ×_{B} X supported

on any finite number of closed fibers of Ẋ_{B}^{d} ×_{B} X → B.

Proof. The existence of an extension of α_{K}^{d} is a straightforward consequence of the Néron mapping property, [BLR], Def. 1, p. 12: since Ẋ_{B}^{d} is smooth over B and has generic fiber X_{K}^{d}, the Abel map α_{K}^{d} admits a unique extension N(α_{K}^{d}) : Ẋ_{B}^{d} → N(Pic^{d} X_{K}).

Since also the Néron map of O_{Ẋ_{B}^{d}×_{B}X} (E^{d} + D), for any D as described in the lemma, extends α_{K}^{d}, the last statement follows from the fact that N_{f}^{d} is separated over B. ■

3. Abel maps to balanced Picard schemes

We want to give a modular interpretation of the Abel–Néron maps and, at the same time, study the problem of completing them. To do this we shall use some results of [C05], where Néron models are glued together over the moduli space of stable curves and are thereby endowed of a geometrically meaningful completion. Our moduli problem is centered around Definition 3.2 below. First, we recall a few concepts.

3.1. Let X be a nodal curve of arithmetic genus g ≥ 2. Denote by ω_{X} its canonical, or dualizing bundle. For each proper subcurve Z ⊊ X, always assumed to be complete, let Z′ := \overline{X ∖ Z} and k_{Z} := #Z ∩ Z′. Also, let w_{Z} := deg_{Z} ω_{X}. If Z is connected, denote by g_{Z} its arithmetic genus, and recall that

(10)  w_{Z} = 2g_{Z} − 2 + k_{Z},

a well-known identity that can be proved using adjunction.

We call X semistable (resp. stable) if k_{Z} ≥ 2 (resp. k_{Z} ≥ 3) for each smooth rational component Z of X. Those Z for which k_{Z} = 2 are called exceptional. A semistable curve is called quasistable if two exceptional components never meet each other. If X is semistable, it follows from (10) that w_{Z} ≥ 0 for each subcurve Z ⊆ X, with equality if and only if Z is a union of exceptional components.

A family of semistable (resp. stable, resp. quasistable) curves is a flat, projective map f : X → B whose geometric fibers are semistable (resp. stable, resp. quasistable) curves. A line bundle of degree d on such a family f : X → B is a line bundle on X whose restriction to each fiber has degree d.

Definition 3.2. Let X be a semistable curve of arithmetic genus g ≥ 2, and let L ∈ Pic^{d} X.

(i) We say that L and its multidegree \underline{de}g L are semibalanced if for each connected proper subcurve Z ⊊ X the Basic Inequality below holds:

(11)  m_{Z}(d) ≤ deg_{Z} L ≤ M_{Z}(d),

where

M_{Z}(d) := \frac{dw_{Z}}{2g − 2} + \frac{k_{Z}}{2}

and

m_{Z}(d) := \{ \begin{matrix}M_{Z}(d) − k_{Z} & if Z is not an exceptional component, \\ 0 & if Z is an exceptional component. \end{matrix}

(ii) We call L and \underline{de}g L balanced if they are semibalanced and if for each exceptional component E ⊂ X we have

deg_{E} L = 1.

(iii) We call L and \underline{de}g L stably balanced if they are balanced and if for each connected proper subcurve Z ⊊ X such that deg_{Z} L = m_{Z}(d) the complement Z′ is a union of exceptional components.

(iv) We denote by B_{X}^{d} the set of balanced multidegrees on X, and by B̃_{X}^{d}

the subset of stably balanced ones.^{1}

(v) A line bundle (of degree d) on a family of semistable curves is called semibalanced (balanced or stably balanced) if its restriction to each geometric fiber of the family is.

1In [C05] B_{X}^{d} denotes the semibalanced multidegrees.

Remark 3.3. Definition 3.2 is 4.6 of [C05], which originates from [C94], Section 3.1 and from [CCC04], 5.1.1. There the Basic Inequality is

| deg_{Z} L − \frac{dw_{Z}}{2g − 2}| ≤ \frac{k_{Z}}{2} ,

and the extra condition for when Z is an exceptional component is imposed separately. For convenience, we presented a set of inequalities, (11), that includes the exceptional cases. Abusing the terminology, we still call (11) the Basic Inequality.

We mention some simple but useful consequences of the definiton. (A) If X is stable, then (i) and (ii) coincide, i.e. a semibalanced line bundle

is always balanced. (B) There can only exist a balanced line bundle on a semistable curve X if X

is quasistable. Indeed, let Z ⊂ X be a connected chain of exceptional components. If L is a semibalanced line bundle on X, then L has degree 0 on every component of Z but possibly one, where L may have degree 1. However, if L is balanced, L cannot have degree 0 on any component of Z. Then Z ≅ ℙ^{1} and deg_{Z} L = 1.

(C) To check whether L is semibalanced it is enough to check whether deg_{Z} L ≥ m_{Z}(d) for each connected proper subcurve Z ⊊ X. Indeed, let Z be such a subcurve, and let Y_{1}, …, Y_{n} denote the connected components of Z′. By hypothesis,

deg_{Y_{i}} L ≥ m_{Y_{i}}(d) ≥ \frac{dw_{Y_{i}}}{2g − 2} − \frac{k_{Y_{i}}}{2}

for each i. Since k_{Y_{1}} + ⋯ + k_{Y_{n}} = k_{Z′} = k_{Z}, summing up the above inequalities we get

deg_{Z′} L ≥ \frac{dw_{Z′}}{2g − 2} − \frac{k_{Z}}{2} .

Now, since deg_{Z} L + deg_{Z′} L = d and w_{Z} + w_{Z′} = 2g − 2, we get

deg_{Z} L = d − deg_{Z′} L ≤ d − \frac{dw_{Z′}}{2g − 2} + \frac{k_{Z}}{2} = \frac{dw_{Z}}{2g − 2} + \frac{k_{Z}}{2} .

3.4. In [C05], Lemma 4.4, it is proved that each multidegree class has a semibalanced representative. More precisely, fix an integer d, and let X be a stable curve. Recall the notation in (4) and 3.2 (iv). Then Lemma 4.4 of [C05] implies that the natural map below is surjective (square brackets denoting classes):

(12)  \begin{aligned}[…] : & B_{X}^{d} & ⟶ & Δ_{X}^{d} \\ & \underline{d} & ↦ & [\underline{d}] \end{aligned}

We shall say that X is “d-general” if the map (12) is bijective; see Definition 3.6 below

3.5. The moduli problem for balanced line bundles was introduced and studied in [C94] to compactify the universal Picard scheme over M_{g}. That compactification was constructed as a GIT-quotient. We do not need to recall the details of the construction here, only a few facts. There are morphisms

H_{d} ⟶^{π_{d}} H_{d}/G ⟶^{ϕ_{d}} \overline{M}_{g}

where H_{d} is an open subscheme of a suitable Hilbert scheme, acted upon by an algebraic group G, the map π_{d} is a GIT-quotient map, and ϕ_{d} is a surjective, projective morphism. The quotient scheme H_{d}/G, denoted by \overline{P}_{d, g}, is integral and projective. The fiber of ϕ_{d} over a general smooth curve X is exactly Pic^{d} X; see [C94], Thm. 6.1.

Denote by U ⊆ \overline{P}_{d, g} the nonempty open subscheme over which π_{d} restricts to a geometric GIT-quotient, i.e. all fibers are orbits, and all stabilizers are finite and reduced.

Definition 3.6. Let X be a stable curve of arithmetic genus g ≥ 2. We say that X is d-general if any of the following equivalent conditions hold:

(i) ϕ_{d}^{−1}(X) ⊂ U. (ii) The class map (12) of 3.4 is bijective. (iii) Every balanced line bundle on X is stably balanced, i.e. B̃_{X}^{d} = B_{X}^{d}.

The equivalence of these three conditions follows from [C94] Lemma 6.1. It is known that every stable curve of genus g is d-general if and only if

the integers d and g satisfy (d − g + 1, 2g − 2) = 1; see [C94], Prop 6.2.

3.7. We need to recall when two semibalanced line bundles are defined to be equivalent. Let X be a stable curve, and X_{1} and X_{2} two semistable curves having X as stable model. For each i = 1, 2 let L_{i} be a semibalanced line bundle on X_{i}. Let Y_{i} be the semistable curve obtained by contracting all exceptional components of X_{i} where L_{i} has degree 0. Then there is a unique line bundle M_{i} on Y_{i} whose pullback to X_{i} is L_{i}. Since L_{i} is semibalanced, M_{i} is balanced and Y_{i} is quasistable. Let F_{i} ⊆ Y_{i} be the union of all the exceptional components of Y_{i}, and let Ỹ_{i} := F′_{i}, the complementary subcurve of F_{i} in Y_{i}. Then L_{1} and L_{2} are equivalent if Y_{1} = Y_{2} and M_{1}|Ỹ_{1} ≅ M_{2}|Ỹ_{2}. Notice that M_{i} is equivalent to L_{i} for i = 1, 2.

Thus, every equivalence class includes always a balanced line bundle N on a quasistable curve Y. The quasistable curve Y is unique, but N is not. What is unique is the restriction of N to Ỹ, the complementary subcurve of the union F of all the exceptional components of Y. The quasistable curve Y and N|Ỹ determine the equivalence class. The restriction N|_{F} is also unique, since a balanced line bundle must have degree 1 on every exceptional component. So, our equivalence relation disregards the gluing data of the bundles over the points in Ỹ ∩ F.

If X → B is a family of semistable curves, then two semibalanced line bundles L_{1} and L_{2} on X → B are called equivalent if and only if their restrictions to every geometric fiber of X → B are equivalent in the sense explained above.

3.8. Let d and g be integers, with g ≥ 2. Assume first that d − g + 1 and 2g − 2 are coprime, so that every stable curve of arithmetic genus g is d-general. Then the construction summarized in 3.5 can be improved, by considering stacks. More precisely, there exist two (modular) Deligne– Mumford stacks \overline{P}_{d,g} and P_{d,g}, each one equipped with a natural, strongly representable morphism to \overline{M}_{g}. (To tie in with 3.5, notice that \overline{P}_{d,g} is the quotient stack [H_{d}/G].) The following properties hold; see [C05], Section 5 for details:

(A) For each (d-general) stable curve X, denote by P_{X}^{d} and \overline{P_{X}^{d}} the fibers of P_{d,g} and \overline{P}_{d,g} over X. Since P_{d,g} and \overline{P}_{d,g} are strongly representable over \overline{M}_{g}, both P_{X}^{d} and \overline{P_{X}^{d}} are quasiprojective schemes.

The first, P_{X}^{d}, is the fine moduli scheme of degree-d balanced line bundles on X. The second, \overline{P_{X}^{d}}, is the coarse moduli scheme of equivalence classes of degree-d semibalanced line bundles on semistable curves having X as stable model; see 3.7. Actually, \overline{P_{X}^{d}} is not far from being a fine moduli scheme; see (C) below.

