CHARACTERIZING CURVES BY THEIR ODD THETA-CHARACTERISTICS

Lucia Caporaso and Edoardo Sernesi

Abstract: We show that a general canonical curve is uniquely determined by the configuration of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.

1. Introduction

Consider a complex, canonical curve X of genus g. Such a curve possesses 2^{2g} line bundles L of degree g − 1 such that L^{⊗2} = K_{X}, the theta-characteristics of X, and precisely N_{g} := 2^{g−1}(2^{g} − 1) of them are odd (i.e. h^{0}(X, L) is odd). To a non-zero section of a theta-characteristic one associates a “half canonical” divisor D, whose double, 2D, is cut on X by a hyperplane H in ℙ^{g−1}. For obvious reasons, such a hyperplane H will be called a theta-hyperplane of X.

Assume that X is general; then every odd theta-characteristic L satisfies h^{0}(X, L) = 1, while the even ones have no non-zero sections. Therefore X determines a distinguished configuration of N_{g} thetahyperplanes, which will be denoted θ(X), and considered as an element of Sym^{N_{g}}(ℙ^{g−1})^{∗}.

A natural question is whether, conversely, such a distinguished set of hyperplanes uniquely determines the curve. The main result of this paper answers the question affirmatively (Theorem 6.1.1)

Main Theorem. Let X and X′ be general canonical curves of genus g ≥ 4. If θ(X) = θ(X′) then X = X′.

The case g = 3, not considered here, has recently been settled in [CS] (see below).

The above Theorem can be put in a different perspective using the classical theory of Riemann surfaces and their jacobians. Let J(X) = P ic^{0}(X) be the jacobian variety of X, fix a line bundle L_{0} of degree g − 1 and let Θ = Θ_{L_{0}} = {ξ ∈ J(X) : h^{0}(ξ ⊗ L_{0}) > 0} be the corresponding theta divisor. If we choose L_{0} to be a theta-characteristic we obtain a symmetric theta-divisor and the 2-torsion points of J(X) correspond, via multiplication by L_{0}, to the theta-characteristics on X. If X is general, Θ contains exactly N_{g} 2-torsion points, corresponding to the odd theta-characteristics, and they are nonsingular by Riemann Singularity Theorem. Therefore, by the standard geometric interpretation of the Gauss map for the theta divisor, we obtain that the set of Gauss images of the 2-torsion points of Θ, call it γ(J(X), Θ_{2}), coincides with θ(X). Therefore our theorem implies the following

Theorem. If X and X′ are general abstract curves of genus g ≥ 4 such that γ(J(X), Θ_{2}) and γ(J(X′), Θ′_{2}) are projectively equivalent, then X ≅ X′.

This result gives, in particular, a refinement of the Theorem of Torelli (in the generic case), because γ(J(X), Θ_{2}) only depends on the first order behaviour at finitely many points of the principally polarized abelian variety (J(X), Θ).

Notice moreover that the definition of γ(J(X), Θ_{2}) can be extended to any principally polarized abelian variety (A, Θ), provided that the 2-torsion points of the divisor Θ, which can be assumed to be symmetric, are all nonsingular. This condition is satisfied on a dense open subset U of the moduli space A_{g} of principally polarized abelian varieties. It is therefore natural to ask whether a result analogous to the above theorem is valid for principally polarized abelian varieties.

This could lead to an “odd counterpart” of the classical coordinatization of A_{g} by theta constants (see [Mu1] or [Mu3]).

Similar issues have also been studied using the theory of theta functions, by R. Salvati Manni in [SM]. Our approach to the Main Theorem is based on the study of degenerations of line bundles and linear

series for families of curves. From this point of view, the first issue is to study theta-characteristics and theta-hyperplanes for singular curves, and to relate the abstract and the projective description. This is the topic of Section 2.

For the abstract question, we use the so-called moduli space of stable spin curves, \overline{S_{g}}, constructed by M. Cornalba in [Co]. \overline{S_{g}} is a geometrically meaningful compactification of the moduli space of thetacharacteristics; it is endowed with a natural, finite morphism π : \overline{S_{g}} ⟶ \overline{M_{g}} onto the moduli space of Deligne-Mumford stable curves \overline{M_{g}} ([DM ]). The fibers of π over singular curves parametrize their “generalized theta-characteristics”.

On the projective side, the configuration θ(X) of theta-hyperplanes can be consistently defined for certain singular curves in ℙ^{g−1}; we thus have a regular morphism

θ : V ⟶ Sym^{N_{g}}(ℙ^{g−1})^{∗}

where V is a suitable subscheme of the Hilbert scheme containing canonical curves in ℙ^{g−1}. The curves parametrized by V include general smooth curves and what we call split curves. A split curve is the union of two rational normal curves meeting transversely at g + 1 points (details in 2.1). In this framework, the crucial result is that split curves satisfy our main Theorem; that is, a split curve X_{0} is uniquely determined, among all curves in V, by the configuration θ(X_{0}) (see 4.4.5).