P_{X}^{d} lies naturally inside \overline{P_{X}^{d}} as an open and dense subscheme. (B) Let f : X → B be any family of (d-general) stable curves of genus g,

and consider the schemes

P_{f}^{d} := B ×_{\overline{M}_{g}} P_{d,g} and \overline{P_{f}^{d}} := B ×_{\overline{M}_{g}} \overline{P}_{d,g}.

(That these are indeed schemes follows, again, from the fact that the maps P_{d,g} → \overline{M}_{g} and \overline{P}_{d,g} → \overline{M}_{g} are strongly representable.) We have a natural inclusion P_{f}^{d} ⊂ \overline{P_{f}^{d}}. An explicit description of P_{f}^{d}, for

when f is a local regular pencil, is (13). As for \overline{P_{f}^{d}}, the following fact holds: to each triple (T, Y → T, L) where T is a B-scheme, Y → T is a family of semistable curves having f_{T} : X_{T} → T as stable model, and L is a semibalanced line bundle of degree d on Y → T, there corresponds a moduli map

μ̂_{L} : T ⟶ \overline{P_{f}^{d}},

taking each geometric point t of T to the equivalence class of the restriction of L to the (geometric) fiber of Y over t. We call μ̂_{L} the moduli map of L.

The image of μ̂_{L} is contained in P_{f}^{d} if and only if L has degree 0 on every exceptional component of every geometric fiber of Y → T.

(C) The scheme P_{f}^{d} is a fine moduli scheme; see [C05], Cor. 5.14 and

Rmk. 5.15. Also, \overline{P_{f}^{d}} is not far from being a fine moduli scheme. In fact,

it is endowed with a quasiuniversal pair (Z → \overline{P_{f}^{d}}, N), where Z → \overline{P_{f}^{d}}

is a family of quasistable curves having f\overline{P_{f}^{d}} : X\overline{P_{f}^{d}} → \overline{P_{f}^{d}} as stable

model, and N is a balanced line bundle of degree d on Z → \overline{P_{f}^{d}} that has a role similar to that of a Poincaré bundle. Indeed, for each triple (T, Y → T, L) where T is a B-scheme, Y → T is a family of semistable curves with stable model f_{T} : X_{T} → T, and L is a semibalanced line bundle of degree d on Y → T, there is a map Y → Z such that the diagram of maps below is commutative,

Y ⟶ Z
↓  ↓
T ⟶^{μ̂_{L}} \overline{P_{f}^{d}},

and such that L is equivalent to the pullback of N to Y; see 3.7. (The map Y → Z is certainly not uniquely determined, which is why we call the pair (Z → \overline{P_{f}^{d}}, N) quasiuniversal.) Furthermore, if Y → T is a family of quasistable curves, and L is balanced, then the map Y → Z can be chosen such that the above diagram is a fibered product diagram.

Remark 3.9. If (d − g + 1, 2g − 2) ≠ 1, almost everything in 3.8 works over the open subset of \overline{M}_{g} parametrizing d-general stable curves. For a proof, it suffices to argue exactly as for Theorem 5.9 in [C05], after replacing \overline{M}_{g} with the substack of d-general curves, and the stacks P_{d,g} and \overline{P}_{d,g} with the corresponding substacks (over d-general curves).

The only assertion in 3.8 that does not hold is the existence of a “Poincaré” line bundle, in (C), which will never be used in this paper.

We are ready to go back to the study of Abel maps.

Proposition 3.10. Let f : X → B be a regular pencil of d-general stable curves. Then there exists a canonical map

α_{f}^{d} : Ẋ^{d} ⟶ P_{f}^{d}

which restricts to the d-th Abel map on the generic fiber.

Proof. We may glue local extensions of the d-th Abel map of X_{K} because they are unique. Thus we may assume f is local; let X be the closed fiber of f. In this case, the explicit description of P_{f}^{d} is (see [C05], Cor. 5.14)

(13)  P_{f}^{d} = \frac{∐_{\underline{d}∈B_{X}^{d}} Pic\underline{d} ; f}{∼_{K}} ,

where, as in 2.3, “∼_{K}” means gluing over the generic fiber.

As we know from (6) in 2.3, N_{f}^{d} is described in a very similar way. Indeed, by [C05], Thm. 6.1, we have a canonical isomorphism

(14)  ϵ_{f}^{d} : P_{f}^{d} ⟶^{≅} N_{f}^{d}.

To describe it precisely is straightforward: for each \underline{d} ∈ B_{X}^{d} the restriction of ϵ_{f}^{d} to Pic\underline{d} ; f is the natural isomorphism

Pic\underline{d} ; f ⟶^{≅} Pic_{f}^{[\underline{d}]} ⊂ N_{f}^{d},

restricting to the identity on the generic fibers. The isomorphism ϵ_{f}^{d} is completely described because, since X is d-general, the class map B_{X}^{d} → Δ_{X}^{d} is bijective, by Definition 3.6. To conclude, use Lemma 2.6 and (14) to define α_{f}^{d} := (ϵ_{f}^{d})^{−1} ∘ N(α_{K}^{d}). ■

We call α_{f}^{d} the d-th Abel map of f. The natural problem now is to describe α_{f}^{d} as the moduli map of a balanced line bundle on π : Ẋ^{d} ×_{B} X → Ẋ^{d}. Since P_{f}^{d} is a fine moduli scheme, this should be possible. In fact, the proof of [C94], Prop. 4.1, p. 621, can be used to produce an algorithm for determining the necessary twisters we need to tensor O_{Ẋ^{d}×_{B}X} (E^{d}) with to get the balanced line bundle.

However the explicit description of this line bundle turns out to be difficult to find in general. In Section 4 we will find it for d = 1. In the next subsection, 3.11, we will do that for every d in a special case.

3.11. Two-component curves. Let X be a stable curve with only two irreducible components, C_{1} and C_{2}. Let g be the arithmetic genus of C and δ := #C_{1} ∩ C_{2}. Set (cf. 3.3)

m := ⌈m_{C_{1}}(d)⌉ = ⌈\frac{dw_{C_{1}}}{2g − 2} − \frac{δ}{2}⌉.

The set B_{X}^{d} of balanced multidegrees of X satisfies

(15) B_{X}^{d} ⊇ {(m, d − m), (m + 1, d − m − 1), …, (m + δ − 1, d − m − δ + 1)},

with equality if and only if m_{C_{1}}(d) is not integer, if and only if X is d-general.

For each integer a such that 0 ≤ a ≤ d define r(a) to be the integer determined by the following two conditions:

0 ≤ r(a) < δ and a − m ≡ r(a) mod δ.

Using this notation we have:

Proposition 3.12. Let X be a stable curve with exactly two irreducible components, C_{1} and C_{2}. Let δ := #C_{1} ∩ C_{2}. For each regular smoothing f : X → B of X, let

L^{(d)} := O_{Ẋ^{d}×_{B}X} (E^{d} + ∑_{a=0}^{d} \frac{a − m − r(a)}{δ}C_{1}^{a} × C_{2}^{d−a} × C_{1})

where, abusing notation, we view

C_{1}^{a} × C_{2}^{d−a} × C_{1} ⊂ X^{d} × X ⊂ X^{d} ×_{B} X

as a Cartier divisor of Ẋ^{d} ×_{B} X, by restriction. Then L^{(d)} is balanced on fẊ^{d} : Ẋ^{d} ×_{B} X → Ẋ^{d}. Furthermore, if X is d-general, then the d-th Abel map α_{f}^{d} : Ẋ^{d} → P_{f}^{d} is the moduli map of L^{(d)}.

Proof. Since each C_{1}^{a} × C_{2}^{d−a} × C_{1} is supported over the closed point of B, the line bundle L^{(d)} coincides with O_{Ẋ^{d}×_{B}X} (E^{d}) over the generic point of B. This implies that L^{(d)} and O_{Ẋ^{d}×_{B}X} (E^{d}) have the same Néron map.

If X is d-general, α_{f}^{d} is defined and coincides with the Néron map of O_{Ẋ^{d}×_{B}X} (E^{d}) on X_{K}^{d}. On the other hand, if the line bundle L^{(d)} is balanced on π : Ẋ^{d} ×_{B} X → Ẋ^{d}, there is an associated moduli map μ̂_{L^{(d)}} : Ẋ^{d} → P_{f}^{d} by 3.8 (B). Since μ̂_{L^{(d)}} coincides with the Néron map of L^{(d)} on X_{K}^{d}, it follows that μ̂_{L^{(d)}} = α_{f}^{d}. Thus, it suffices to prove that L^{(d)} is balanced.

To verify this, we must compute the multidegree of the restriction of L^{(d)} to every singular fiber of π : Ẋ^{d} × X ⟶ Ẋ^{d}. Of course, all singular fibers lie over Ẋ^{d}. So, let p ∈ Ẋ^{d}, and denote by X_{p} the fiber π^{−1}(p). The point p determines a unique pair of nonnegative integers (a_{0}, b_{0}) such that a_{0} + b_{0} = d and p ∈ C_{1}^{a_{0}} × C_{2}^{b_{0}}. Then

(16)  (E^{d} · C_{1}, E^{d} · C_{2}) = (a_{0}, b_{0}) = (a_{0}, d − a_{0}).

To compute the intersection degrees with C_{1} and C_{2} of the remaining summands defining L^{(d)}, notice first that, since p ∈ C_{1}^{a_{0}} × C_{2}^{b_{0}}, the only nonzero degrees come from the summand indexed by a = a_{0}. Now, using

(C_{1}^{a_{0}} × C_{2}^{b_{0}} × C_{1}) · (C_{1}, C_{2}) = (−δ, δ)

and (16), we get

\underline{de}g_{X_{p}} L^{(d)} = (m + r(a_{0}), d − m − r(a_{0})),

which is balanced because 0 ≤ r(a_{0}) < δ; see (15). ■

Example 3.13. Let X be a “split” curve of arithmetic genus g, that is, X = C_{1}∪C_{2} with C_{i} ≅ ℙ^{1} and #C_{1}∩C_{2} = g+1. Then, for each d = 1, …, g and any regular smoothing f : X → B of X, the map α_{f}^{d} is the moduli map of O_{Ẋ^{d}×_{B}X} (E^{d}).

In particular, for any p_{1}, …, p_{d} ∈ Ẋ we have, independently of f,

α_{f}^{d}(p_{1}, …, p_{d}) = O_{X}(p_{1} + ⋯ + p_{d}).

A split curve is d-general if and only if d ≡ g mod 2. Actually, the conclusion of Example 3.13 is valid regardless of X being d-general, because at any rate O_{Ẋ^{d}×_{B}X} (E^{d}) is stably balanced on fẊ^{d}.

Remark 3.14. The case of split curves is in a sense special. In general, we should expect the restriction α_{f}^{d}|_{X} of the d-th Abel map of Proposition 3.12 to depend on the choice of smoothing f. For a simple concrete example of this dependence, consider the case d = 2 and δ = 2. Then X is stable and 2-general if C_{1} and C_{2} have distinct positive arithmetic genera. Suppose C_{1} has smaller genus. Then m = 0, and thus r(0) = 0, r(1) = 1, but r(2) = 0. Let p, q ∈ C_{1} ∖ C_{1} ∩ C_{2}. Then

α_{f}^{2}(p, q) = O_{X}(p + q) ⊗ O_{X} (C_{1})|_{X}.

Now, as f varies through all smoothings of X, the restriction O_{X} (C_{1})|_{X} varies through all line bundles restricting to O_{C_{1}}(−r_{1} −r_{2}) and O_{C_{2}}(r_{1} +r_{2}), where r_{1} and r_{2} are the nodes of X. So α_{f}^{2}(p, q) depends on the choice of f.

Denote by Σ_{g}^{d} the locus in \overline{M}_{g} of curves that are not d-general. Then Σ_{g}^{d} is a proper closed subset of \overline{M}_{g}; in fact, with the notation of 3.5, Σ_{g}^{d} is the image via ϕ_{d} of the complement of U. As we mentioned in 3.5, Σ_{g}^{d} empty if and only if (d − g + 1, 2g − 2) = 1. Since the rest of our paper is devoted to Abel maps for d = 1, we conclude this section by describing the locus of curves that are not 1-general.

Proposition 3.15. Let g ≥ 2.

(i) If g is odd then Σ_{g}^{1} is empty. (ii) If g is even then Σ_{g}^{1} is the closure in \overline{M}_{g} of the locus of curves X

such that X = C_{1} ∪ C_{2}, with C_{1} and C_{2} smooth of the same genus and #C_{1} ∩ C_{2} odd.

Proof. If g is odd then

(1 − g + 1, 2g − 2) = (g − 2, g − 1) = 1.

So Part (i) follows; see 3.6.

For Part (ii), let X be a stable curve. Suppose first that X has the description given in (ii). Then

(\frac{1 − k}{2} , \frac{1 + k}{2} ) ∈ B_{X}^{1} ∖ B̃_{X}^{1},

where k := k_{C_{1}} = k_{C_{2}}. So X is in Σ_{g}^{1}. Since Σ_{g}^{1} is closed in \overline{M}_{g}, it contains the closure of the locus defined in (ii).

Suppose now that X is in Σ_{g}^{1}, i.e. there is a line bundle L on X such that \underline{de}g L ∈ B_{X}^{1} ∖ B̃_{X}^{1}. Then there is a connected, proper subcurve Z ⊊ X such that either m_{Z}(1) or M_{Z}(1) is equal to deg_{Z} L. Then both m_{Z}(1) and M_{Z}(1) are integers. Thus

(17)  \frac{w_{Z}}{w} + \frac{k_{Z}}{2} ∈ ℤ

where w := deg ω_{X} = 2g − 2. Now, since X is stable,

(18)  0 < \frac{w_{Z}}{w} < 1.

In particular, \frac{w_{Z}}{w} is never an integer, and thus (17) implies that k_{Z} is odd. Since k_{Z} is odd, (17) and (18) immediately yield \frac{w_{Z}}{w} = \frac{1}{2}.

If Z′ is connected, then, since \frac{w_{Z′}}{w} = \frac{1}{2} as well, we have g_{Z} = g_{Z′}. Since both Z and Z′ are limits of smooth curves, X lies in the closure of the locus described in (ii).

Thus it remains to show that Z′ is also connected. Let Z′_{1}, …, Z′_{m} be the connected components of Z′. Notice that

(19)  k_{Z′_{1}} + ⋯ + k_{Z′_{m}} = k_{Z′} = k_{Z}.

Suppose by contradiction that m > 1. Then

(20)  0 < \frac{w_{Z′_{i}}}{w} < \frac{1}{2}

for each i. Since \underline{de}g L ∈ B_{X}^{1}, we have

\frac{w_{Z′_{i}}}{w} − \frac{k_{Z′_{i}}}{2} ≤ deg_{Z′_{i}} L ≤ \frac{w_{Z′_{i}}}{w} + \frac{k_{Z′_{i}}}{2}

for each i. Using (20), we get

\frac{1 − k_{Z′_{i}}}{2} ≤ deg_{Z′_{i}} L ≤ \frac{k_{Z′_{i}}}{2}

for each i. Summing up, and using (19), we get

\frac{m − k_{Z}}{2} ≤ deg_{Z′} L ≤ \frac{k_{Z}}{2} .

Now, since deg L = 1, we must have

(21)  \frac{2 − k_{Z}}{2} ≤ deg_{Z} L ≤ \frac{2 − m + k_{Z}}{2} .

Suppose first that deg_{Z} L = M_{Z}(1), that is, deg_{Z} L = (1 + k_{Z})/2. Then (21) implies 1 + k_{Z} ≤ 2 − m + k_{Z}, and hence m ≤ 1, a contradiction.

Finally, suppose deg_{Z} L = m_{Z}(1), that is, deg_{Z} L = (1 − k_{Z})/2. Then (21) implies 2 − k_{Z} ≤ 1 − k_{Z}, a contradiction as well. ■

4. Modular interpretation of the first Abel map

The following diagram represents the families we shall deal with in this section, starting from a regular pencil of stable curves f : X → B:

(22)  \begin{matrix}Ẋ ×_{B} X & ↪ & X_{B}^{2} & ⟶ & X \\ ↓ π̇ & & ↓ π & & ↓ f \\ Ẋ & ↪ & X & ⟶ & B \end{matrix}

where π is the projection onto the first factor.

We denote by Δ ⊂ X_{B}^{2} the diagonal. Its restriction to Ẋ ×_{B} X is a Cartier divisor. Denote by O_{Ẋ ×_{B}X} (Δ) the associated line bundle. We may view O_{Ẋ ×_{B}X} (Δ) as a family of degree-1 line bundles on the fibers of π̇. Recall that the first Abel map of the generic fiber of f is the moduli map of the restriction of O_{Ẋ ×_{B}X} (Δ); see 2.5. We want to interpret the first Abel map α_{f}^{1}, defined in Proposition 3.10, as the moduli map of a balanced line bundle on π̇, which will necessarily be a (possibly trivial) twist of O_{Ẋ ×_{B}X} (Δ).

In fact, we shall see that O_{Ẋ ×_{B}X} (Δ) fails to be balanced over points of a singular fiber X of f only if X has a separating node. To fix this, we will tensor O_{Ẋ ×_{B}X} (Δ) by twisters supported on so-called “tails”.

We need a few preliminary results which hold for any curve X, possibly having singularities other than nodes. For the sake of future applications of the techniques developed in this paper, from now until 4.5, and in 4.8, 4.11 and 4.13, we shall be in this more general situation, i.e. X will be any (reduced, connected and projective) curve over an algebraically closed field.

Let r be a node of X and X_{r}^{ν} → X be the normalization of X at r only. If X_{r}^{ν} is not connected, r is called a separating node of X.

Definition 4.1. Let X be a curve of arithmetic genus g. A proper subcurve Q ⊊ X will be called a tail of X if Q intersects the complementary subcurve Q′ in a separating node r of X. We say that Q is attached to r or that r generates Q. A tail Q of X will be called small if g_{Q} < g/2 and large if g_{Q} > g/2. Let

Q(X) := {Q ⊂ X : Q is a small tail of X}.

If X has no separating node, for instance if X is smooth, then Q(X) = ∅.

If r is a separating node of X, then X_{r}^{ν} has two connected components, isomorphic to the two tails generated by r; hence every tail is connected.

For every tail Q ⊂ X we have that g = g_{Q} + g_{Q′}. So, at least one of the two tails attached to a separating node has arithmetic genus at most g/2. If the curve X is stable and 1-general, it follows from Proposition 3.15 that no tail of X can have genus equal to g/2, or in other words that every tail of X is either small or large.

Remark 4.2. Let r be a separating node of X generating the tails Q and Q′. If Z ⊂ X is a connected subcurve not containing r, then Z is entirely contained in either Q or Q′.

Lemma 4.3. Let X be a curve and Q_{1} and Q_{2} two tails of X. Then

Q_{1} ∪ Q_{2} = X or Q_{1} ∩ Q_{2} = ∅ or Q_{1} ⊆ Q_{2} or Q_{2} ⊆ Q_{1}.

Proof. For each i = 1, 2 let r_{i} be the separating node of X generating Q_{i}. If r_{1} = r_{2} then either Q_{1} = Q′_{2}, and hence Q_{1} ∪ Q_{2} = X, or Q_{1} = Q_{2}. So we may assume that r_{1} ≠ r_{2}.

Thus r_{1} ∉ Q_{2} or r_{1} ∉ Q′_{2}. Suppose first that r_{1} ∉ Q_{2}. Since Q_{2} is connected, either Q_{2} ⊂ Q_{1} or Q_{2} ⊂ Q′_{1} by Remark 4.2. If Q_{2} ⊂ Q′_{1} then Q_{1} ∩ Q_{2} ⊆ Q_{1} ∩ Q′_{1} = {r_{1}}, and hence Q_{1} ∩ Q_{2} = ∅ because r_{1} ∉ Q_{2}. So either Q_{2} ⊂ Q_{1} or Q_{1} ∩ Q_{2} = ∅.

The case where r_{1} ∉ Q′_{2} is treated similarly. In this case, either Q′_{2} ⊂ Q′_{1}, and hence Q_{1} ⊂ Q_{2}, or Q′_{1} ∩ Q′_{2} = ∅, and hence Q_{1} ∪ Q_{2} = X. ■

Lemma 4.4. Let X be a curve, and Q a tail of X. Then, for any two line bundles L_{1} on Q and L_{2} on Q′, there is, up to isomorphism, a unique line bundle L on X such that L|_{Q} ≅ L_{1} and L|_{Q′} ≅ L_{2}.

Proof. Let r be the separating node of X to which Q is attached. For each isomorphism μ : L_{1}|_{{r}} → L_{2}|_{{r}}, let L be the kernel of the composition

ϕ_{μ} : L_{1} ⊕ L_{2} ⟶ L_{1}|_{{r}} ⊕ L_{2}|_{{r}} ⟶^{μ̃} L_{2}|_{{r}},

where μ̃ := (−μ, 1). Since ϕ_{μ} is surjective, L ≠ L_{1} ⊕ L_{2}. Also, since μ is an isomorphism, L ≠ L_{1}(−r) ⊕ L_{2} and L ≠ L_{1} ⊕ L_{2}(−r). Since r is a node of X, it follows that L is a line bundle, and L|_{Q} ≅ L_{1} and L|_{Q′} ≅ L_{2}.