Here are the outline of the argument and a summary of the paper. Consider a general family X ⟶ T (T a smooth curve) of non-singular canonical curves specializing to a split curve X_{0}; denote by X_{t} the generic fiber. If the Theorem were false, there would exist a second family of canonical curves X′ ⟶ T with nonsingular generic fiber X′_{t}, such that θ(X_{t}) = θ(X′_{t}). The first difficulty is to control the special fiber of the second family, X′_{0}, over which there is no a priori information. In particular, θ(X′_{0}) might fail to be well defined (if it were defined, it would obviously equal θ(X_{0})). To handle the situation, we study the stable reduction of X′ ⟶ T and transfer our projective problem into an abstract one, using the existence of \overline{S_{g}} and the properties established in Section 2.

The goal is to use the (given!) family of configurations of hyperplanes to control the central fiber of the stable reduction of X′ ⟶ T. To do that, we must characterize split curves abstractly by means of their (generalized) theta-characteristics; loosely speaking, we are after an abstract counterpart of the projective characterization by theta-hyperplanes mentioned above. This is the topic of Section 3; a purely combinatorial invariant for an abstract nodal curve (the “set of exponents”) is defined and it is proved to uniquely determine split curves among all stable curves (3.4.1, 3.4.2). The set of exponents encodes the numerical data of the scheme-theoretic fibers of the structural morphism π : \overline{S_{g}} ⟶ \overline{M_{g}}. Whence its relevance for our central problem.

From this analysis, we deduce that the stable reduction of X′ ⟶ T has a split curve as special fiber. Assume now, for simplicity, that X′ ⟶ T is the canonical model of its stable reduction. Then X′_{0} is

a split curve and we can apply the result mentioned before (4.4.5) to infer that, since θ(X′_{0}) = θ(X_{0}), then X′_{0} = X_{0}. The technical problem, behind the simplification, is dealt with in Section 4, by studying the natural action of P GL(g) on the spaces of configurations of hyperplanes; we here use methods of Geometric Invariant Theory.

To conclude that X_{t} = X′_{t}, it suffices to show that the morphism θ is étale locally at a split curve. This is done in Section 5, by applying M. Maruyama’s theory of elementary transformations of vector bundles, to standard deformation theory.

We conclude this introduction with a few words about the case of genus 3. In [CS] we proved that a general plane quartic is uniquely determined by its 28 bitangent lines, using a far more simple version of a similar approach. In that case, there is no need to consider the moduli theoretic aspect of the problem: the projective analysis suffices, using Mumford’s Geometric Invariant Theory. Just recently, D. Lehavi in [L] extended our result to every smooth quartic; we thank him for sending us his preprint. The combinatorial Theorem 3.4.1 is a strengthening of our original statement, found by C. Casagrande, to whom we are grateful.

1.1. Notations and Conventions. We work over the field of complex numbers. A semistable curve in the sense of Deligne and Mumford is a connected, reduced, projective curve Y having at most nodes as singularities and such that if E ⊂ Y is a smooth rational component, then E meets the union of the remaining components of Y in at least 2 points. An E such that E ∩ \overline{Y − E} = 2 is called a destabilizing component of Y. If Y is semistable and has no destabilizing component, Y is called a stable curve (in the sense of Deligne and Mumford). Stable curves of genus g admit a coarse moduli space, denoted by \overline{M_{g}}; it is a projective, integral scheme.

If Y is semistable and no two of its destabilizing components meet, then Y is called a quasistable curve (a “decent” curve in the terminology of M. Cornalba, in [Co]) .

An abstract, stable curve Y of genus g is a split curve if Y is the union of two rational, nonsingular curves meeting transversely at g + 1 points. A curve X ⊂ ℙ^{g−1} is a projective split curve if and only if X is the union of two rational normal curves meeting transversely at g + 1 points.

Clearly, a projective split curve is the canonical image of an abstract split curve.

We shall consider “families” (of curves, almost always) over a one-dimensional pointed base T, where T denotes a nonsingular, connected, affine, curve of finite type and t_{0} ∈ T a marked point. A family U ⟶ T is a flat, proper morphism of schemes; the fiber over the marked point t_{0} will be denoted by U_{0} or simply by U, and called the “special” fiber. The fiber over t ≠ t_{0} will be denoted bt U_{t} and called the “generic” fiber.

In certain contexts, a family U ⟶ T as above will be called a one parameter deformation of its central fiber U_{0}.

We shall denote P_{N_{g}} := Sym^{N_{g}}(ℙ^{g−1})^{∗}.

For S a topological space, γ_{c}(S) is the number of its connected components.

1.2. Let L and M be two finite sets of integers. We say that L dominates M (in symbols L ≥ M) iff there exists a surjective map α : L ⟶ M such that for every l ∈ L we have α(l) ≥ l. It is very easy to see that, if L ≥ M and M ≥ L, then L = M. In particular, the above definition gives a partial ordering on the set of all finite sets of integers.