Conversely, if N is a line bundle on X for which there are isomorphisms λ_{1} : N|_{Q} → L_{1} and λ_{2} : N|_{Q′} → L_{2}, then N is the kernel of ϕ_{μ}, where μ := λ_{2}|_{{r}} ∘ λ_{1}^{−1}|_{{r}}.

Finally, if μ′ : L_{1}|_{{r}} → L_{2}|_{{r}} is another isomorphism, the kernel of ϕ_{μ} is carried isomorphically to the kernel of ϕ_{μ′} by the autoomorphism

(a, 1) : L_{1} ⊕ L_{2} ⟶ L_{1} ⊕ L_{2},

where a is the unique scalar such that μ = aμ′. ■

4.5. Twisters on tails. Let X be a curve. By Lemma 4.4, for each tail Q of X there is a unique, up to isomorphism, line bundle on X whose restrictions to Q and Q′ are O_{Q}(−r) and O_{Q′}(r), where r is the separating node of X generating Q. Denote this bundle by O_{X}(Q).

For each formal sum ∑ a_{Q}Q of tails Q with coefficients a_{Q} ∈ ℤ, set

O_{X}(∑a_{Q}Q) := ⨂ O_{X}(Q)^{⊗a_{Q}}.

If X is a nodal curve, and a closed fiber of a regular pencil f : X → B, then

(23)  O_{X} (∑ a_{Q}Q)|_{X} ≅ O_{X}(∑ a_{Q}Q).

So twisters supported on tails do not depend on the chosen regular pencil. To check (23) it is enough to observe that, for each tail Q of X, since

O_{X} (Q)|_{Q} ≅ O_{Q}(−r) and O_{X} (Q)|_{Q′} ≅ O_{Q}(r),

Lemma 4.4 implies that O_{X} (Q)|_{X} ≅ O_{X}(Q).

To state the main result of this section we need some notation, similar to the one used in Proposition 3.12. Let f : X → B be a regular pencil. Let Z be a subcurve of X, where X ⊂ X is a singular fiber of f. Then Z is a divisor of X. Now, the restriction π_{Z} of the first projection π : X_{B}^{2} → X over Z is the trivial family

X_{B}^{2} ⊃ Z × X ⟶^{π_{Z}} Z.

Thus, for any other subcurve Z_{1} ⊆ X, the product Z × Z_{1} can be viewed as a Weil divisor of X_{B}^{2}. Now, since the open subscheme Ẋ ×_{B} X ⊂ X_{B}^{2} is regular, the restriction of Z×Z_{1} to it is a Cartier divisor. Let O_{Ẋ ×_{B}X} (Z×Z_{1}) denote the associated line bundle. Using this notation, we have an explicit description of the map α_{f}^{1} : Ẋ ⟶ P_{f}^{1} defined in Proposition 3.10.

Theorem 4.6. Let f : X → B be a regular pencil of stable curves. Then the line bundle

L^{(1)} := O_{Ẋ ×_{B}X} (Δ + ∑_{Q∈Q(X_{b})^{b∈B}} Q × Q).

is balanced on π̇ : Ẋ ×_{B} X → Ẋ. Furthermore, assume that the fibers of f are 1-general. Then the following two statements hold.

(i) The morphism α_{f}^{1} : Ẋ → P_{f}^{1} is the moduli map of L^{(1)}. (ii) If M is a balanced line bundle on π̇ : Ẋ ×_{B} X → Ẋ having α_{f}^{1} as moduli

map, then M ≅ L^{(1)}, up to pullbacks from Ẋ.

Remark 4.7. But for Part (ii), the theorem generalizes to curves that are not 1-general. See 5.10 and Remark 5.13.

Proof. The divisor ∑ Q × Q is entirely supported on a union of closed fibers of X ×_{B} X → B. By Lemma 2.6, the Néron maps of L^{(1)} and O(Δ) are equal. Now, if the fibers of f are 1-general, α_{f}^{1} agrees with the Néron map of O(Δ) on X_{K}. On the other hand, if L^{(1)} is balanced on π̇ : Ẋ ×_{B} X → Ẋ, there is an associated moduli map μ̂_{L^{(1)}} : Ẋ → P_{f}^{1} by 3.8 (B). Since μ̂_{L^{(1)}} coincides with the Néron map of L^{(1)} on X_{K}, it follows that μ̂_{L^{(1)}} = α_{f}^{1}. Thus, to prove Part (i) it is enough to prove that L^{(1)} is balanced on π̇.

To prove Part (ii), it is also enough to show that L^{(1)} is balanced, since P_{f}^{1} is a fine moduli scheme; see 3.8 (C).

Let us now prove that L^{(1)} is indeed balanced. We need only check this on each singular fiber of f, whence we may assume f is local. Let X be the closed fiber. It suffices to consider the singular fibers of the first projection

π̇ : Ẋ ×_{B} X ⟶ Ẋ ,

which are all isomorphic to X. Let p ∈ X be a nonsingular point, and set L_{p}^{(1)} := L^{(1)}|_{π̇^{−1}(p)}. Then

(24)  L_{p}^{(1)} ≅ O_{X}(p) ⊗ O_{X}( ∑_{p∈Q^{Q∈Q(X)}} Q).

We conclude by Lemma 4.9 (ii), observing that, since X is stable, a semibalanced line bundle on X is necessarily balanced; see 3.3 (A). ■

The next two lemmas are needed to finish the proof of Theorem 4.6.

Lemma 4.8. Let X be a curve, and Z a connected, proper subcurve. Let

(25)  Q_{1} ⊂ Q_{2} ⊂ ⋯ ⊂ Q_{n−1} ⊂ Q_{n}

be a chain of tails of X, and let r_{i} be the separating node of X generating Q_{i} for each i = 1, …, n. Then

−1 ≤ deg_{Z} O_{X}(∑ Q_{i}) ≤ 1.

Furthermore, the extremes are attained if and only if there is a unique j such that r_{j} ∈ Z ∩ Z′. In this case, the lower bound is attained if Z ⊆ Q_{j}, and the upper bound is attained if Z ⊆ Q′_{j}.

Proof. If r_{ℓ} ∉ Z for any ℓ = 1, …, n, then deg_{Z} O_{X}(∑ Q_{i}) = 0. Suppose now that Z contains at least one r_{ℓ}. Let i and j be the smallest and greatest integers such that r_{i} ∈ Z and r_{j} ∈ Z, respectively. Since Z is connected, Z contains as well all the irreducible components of X containing r_{i+1}, …, r_{j−1}. In particular, r_{ℓ} ∉ Z ∩ Z′ for any ℓ = i + 1, …, j − 1.

If Z ∩ Z′ contains both r_{i} and r_{j} or neither of them, deg_{Z} O_{X}(∑ Q_{i}) = 0. If Z ∩ Z′ contains r_{i} but not r_{j}, then deg_{Z} O_{X}(∑ Q_{i}) = 1 and Z ⊆ Q′_{i}. At last, if Z ∩ Z′ contains r_{j} but not r_{i}, then deg_{Z} O_{X}(∑ Q_{i}) = −1 and Z ⊆ Q_{j}. ■

Lemma 4.9. Let X be a semistable curve, and p a nonsingular point. Then the following two statements hold.

(i) The line bundle O_{X}(p) is semibalanced if and only if p does not belong to any small tail of X.

(ii) The line bundle

L_{p}^{(1)} := O_{X}(p) ⊗ O_{X}( ∑_{p∈Q^{Q∈Q(X)}} Q)

is semibalanced.

Proof. The “if part” of (i) is a consequence of (ii), as the sum of tails in (ii) is zero when p does not belong to any small tail. As for the “only-if part”, recall from 3.1 that

(26)  w_{Z} = 2g_{Z} − 2 + k_{Z}

for every connected proper subcurve Z ⊂ X. In particular,

(27)  w_{Q} < g − 1 for every small tail Q of X.

So, if p is contained in a small tail Q, then

M_{Q}(1) < 1 = deg_{Q} O_{X}(p).

Hence the Basic Inequality (11) is not satisfied for Q. So O_{X}(p) is not semibalanced.

We need only prove (ii) now. First, since X is semistable, w_{Z} ≥ 0 for every subcurve Z ⊆ X; see 3.1. As a consequence,

(28)  w_{Z_{1}} ≤ w_{Z_{2}} for all subcurves Z_{1} and Z_{2} of X with Z_{1} ⊆ Z_{2}.

Let Q_{1}, …, Q_{n} be the small tails of X containing p, and r_{1}, …, r_{n} their generating nodes. Since w_{Q_{i}} + w_{Q_{j}} < 2g − 2 by (27), we have Q_{i} ∪ Q_{j} ≠ X for each i and j. By Lemma 4.3, up to reordering, we may assume that

Q_{1} ⊂ Q_{2} ⊂ ⋯ ⊂ Q_{n−1} ⊂ Q_{n}.

Let N := O_{X}(∑_{1}^{n} Q_{i}); so L_{p}^{(1)} = O_{X}(p) ⊗ N. Let Z be any connected, proper subcurve of X. Then deg_{Z} N ≥ −1 by Lemma 4.8, and hence deg_{Z} L_{p}^{(1)} ≥ −1. As pointed out in Remark 3.3 (C), we need only show that deg_{Z} L_{p}^{(1)} ≥ m_{Z}(1).

First, suppose deg_{Z} N = −1. By Lemma 4.8, there is j such that r_{j} ∈ Z and Z ⊆ Q_{j}. By (27) and (28),

(29)  w_{Z} ≤ w_{Q_{j}} < g − 1,

and hence m_{Z}(1) ≤ 0. Thus, if p ∈ Z,

deg_{Z} L_{p}^{(1)} = 0 ≥ m_{Z}(1).

Suppose p ∉ Z. Then Z ≠ Q_{j}. Since r_{j} ∈ Z, either k_{Z} ≥ 3 or Z is a tail of Q_{j}. Now, if Z were a tail of Q_{j}, then \overline{Q_{j} ∖ Z} would be a tail of X contained in Q_{j}, whence a small tail. Since p ∈ \overline{Q_{j} ∖ Z}, we have \overline{Q_{j} ∖ Z} = Q_{i} for some i < j, or Z = \overline{Q_{j} ∖ Q_{i}}. But then deg_{Z} N = 0, a contradiction. Thus k_{Z} ≥ 3. In particular, Z is not an exceptional component of X. It follows now from (29) that m_{Z}(1) < −1, and hence

deg_{Z} L_{p}^{(1)} ≥ −1 > m_{Z}(1).

Second, suppose deg_{Z} N ≥ 0. Then deg_{Z} L_{p}^{(1)} ≥ 0. By (28) we have that w_{Z} ≤ w_{X} = 2g − 2. So, if Z is not a large tail of X, then m_{Z}(1) ≤ 0, and hence

deg_{Z} L_{p}^{(1)} ≥ 0 ≥ m_{Z}(1).

On the other hand, suppose that Z is a large tail. At any rate, m_{Z}(1) ≤ 1/2. Thus, if p ∈ Z,

(30)  deg_{Z} L_{p}^{(1)} ≥ 1 ≥ 1/2 ≥ m_{Z}(1).

Finally, suppose p ∉ Z. Then p lies on Z′, which is a small tail of X. Thus Z′ = Q_{j} for some j, and hence Z = Q′_{j}. It follows that deg_{Z} N = 1, and hence (30) holds as well. ■

Let X be a 1-general stable curve. Let Ẋ := X ∖ X_{sing}. For any regular smoothing f of X, let

α_{X}^{1} := α_{f}^{1}|Ẋ : Ẋ ⟶ P_{X}^{1}.

The notation is not ambiguous by the following consequence of Theorem 4.6.

Corollary 4.10. Let X be a 1-general stable curve. Then α_{X}^{1} does not depend on f. In fact, for each nonsingular point p ∈ X we have

α_{X}^{1}(p) = O_{X}(p) ⊗ O_{X}( ∑_{p∈Q^{Q∈Q(X)}} Q).

Proof. The expression of α_{X}^{1}(p) follows from (24) in the proof of Theorem 4.6. By 4.5 the map α_{X}^{1} does not depend on f. ■

If X is free from separating nodes then α_{X}^{1} is injective. This follows immediately from Lemma 4.13 below. The same lemma will be used in the proof of Proposition 5.9, a more general and precise statement. For the lemma and the proposition, the definition below is used.

Definition 4.11. Let X be a curve. A rational, smooth component C of X is called a separating line if C intersects \overline{X ∖ C} in separating nodes of X. More generally, a connected subcurve Z ⊆ X of arithmetic genus 0 is called a separating tree of lines if Z intersects \overline{X ∖ Z} in separating nodes of X.

4.12. Let X be a curve, and Z ⊊ X a proper connected subcurve such that Z intersects \overline{X ∖ Z} in separating nodes of X. Then the connected components of \overline{X ∖ Z} are tails of X. In addition, if r is a separating node of Z, then r is a separating node of X.

A curve of arithmetic genus 0 is a curve of compact type, i.e. a nodal curve with every node separating, whose irreducible components are smooth and rational. So, if Z is a separating tree of lines, every node of Z is a separating node of Z, and hence of X. It follows that every connected subcurve of Z is also a separating tree of lines. In particular, every irreducible component of Z is a separating line.

We shall later need the following lemma.

Lemma 4.13. Let X be a curve, and p and q distinct nonsingular points of X. Let C ⊆ X be the irreducible component containing p. Then there is an isomorphism O_{X}(p) ≅ O_{X}(q) if and only if C contains q and is a separating line of X.

Proof. Assume first that C contains q and is a separating line of X. Since C is smooth and rational, O_{C}(p) ≅ O_{C}(q). We may thus assume C ≠ X. Since C meets C′ := \overline{X ∖ C} in separating nodes, applying Lemma 4.4 a few times, we can show that a line bundle on X is uniquely determined by its restrictions to C and to C′. Since O_{X}(p) and O_{X}(q) restrict to isomorphic line bundles on C and to the trivial line bundle on C′, it follows that O_{X}(p) ≅ O_{X}(q).

Conversely, suppose O_{X}(p) ≅ O_{X}(q). Since O_{X}(p) has degree 1 on C, so has O_{X}(q), and hence q ∈ C as well. Now, since O_{X}(p) ≅ O_{X}(q), in particular O_{C}(p) ≅ O_{C}(q). Since p ≠ q, it follows from [AK], Thm. 8.8, p. 108 that C ≅ ℙ^{1}.

If C = X we are done. Suppose thus that C ≠ X, and let C′ := \overline{X ∖ C}. Also, suppose by contradiction that C ∩ C′ is not made of separating nodes of X. Then there is a connected subcurve Z ⊆ C′ such that C ∩ Z is a scheme of length at least 2.

Since X is connected, the restriction τ : H^{0}(X, O_{X}(p)) → H^{0}(C, O_{C}(p)) is injective. But it is not surjective. Indeed, if a nonconstant σ ∈ H^{0}(C, O_{C}(p)) could be extended to σ̃ ∈ H^{0}(X, O_{X}(p)), then σ̃ would have to be constant on Z and hence σ would be constant on C ∩ Z. Since C ≅ ℙ^{1}, this is impossible, σ being a nonconstant section of O_{ℙ^{1}}(1) So τ is injective, but not surjective, and hence h^{0}(X, O_{X}(p)) = 1. Since O_{X}(p) ≅ O_{X}(q), it follows that p = q, an absurd. ■

5. Completing the first Abel map

The main result of this section is Theorem 5.5. We shall prove it first in a simpler case in Proposition 5.2, where we have a neater statement concerning the modularity, see 5.3.

As in Section 4, certain basic results of this section hold in more generality for curves having singularities other than nodes. Apart from the notation set in 5.1 below, these results are concentrated in 5.4.

5.1. Let X be a curve. For each node r ∈ X, let X_{r}^{ν} → X denote the partial normalization of X at r, and let X̂_{r} be the curve obtained by adding to X_{r}^{ν} a smooth rational curve E_{r} connecting the two branches over r. Thus

X̂_{r} = X_{r}^{ν} ∪ E_{r},

and there is a natural surjection σ_{r} : X̂_{r} → X such that σ_{r}(E_{r}) = {r}, and such that σ_{r} is an isomorphism away from E_{r}.

Assume now that X is a 1-general stable curve, and let r be a nonseparating node of X. Let \overline{r} ∈ E_{r} ⊂ X̂_{r} such that \overline{r} is a nonsingular point of X̂_{r}. Then the line bundle OX̂_{r}(\overline{r}) ∈ Pic^{1} X̂_{r} is balanced by Lemma 4.9, and hence determines a point of \overline{P_{X}^{1}} ∖ P_{X}^{1}; see 3.8 (B). This point does not depend on the choice of \overline{r}; see 3.7. Thus we shall denote it by ℓ_{r}.

Proposition 5.2. Let f : X → B be a regular pencil of 1-general stable curves free from separating nodes. Then α_{f}^{1} : Ẋ → P_{f}^{1} extends to an injection

\overline{α_{f}^{1}} : X ⟶ \overline{P_{f}^{1}}

such that \overline{α_{f}^{1}}(r) = ℓ_{r} ∈ \overline{P_{X}^{1}} for each node r of each closed fiber X of f.

Remark 5.3. More precisely, the proof will show that \overline{α_{f}^{1}} is the moduli map of the line bundle O_{Y}(Δ̃), where Y → X_{B}^{2} is a partial resolution of singularities, and Δ̃ is the proper transform in Y of the diagonal Δ; see 5.4.

Proof. Denote by ρ : Y → X_{B}^{2} the partial resolution of singularities described in 5.4, from where we take some of the properties mentioned below. The map ρ is an isomorphism away from the points (r, r) for r ∈ X ∖ Ẋ. On the other hand, if r ∈ X ∖ Ẋ, then ρ^{−1}(r, r) is a copy of ℙ^{1}. In addition, composing ρ with the first projection π,

Y ⟶^{ρ} X_{B}^{2} ⟶^{π} X ,

we obtain a family of quasistable curves Y → X having π : X_{B}^{2} → X as stable model.

For each closed fiber X of f, and each r ∈ X_{sing} ⊂ X, let Y_{r} be the fiber of π ∘ ρ over r. Then Y_{r} = X̂_{r}, where X̂_{r} is as defined in 5.1. On the other hand, each fiber of π ∘ ρ over Ẋ is the same as the corresponding fiber of π.

Let Δ̃ ⊂ Y be the proper transform of Δ. By Property 5.4 (B), the map ρ restricts to an isomorphism between Δ̃ and Δ. Also, Δ̃ meets each fiber Y_{r} = X̂_{r} of π ∘ρ over X ∖Ẋ transversally at a nonsingular point \overline{r} contained in the exceptional component E_{r}.

To prove that α_{f}^{1} extends, we prove two claims: first, that O_{Y}(Δ̃) is balanced on π ∘ ρ : Y → X, so it induces a morphism \overline{α_{f}^{1}} : X → \overline{P_{f}^{1}}, its moduli map; and second, to show that \overline{α_{f}^{1}} extends α_{f}^{1}, that the restriction of O_{Y}(Δ̃) to the each fiber of π ∘ ρ over Ẋ is isomorphic to the corresponding restriction of L^{(1)}, whose moduli map is α_{f}^{1} by Theorem 4.6.

We may now assume that f is local. Let X be its closed fiber. For each r ∈ X_{sing}, since Δ̃ intersects Y_{r} transversally at \overline{r}, we have

(31)  O_{Y}(Δ̃)|_{Y_{r}} ≅ OX̂_{r}(\overline{r}),

which is balanced by Lemma 4.9. In addititon, for each nonsingular point p ∈ X we have

O_{Y}(Δ̃)|_{Y_{p}} ≅ O_{X}(p)

which is balanced and isomorphic to the corresponding restriction of L^{(1)}, also by Lemma 4.9. Therefore O_{Y}(Δ̃) induces a moduli map

\overline{α_{f}^{1}} : X ⟶ \overline{P_{f}^{1}}

which extends α_{f}^{1}. Notice that (31) also shows that \overline{α_{f}^{1}}(r) = ℓ_{r} for each r ∈ X_{sing}.

To show that \overline{α_{f}^{1}} is injective it suffices to consider singular points of X,

by Corollary 4.10 and by the fact that \overline{α_{f}^{1}}(r) ∈ \overline{P_{X}^{1}} ∖ P_{X}^{1} for every node

r ∈ X. Now, if r ∈ X_{sing}, then \overline{α_{f}^{1}}(r) represents a balanced line bundle on X̂_{r}. Hence, two different nodes r and r′ of X are mapped to two points of \overline{P_{X}^{1}} corresponding to balanced line bundles on different quasistable curves, namely X̂_{r} and X̂_{r′}. Thus \overline{α_{f}^{1}}(r) ≠ \overline{α_{f}^{1}}(r′); see 3.7. ■

5.4. Resolution of singularities. In the proof of Proposition 5.2 we used a partial resolution of singularities of X_{B}^{2} which we are now going to describe in detail, and in more generality.

Let f : X → B be a regular pencil. The threefold X_{B}^{2} is singular at the points (r_{1}, r_{2}), where r_{1} and r_{2} are (not necessarily distinct) singular points of the same closed fiber of f.

Let X be a closed fiber of f, and r_{1} and r_{2} nodes of X. Since f is regular, locally around r_{i} the surface X is formally equivalent to the surface in 𝔸^{3} given by the equation x_{i}y_{i} = t, where t denotes a local parameter of B at the closed point covered by X. Pulling back these local equations to X_{B}^{2} under the two projection maps X_{B}^{2} → X, and abusing of the same notation, we get that X_{B}^{2} is formally equivalent, locally around (r_{1}, r_{2}), to the threefold in 𝔸^{5} given the equations

\{ \begin{matrix}x_{1}y_{1} = t, \\ x_{2}y_{2} = t. \end{matrix}

If r_{1} = r_{2}, then the diagonal Δ ⊂ X_{B}^{2} contains (r_{1}, r_{2}), and we may assume that it is given locally around (r_{1}, r_{2}) by

\{ \begin{matrix}x_{1}y_{1} = t, \\ x_{2} = x_{1}, \\ y_{2} = y_{1}. \end{matrix}

Locally around (r_{1}, r_{2}) we may eliminate t, and view X_{B}^{2} as the cone C ⊂ 𝔸^{4} over the smooth quadric in ℙ^{3} given by x_{1}y_{1} = x_{2}y_{2}. Also, if r_{1} = r_{2}, we may view Δ as the plane D ⊂ 𝔸^{4} given by x_{2} = x_{1} and y_{2} = y_{1}.

Notice that C is singular only at the origin. To resolve this singularity we need only blow up a plane in C containing the origin. Any plane will do, but let us blow up the plane given by x_{1} = x_{2} = 0. The blowup is the nonsingular threefold C̃ ⊂ ℙ^{1} × 𝔸^{4} given by the equations

\{ \begin{matrix}ξ_{2}x_{1} = ξ_{1}x_{2}, \\ ξ_{1}y_{1} = ξ_{2}y_{2}, \end{matrix}

where ξ_{1}, ξ_{2} are homogeneous coordinates of ℙ^{1}. The blowup γ : C̃ → C is isomorphic to C away from the origin. In addition, the fiber F over the origin is given by x_{1} = x_{2} = y_{1} = y_{2} = 0, and hence is isomorphic to ℙ^{1}.

The exceptional divisor E of the blow up C̃ is given by x_{2} = 0 where ξ_{2} ≠ 0, and x_{1} = 0 where ξ_{1} ≠ 0. In particular, F ⊂ E. Now, since ξ_{2}x_{1} = ξ_{1}x_{2}, summing the divisor given by ξ_{1} = 0 to E we get the principal divisor given by x_{1} = 0. Thus E · F = −1.

Suppose r_{1} = r_{2}. Then γ^{−1}(D) is given by x_{1}(ξ_{1} − ξ_{2}) = y_{2}(ξ_{1} − ξ_{2}) = 0 where ξ_{1} ≠ 0, and by x_{2}(ξ_{1} − ξ_{2}) = y_{1}(ξ_{1} − ξ_{2}) = 0 where ξ_{2} ≠ 0. Thus γ^{−1}(D) is the union of the Cartier divisor given by ξ_{1} = ξ_{2} and the fiber F. The strict transform D̃ of D is thus a Cartier divisor intersecting F transversally at a point.

For i = 1, 2, let ϕ_{i} : C̃ → 𝔸^{2} be the composition of γ with the projection onto the plane with coordinates x_{i}, y_{i}. Its fiber over the origin is given by x_{1} = y_{1} = ξ_{1}x_{2} = ξ_{2}y_{2} = 0. It is the union of F and the affine lines N_{1}, given by x_{1} = y_{1} = ξ_{1} = y_{2} = 0, and N_{2}, given by x_{1} = y_{1} = ξ_{2} = x_{2} = 0. The lines N_{1} and N_{2} do not meet, and F intersects each N_{i} transversally at a single point. Also, ϕ_{2} maps N_{1} and N_{2} isomorphically onto the lines y_{2} = 0 and x_{2} = 0, respectively.

The exceptional divisor E contains N_{2}, and intersects N_{1} transversally. Since ξ_{1} ≠ 0 on N_{2}, we have E · N_{2} = 0. If r_{1} = r_{2}, the strict transform D̃ does not meet either N_{1} or N_{2}, and intersects F transversally.

(An analogous description holds if we reverse the roles of ϕ_{1} and ϕ_{2}.) We will now consider the global picture. Let I_{Δ} denote the ideal sheaf of

the diagonal Δ ⊂ X_{B}^{2}, and let Ǐ_{Δ} denote the dual sheaf, i.e.

Ǐ_{Δ} := Hom(I_{Δ}, O_{X_{B}^{2}}).

Since I_{Δ} is a sheaf of ideals, Ǐ_{Δ} is a sheaf of fractional ideals of X_{B}^{2}. A piece of notation: for each open subscheme U ⊆ X_{B}^{2} and each sheaf of fractional ideals M of U, consider its powers M^{n}, and form the associated sheaf of Rees algebras:

R(M) := O_{U} ⊕ M ⊕ M^{2} ⊕ ⋯ ⊕ M^{n} ⊕ · · · .

Set Y := Proj_{X_{B}^{2}}(R(Ǐ_{Δ})), and let ρ : Y → X_{B}^{2} be the structure map.

We may view ρ as a blowup. In fact, for any open subscheme U ⊆ X_{B}^{2} over which there is an embedding ι : Ǐ_{Δ}|_{U} → L into an invertible sheaf L, we may view ρ : ρ^{−1}(U) → U as the blowup of U along the closed subscheme V ⊆ U whose sheaf of ideals I_{V |U} satisfies ι(Ǐ_{Δ}|_{U}) = I_{V |U}L. In other words, ι induces an isomorphism over U:

Proj_{U}(R(I_{V |U})) ⟶ Proj_{U}(R(Ǐ_{Δ})|_{U}).

In the same vein, for each invertible sheaf of ideals J ⊆ O_{U} we have that Hom(I_{Δ}|_{U}, J) = Ǐ_{Δ}|_{U}J, and hence we obtain a canonical isomorphism over U:

Proj_{U}(R(Ǐ_{Δ})|_{U}) ⟶ Proj_{U}(R(Hom(I_{Δ}|_{U}, J))).

Since I_{Δ} is invertible away from the points (r, r) for r ∈ X ∖ Ẋ, it follows from the above description that ρ is an isomorphism away from these same points. In addition, around the points (r, r), where r is a node of a closed fiber of f, the map ρ is formally equivalent to the blowup described above, because

Hom_{A}((x_{1} −x_{2}, y_{1} −y_{2}), (x_{1} −x_{2})) = (x_{1}, x_{2}), where A := \frac{ℂ[[x_{1}, x_{2}, y_{1}, y_{2}]]}{(x_{1}y_{1} − x_{2}y_{2})}.

Then all of the properties above, verified locally, yield global properties of ρ. Indeed, assume that the fibers of f are nodal. (It would actually be enough to assume that the fibers are Gorenstein.) Then, recalling that π : X_{B}^{2} → X denotes the first projection, the following statements hold: (A) The composition

π ∘ ρ : Y ⟶ X

is a family of curves whose fiber Y_{r} over a point r of a closed fiber X of f is X, if r is nonsingular, and X̂_{r}, described in 5.1, if r is a node.

(B) Let Δ̃ ⊂ Y denote the proper transform of Δ. For each node r of each closed fiber X of f, the transform Δ̃ intersects the fiber Y_{r} transversally at a point lying in the exceptional component E_{r} = ρ^{−1}(r, r).

(C) Let Q be a tail of a closed fiber X of f, and r the node of X generating

Q. Let

Q̃^{2} := ρ^{−1}(Q × Q).

Then Q̃^{2} is a Cartier divisor of Y containing E_{r}. Furthermore,

Q̃^{2} · E_{r} = −1, Q̃^{2} · Q̂ = −1, and Q̃^{2} · Q̂′ = 1,

where, using the notation in 5.1, Q̂ := σ_{r}^{−1}(Q) and Q̂′ := \overline{X̂_{r} ∖ Q̂}, i.e. Q̂ is the tail of X̂_{r} mapping to Q and containing E_{r}, and Q̂′ is the complementary tail.

We may now generalize Proposition 5.2.

Theorem 5.5. Let f : X → B be a regular pencil of 1-general stable curves. Then there exists a morphism

\overline{α_{f}^{1}} : X ⟶ \overline{P_{f}^{1}}

extending α_{f}^{1} : Ẋ → P_{f}^{1}. If r is a node of a closed fiber X of f, then \overline{α_{f}^{1}}(r) ∈ P_{X}^{1} if and only if r is a separating node of X.

Remark 5.6. The result extends to curves that are not 1-general. See 5.10 and 5.13.

Proof. As in the proof of Proposition 3.10 we may work locally around each singular fiber. So, assume f is local, and let X denote its closed fiber.

The new difficulty with respect to Proposition 5.2 is that, if X has separating nodes, α_{f}^{1} is the moduli map of a nontrivial “twist” of the diagonal by Theorem 4.6, and thus the same must hold for its completion. Fortunately, however, the divisors we need for the “twist” are already present in the partial resolution of singularities ρ : Y → X_{B}^{2} described in 5.4.

Namely, let Q_{1}, …, Q_{m} be all the small tails of X. Let Δ̃ ⊂ Y be the strict transform of Δ, and set Q̃_{i}^{2} := ρ^{−1}(Q_{i} × Q_{i}) for i = 1, …, m. As seen in 5.4, all the Q̃_{i}^{2} and Δ̃ are Cartier divisors. Define the line bundle

(32)  M := O_{Y}(Δ̃ + Q̃_{1}^{2} + ⋯ + Q̃_{m}^{2}).

We claim that M is semibalanced on the composition π ∘ ρ : Y → X of ρ with the first projection π : X_{B}^{2} → X. Once the claim is proved, we may let \overline{α_{f}^{1}} : X ⟶ \overline{P_{f}^{1}} be the moduli map of M; see 3.8 (B).

To prove the claim, first observe that ρ is an isomorphism over Ẋ ×_{B} X, whence

M|_{Ẋ ×_{B}X} ≅ O_{Ẋ ×_{B}X} (Δ + Q_{1} × Q_{1} + ⋯ + Q_{m} × Q_{m}),

which is balanced, by Theorem 4.6, and defines α_{f}^{1}. Thus, once M is shown to be semibalanced, we have that \overline{α_{f}^{1}}|Ẋ = α_{f}^{1}.

Now, let r ∈ X_{sing}. The fiber Y_{r} := (π ∘ ρ)^{−1}(r) is equal to X̂_{r} by Property 5.4 (A). Also, Δ̃ intersects Y_{r} transversally at a point \overline{r} of the exceptional component E_{r} = ρ^{−1}(r, r), by Property 5.4 (B).

For each i = 1, …, m, let r_{i} be the separating node of X generating Q_{i}. Let

Q̂_{i} := σ_{r}^{−1}(Q_{i}) ⊂ X̂_{r},

and let Q̂′_{i} be its complement in X̂_{r}. Then Q̂_{i} is a small tail of X̂_{r} dominating Q_{i}, and containing E_{r} if and only if r ∈ Q_{i}. If r = r_{i} then also \overline{Q̂_{i} ∖ E_{r}} is a small tail of X̂_{r}. These are all the small tails of X̂_{r}: the subcurves Q̂_{1}, …, Q̂_{m}, together with \overline{Q̂_{i} ∖ E_{r}} in case r = r_{i}.

For each i = 1, …, m, the subscheme Q_{i} × Q_{i} ⊂ X_{B}^{2} is a Cartier divisor away from (r_{i}, r_{i}). Identifying Y_{r} with X̂_{r}, we claim that

(33)  O_{Y}(Q̃_{i}^{2})|_{Y_{r}} ≅ \begin{cases}OX̂_{r} & if r ∉ Q_{i}, \\ OX̂_{r}(Q̂_{i}) & if r ∈ Q_{i}. \end{cases}

In fact, if r ∉ Q_{i}, then Q̃_{i}^{2} does not meet Y_{r}, and hence (33) holds. Suppose now that r ∈ Q_{i}. Recall that Q̂_{i} is a tail of X̂_{r}. Let s_{i} denote its generating node. If r ≠ r_{i} then, since Q_{i} × Q_{i} is a Cartier divisor of X_{B}^{2} at (r, r_{i}), we have

(34)  O_{Y}(Q̃_{i}^{2})|Q̂_{i} ≅ OQ̂_{i}(−s_{i}) and O_{Y}(Q̃_{i}^{2})|Q̂′_{i} ≅ OQ̂′_{i}(s_{i}).

The same restrictions are achieved with OX̂_{r}(Q̂_{i}). Thus (33) follows from Lemma 4.4. Finally, if r = r_{i} then (34) still holds, by Property 5.4 (C), and hence (33) follows in the same way. The proof of (33) is complete.

Now, notice that Q_{i} contains r if and only if Q̂_{i} contains \overline{r}. In addition, if r = r_{i} then \overline{r} ∉ \overline{Q̂_{i} ∖ E_{r}}. Since Q̂_{1}, …, Q̂_{m}, and \overline{Q̂_{i} ∖ E_{r}} if r = r_{i}, are all the small tails of X̂_{r}, it follows from (33) that

M|_{Y_{r}} ≅ OX̂_{r}(\overline{r}) ⊗ OX̂_{r}( ∑_{\overline{r}∈Q^{Q∈Q(X̂r)}} Q),

which is semibalanced by Lemma 4.9. Our claim is proved, and thus we finish the proof of the existence of \overline{α_{f}^{1}}.

To prove the second statement of the theorem, it suffices to prove that for any node r ∈ X we have

deg_{E_{r}} M = \{ \begin{matrix}1 & if r is not separating, \\ 0 & otherwise. \end{matrix}

To prove this, notice that, if r ≠ r_{i} then Q̃_{i}^{2} · E_{r} = 0, whereas if r = r_{i} then Q̃_{i}^{2} · E_{r} = −1 by Property 5.4 (C). Since at any rate Δ̃ · E_{r} = 1, the degree of M|_{E_{r}} is 1, unless r = r_{i} for some i, in which case the degree is 0. Since X is 1-general, each separating node of X generates a small tail, and hence is equal to r_{i} for some i. So \overline{α_{f}^{1}}(r) ∈ P_{X}^{1} if and only if r is a separating node. ■

Example 5.7. Let X be a curve of compact type with two components, C_{1} and C_{2}. Then C_{1} and C_{2} are smooth, and C_{1} ∩ C_{2} = {r}, where r is the unique node of X. Assume g_{C_{1}} < g_{C_{2}}. Then X̂_{r} = C_{1} ∪ E ∪ C_{2} and Q(X̂_{r}) = {C_{1}, C_{1} ∪ E}, where E = ℙ^{1}. The line bundle M in the proof of Theorem 5.5, whose moduli map is \overline{α_{f}^{1}}, satisfies

M = O_{Y}(Δ̃ + C̃_{1}^{2}).

In this case, it is easy to describe the completed Abel map. First notice that there is a canonical isomorphism \overline{P_{X}^{1}} ≅ Pic^{0} C_{1} × Pic^{1} C_{2}, essentially by Lemma 4.4. Hence, a point ℓ ∈ \overline{P_{X}^{1}} is represented by a pair (L_{1}, L_{2}) with L_{i} ∈ Pic C_{i}. For i = 1, 2 let q_{i} ∈ C_{i} lying above r. Then

\overline{α_{f}^{1}}(p) = \{ \begin{matrix}(O_{C_{1}}(p − q_{1}), O_{C_{2}}(q_{2})) & if p ∈ C_{1}, \\ (O_{C_{1}}, O_{C_{2}}(p)) & if p ∈ C_{2}. \end{matrix}

In particular, \overline{α_{f}^{1}}(r) = (O_{C_{1}}, O_{C_{2}}(q_{2})). Thus, composing \overline{α_{f}^{1}}|_{C_{1}} with the projection Pic^{0} C_{1} × Pic^{1} C_{2} → Pic^{0} C_{1} we obtain the classical Abel–Jacobi map of C_{1} with base point q_{1}, i.e.

\begin{matrix}C_{1} & ⟶ & Pic^{0} C_{1} \\ p & ↦ & O_{C_{1}}(p − q_{1}). \end{matrix}

The analogous composition for C_{2} gives the first Abel map C_{2} → Pic^{1} C_{2}.

Let X be a 1-general stable curve, and f a regular smoothing of X. The restriction \overline{α_{f}^{1}}|Ẋ coincides with α_{f}^{1}|Ẋ, whence does not depend on f

by Corollary 4.10. So \overline{α_{f}^{1}}|_{X} does not depend on f either, and we may set

\overline{α_{X}^{1}} := \overline{α_{f}^{1}}|_{X}.

5.8. Explicit description of the complete Abel map. Let X be a 1-general stable curve and p ∈ X. We shall now explicitely describe \overline{α_{X}^{1}}(p), in formulas (35) and (36) below, following the proof of Theorem 5.5.

First some notation. Let P_{1}, …, P_{m} be all the small tails of X containing p. (The unusual naming of the tails using “P ” rather than “Q” is to match the notation of the proof of Proposition 5.9.) By Lemma 4.3 we can write

P_{m} ⊂ P_{m−1} ⊂ ⋯ ⊂ P_{2} ⊂ P_{1}.

Set Z_{i} := \overline{P_{i} − P_{i+1}} for each i = 1, …, m − 1 and Z_{m} := P_{m}, so that P_{1} = ∪_{1}^{m}Z_{i}. Also, put Q := P′_{m}, a large tail of X. Hence

X = P_{1} ∪ Q = ∪_{1}^{m}Z_{i} ∪ Q.

Let r_{1}, …, r_{m} be the separating nodes of X generating P_{1}, …, P_{m}. Notice that Z_{i} ∩Z′_{i} = {r_{i}, r_{i+1}} if i = 1, …, m−1, and Z_{m} ∩Z′_{m} = {r_{m}}. Therefore each of the Z_{i} and Q meets the complementary curve in separating nodes of X. Hence, by iterated use of Lemma 4.4, to give a line bundle on X it suffices to give its restrictions to all the Z_{i} and to Q.

We are now ready to describe \overline{α_{X}^{1}}(p) if p is a nonsingular point or a separating node of X (in which case of course p = r_{m}). Recall that by Theorem 5.5, \overline{α_{X}^{1}}(p) corresponds to a line bundle on X. We have

(35) \overline{α_{X}^{1}}(p) = \{O_{Q}(r_{1}), O_{Z_{1}}(r_{2}−r_{1}), …, O_{Z_{m−1}}(r_{m}−r_{m−1}), O_{Z_{m}}(p−r_{m})\}.

Now, suppose that p is a nonseparating node of X. Then we know that \overline{α_{X}^{1}}(p) corresponds to a line bundle on X̂_{p}. Let E ⊂ X̂_{p} be the exceptional component of X̂_{p}, and let Z̃_{m} denote the normalization of Z_{m} at p only. Keeping the above notation we have

X̂_{p} = Q ∪ Z_{1} ∪ ⋯ ∪ Z_{m−1} ∪ Z̃_{m} ∪ E.

Now, recall from 3.7 that \overline{α_{X}^{1}}(p) is uniquely determined by a line bundle L, of degree 0, on the complementary curve of E; that is, arguing as above, by the string of the restrictions of L to Q, Z_{1}, …, Z_{m−1}, Z̃_{m}. We have

(36) \overline{α_{X}^{1}}(p) = \{O_{Q}(r_{1}), O_{Z_{1}}(r_{2} − r_{1}), …, O_{Z_{m−1}}(r_{m} − r_{m−1}), OZ̃_{m}(−r_{m})\}

Proposition 5.9. Let X be a 1-general stable curve. Let p and q be distinct points of X. Then \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q) if and only if p and q belong to the same separating tree of lines of X.

A similar result for the Abel–Jacobi map to the (degree-0) Jacobian is proved by B.Edixhoven in [E98], Prop. 9.5. His statement (necessarily) excludes the case where p or q is a nonseparating node, since there the target space of the map is a noncompactified Néron model.

Proof. Suppose first that p and q belong to a separating tree of lines of X, call it Z. Since Z is connected, to prove that \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q) it is enough to consider the case where p and q are nonsingular points of X in the same irreducible component C of Z. Now, C is a separating line of X; see 4.12. Thus O_{X}(p) ≅ O_{X}(q) by Lemma 4.13. Since p and q lie on the same component, it follows that α_{X}^{1}(p) = α_{X}^{1}(q), and hence \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q).

Conversely, suppose \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q). We claim that p and q are either nonsingular points or separating nodes of X. Indeed, suppose by contradiction, and without loss of generality, that p is a nonseparating node of X. Then \overline{α_{X}^{1}}(p) ∈ \overline{P_{X}^{1}} ∖ P_{X}^{1} by Theorem 5.5. Since \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q), it follows from Theorem 5.5 as well that q is also a nonseparating node of X. However, \overline{α_{X}^{1}}(p) and \overline{α_{X}^{1}}(q) correspond to balanced line bundles on different quasistable curves, X̂_{p} and X̂_{q}. So \overline{α_{X}^{1}}(p) ≠ \overline{α_{X}^{1}}(q); see 3.7. The contradiction proves the claim.

Since p and q are nonsingular or separating nodes of X, both \overline{α_{X}^{1}}(p) and \overline{α_{X}^{1}}(q) are line bundles on X. Let P_{1}, …, P_{m} be the small tails containing p and Q_{1}, …, Q_{n} the small tails containing q. (We may have m = 0 or n = 0.) It follows from Lemma 4.3, as in the proof of Theorem 4.6, that, up to reordering the tails,

P_{m} ⊂ P_{m−1} ⊂ ⋯ ⊂ P_{2} ⊂ P_{1} and Q_{n} ⊂ Q_{n−1} ⊂ ⋯ ⊂ Q_{2} ⊂ Q_{1}.

Set P_{0} := Q_{0} := X. Let r_{1}, …, r_{m} be the separating nodes of X generating P_{1}, …, P_{m}, and s_{1}, …, s_{n} those generating Q_{1}, …, Q_{n}. In addition, set P_{m+1} := Q_{n+1} := ∅, and put r_{m+1} := p and s_{n+1} := q.

We may assume m ≤ n, without loss of generality. Let i be the largest nonnegative integer such that i ≤ m and P_{j} = Q_{j} for j = 0, 1, …, i. Then also r_{j} = s_{j} for j = 0, 1, …, i. We claim that P_{i+1} ∩ Q_{i+1} = ∅. Indeed, if i = m then P_{i+1} is already empty. Suppose i < m. If P_{i+1} ⊆ Q_{i+1}, then

P_{i+1} ⊆ Q_{i+1} ⊂ Q_{i} = P_{i},

and hence P_{i+1} = Q_{i+1}, contradicting the maximality of i. In a similar way, Q_{i+1} ⊈ P_{i+1}. Since P_{i+1} ∪ Q_{i+1} ≠ X, because P_{i+1} and Q_{i+1} are small tails, it follows from Lemma 4.3 that P_{i+1} ∩ Q_{i+1} = ∅, proving our claim. In particular, r_{i+1} ≠ s_{i+1}.

As P_{i} = Q_{i}, we may consider Y := \overline{P_{i} ∖ (P_{i+1} ∪ Q_{i+1})}. As P_{i+1} and Q_{i+1} do not meet, their union cannot be P_{i}, a connected subcurve of X. Thus Y is a subcurve of X. It is also connected, being either equal to, or a tail of, \overline{P_{i} ∖ P_{i+1}}, which in turn is either equal to, or a tail of, P_{i}, a tail of X.

Since Y ⊆ \overline{P_{i} ∖ P_{i+1}}, the restriction of \overline{α_{X}^{1}}(p) to Y is O_{Y} (r_{i+1} −r_{i}). Analogously, \overline{α_{X}^{1}}(q) restricts to O_{Y} (s_{i+1} − s_{i}). Since \overline{α_{X}^{1}}(p) = \overline{α_{X}^{1}}(q) and r_{i} = s_{i}, it follows that O_{Y} (r_{i+1}) ≅ O_{Y} (s_{i+1}). Since r_{i+1} ≠ s_{i+1}, by Lemma 4.13 applied to the curve Y, we see that r_{i+1} and s_{i+1} are contained in a separating line C of Y. Since Y ∩ Y′ is made of separating nodes of X, so is C ∩ C′; see 4.12. In other words, C is a separating line of X.

For ℓ = 1, …, m − i let Y_{ℓ} := \overline{P_{i+ℓ} − P_{i+ℓ+1}}. Then \overline{α_{X}^{1}}(p) restricts to O_{Y_{ℓ}}(r_{i+ℓ+1} − r_{i+ℓ}). On the other hand, since Y_{ℓ} ⊂ \overline{Q_{i} ∖ Q_{i+1}} for each ℓ, but neither s_{i} ∈ Y_{ℓ} nor s_{i+1} ∈ Y_{ℓ}, we have that \overline{α_{X}^{1}}(q) restricts to the trivial bundle O_{Y_{ℓ}}. Applying Lemma 4.13 to the curve Y_{ℓ}, we get that r_{i+ℓ} and r_{i+ℓ+1} are contained in a separating line C_{ℓ}^{p} of Y_{ℓ}. As before, C_{ℓ}^{p} is also a separating line of X.

Similarly, for each ℓ = 1, …, n−i the points s_{i+ℓ} and s_{i+ℓ+1} are contained in a separating line C_{ℓ}^{q} of X. The union of all the separating lines, namely,

C_{m−i}^{p}, …, C_{2}^{p}, C_{1}^{p}, C, C_{1}^{q}, C_{2}^{q}, …, C_{n−i}^{q},

is a separating tree of lines containing p and q. ■

5.10. Curves that are not 1-general. We conclude by discussing the case of stable curves which are not 1-general. Recall that such curves form a proper closed subset of \overline{M}_{g}, nonempty if and only if g is even, and their combinatorial structure is described in Proposition 3.15.

What kind of complications occur for curves that are not 1-general, or not d-general?

The stack \overline{P}_{d,g} introduced in 3.8 in the case (d − g + 1, 2g − 2) = 1 is constructed as a quotient stack, i.e. \overline{P}_{d,g} := [H_{d}/G]; notation as in 3.5. The same definition can be given for each d, obtaining in this way a quotient stack [H_{d}/G]. However, when non-d-general curves appear, this stack presents some pathologies.

More precisely, recall from 3.5 that the scheme-theoretic quotient H_{d}/G is endowed with a natural surjective morphism ϕ_{d} : H_{d}/G → \overline{M}_{g}. The open subset of \overline{M}_{g} over which the quotient map π_{d} : H_{d} → H_{d}/G is a geometric quotient is exactly the locus of d-general curves. The problem is that, as soon as π_{d} : H_{d} → H_{d}/G fails to be a geometric quotient, the following pathologies occur:

(i) [H_{d}/G] fails to be a Deligne–Mumford stack. (ii) The natural map of stacks [H_{d}/G] → \overline{M}_{g} fails to be representable. (iii) Néron models are not parametrized by [H_{d}/G].

However, when studying Abel maps, we can still obtain some results. Since the stack \overline{P}_{d,g} behaves badly, let us consider the scheme \overline{P}_{d, g} := H_{d}/G introduced in 3.5. As we mentioned above, there is always a surjective morphism ϕ_{d} : \overline{P}_{d, g} → \overline{M}_{g}. By [C94], Thm. 6.1, p. 641, \overline{P}_{d, g} is an integral projective scheme. It is also normal, being a GIT-quotient of H_{d}, which is nonsingular by [C94], Lemma 2.2, p. 609.

Although \overline{P}_{d, g} is not a coarse moduli space, not even away from curves with nontrivial automorphisms, \overline{P}_{d, g} does satisfy useful functorial properties.

Thus, let f : X → B be a regular pencil of stable curves. Let B → \overline{M}_{g} be the associated map, and define

\overline{P_{f}^{d}} := B ×_{\overline{M}_{g}} \overline{P}_{d, g}.

If X is a closed fiber of f, denote by \overline{P_{X}^{d}} the corresponding fiber of \overline{P_{f}^{d}} over B.

As mentioned above, \overline{P_{f}^{d}} may fail to contain the Néron model N_{f}^{d}. However, a functorial property holds: the moduli property given in 3.8 (B) holds exactly as stated. More precisely, to any semibalanced line bundle L on a family of semistable curves Y → T having X_{T} → T as stable model, where T is any B-scheme, we can associate a canonical moduli map μ̂_{L} : T ⟶ \overline{P_{f}^{d}}; see [C94], Prop. 8.1, p. 653.

The main weakness, when nongeneral curves are present, is that different balanced line bundles on the same quasistable, or even stable, curve may be mapped to the same point in \overline{P_{f}^{d}}. Let us give an example of this behavior with regard to Abel maps.

Example 5.11. Let X = C_{1} ∪ C_{2} be a curve of compact type as in Example 5.7. However, assume now that C_{1} and C_{2} have the same genus. Then X is not 1-general. As before, let r be the unique node of X, and let q_{i} be the point of C_{i} lying over r for i = 1, 2.

We shall now exhibit three nonequivalent balanced line bundles that correspond to the same point of \overline{P_{X}^{1}}. Notice that, by Lemma 4.4, to give a line bundle on a curve of compact type is equivalent to give a line bundle on each irreducible component of the curve.

Let p ∈ C_{1} ∖ {q_{1}}. Our first line bundle is L_{1} ∈ Pic X corresponding to the pair

(O_{C_{1}}(p), O_{C_{2}}).

Let Y := X̂_{r}; so Y = C_{1} ∪ E ∪ C_{2}, where E is the exceptional component. Our second line bundle is L_{2} ∈ Pic Y corresponding to the triple

(O_{C_{1}}(p − q_{1}), O_{E}(q_{E}), O_{C_{2}}),

where q_{E} is the point of E glued to q_{1} in Y. Finally, our third line bundle is L_{3} ∈ Pic X corresponding to the pair

(O_{C_{1}}(p − q_{1}), O_{C_{2}}(q_{2})).

We leave out all proofs, referring to [C94], 7.2, Example 2, p. 645 for more details.

The above example shows that, if f : X → B is a regular pencil with a non-1-general fiber X, then \overline{P_{f}^{1}} and \overline{P_{X}^{1}} are not coarse moduli schemes for balanced line bundles.

However, we can still get a map \overline{α_{f}^{1}} : X → \overline{P_{f}^{1}} restricting to the classical Abel map of X_{K}, by using our modular interpretation of the Abel map. In fact, essentially the same line bundle M given in (32) can be used to produce a moduli map \overline{α_{f}^{1}} : X → \overline{P_{f}^{1}}. Most of the results in Sections 4 and 5 hold, provided we change one definition, as explained in Remark 5.12 below.

5.12. Small tails, again. Let X be a stable curve of arithmetic genus g. Taking into account the case where X is not 1-general, we need to adjust the definition of the set Q(X) in 4.1. Suppose that X has a separating node that generates two tails Q and Q′ of equal genus. It is easy to see that, if such a node exists, then it is unique. (A 1-general curve will never admit such a node by Proposition 3.15.) We must add to the set Q(X) of small tails of X either Q or Q′, thus making an arbitrary choice between Q and Q′, which nonetheless turns out to be completely irrelevant.

So, Q(X) is defined as the set of all small tails of X together with one tail of genus g/2, if any such tail exists.

Remark 5.13. The following results of the paper hold with essentially the same proof, as long as we use the modified definition of Q(X) for stable curves X of 5.12:

(i) Theorem 4.6, excluding Part (ii). (ii) Corollary 4.10. (iii) Theorem 5.5.

What will certainly fail is the possibility to interpret the Abel map in a unique way. In other words, if f : X → B is a regular pencil, an extension \overline{α_{f}^{1}} : X → \overline{P_{f}^{1}} of the Abel map of X_{K} is obtained as the moduli map of a semibalanced line bundle, however the line bundle is not uniquely determined.

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L. Caporaso (caporaso@mat.uniroma3.it) Dipartimento di Matematica, Università Roma Tre Largo S. L. Murialdo 1 00146 Roma - Italy

E. Esteves (esteves@impa.br) IMPA Est. D. Castorina 110 22460-320 Rio de Janeiro - Brazil