Let S be a purely dimensional scheme (that is, every irreducible component of S_{red} has the same dimension). We associate to S its multiplicity set L(S) as follows

L(S) := {n : ∃ some irreducible component Z of S such that mult_{Z}S = n}

2. The moduli theoretic framework

2.1. The basic projective set up. Fix g ≥ 3 and consider the Hilbert scheme Hilb^{p(x)}[ℙ^{g−1}] of curves in ℙ^{g−1} having Hilbert polynomial p(x) = (2g − 2)x − g + 1. In it we find the locus of connected curves X of degree 2g − 2 and arithmetic genus g, satisfying the following three conditions:

1. X is reduced and has at most nodes as singularities.

2. X is embedded in ℙ^{g−1} by the complete linear series |ω_{X}| (where ω_{X} denotes the dualizing line bundle of X).

3. No irreducible component of X is contained in a hyperplane.

A curve satisfying 1. and 2. above is, clearly, stable in the sense of Deligne and Mumford; we shall call it a canonical curve throughout the paper.

Definitions. Let X ⊂ ℙ^{g−1} be a curve satisfying 1, 2 and 3 above. A hyperplane H ⊂ ℙ^{g−1} will be called a theta-hyperplane of X if H ∩ X is everywhere non reduced and supported at g − 1 distinct points. Let i be an integer with 0 ≤ i ≤ g − 1. We shall say that the theta-hyperplane H is of type i if H contains exactly i singular points of X. Such a curve X will be called theta-generic if it has finitely many theta-hyperplanes.

As noticed in the introduction, a general nonsingular canonical curve has N_{g} = 2^{g−1}(2^{g} − 1) distinct thetahyperplanes, corresponding bijectively to its odd theta characteristics. Therefore it is theta-generic.

We shall say that an abstract stable curve is theta-generic if its dualizing line bundle is very ample and if its canonical model is theta-generic. Notice that theta-generic curves can be of only two topological types: either they are irreducible, or they are the union of two rational components meeting at g +1 distinct points, i.e. they are split curves.

We shall denote by V ⊂ Hilb^{p(x)}[ℙ^{g−1}] the open subset parametrizing theta-generic curves and by V_{0} ⊂ V the open subset parametrizing nonsingular theta-generic curves. As in [C] and [CS] we define a morphism θ : V_{0} → Sym^{N_{g}}(ℙ^{g−1})^{∗} by θ(X) = {H_{1}, …, H_{N_{g}}}, where H_{1}, …, H_{N_{g}} are the (distinct) thetahyperplanes of X. Then, arguing as in [CS], Lemma 2.3.1, it is easy to see that θ can be extended to a morphism (called again θ)

θ : V → Sym^{N_{g}}(ℙ^{g−1})^{∗}

It will often be convenient to view θ(X) as a (not necessarily reduced) hypersurface of degree N_{g} in ℙ^{g−1}, all of whose irreducible components are (possibly multiple) hyperplanes; such a hypersurface will be also denoted θ(X), by abuse of notation, and called the theta-hypersurface of X. From [C] and [CS] we get that θ(X) is reduced if and only if X is smooth; if X is singular, all hyperplanes of type i appear in θ(X) with multiplicity 2^{i}.

For a given X in V we will denote by t_{i}(X) the number of distinct theta-hyperplanes of type i. The numbers t_{i}(X) only depend on the number of nodes and of irreducible components of X, and have been computed in [C].

Let

X ↪ T × ℙ^{g−1}
↓
T

be a family of curves whose Hilbert polynomial is p(x); the natural morphism to the Hilbert scheme is denoted by

ψ_{X} : T ⟶ Hilb^{p(x)}[ℙ^{g−1}].

Assume that the generic fiber is in V, that is ψ_{X} (T − {t_{0}}) ⊂ V. Since T is a nonsingular curve and P_{N_{g}} := Sym^{N_{g}}(ℙ^{g−1})^{∗} is projective the composition θ ∘ ψ_{X} extends to the whole of T; it will be denoted by

θ_{X} : T ⟶ P_{N_{g}}.

If ψ_{X} (T − {t_{0}}) ⊂ V_{0} we consider the curve

J_{X} ⊂ T × (ℙ^{g−1})^{∗}

defined as the closure of the incidence correspondence

{(t, H) : H ⊂ θ(X_{t}), t ≠ t_{0}} ⊂ T × (ℙ^{g−1})^{∗}

Away from t_{0}, J_{X} ⟶ T is an unramified covering of degree N_{g}.

In the next section, we describe the abstract counterpart of the above set-up.

2.2. Cornalba’s moduli space of spin curves. In [Co] M. Cornalba constructed a geometrically meaningful compactification \overline{S_{g}}, over \overline{M_{g}}, of the moduli space of theta-characteristics of smooth curves of genus g. We need to recall some basic features of \overline{S_{g}}, which is called the moduli space of stable spin curves. Both the scheme description and the stack description are available, we shall confine ourselves to the schematic definition, which suffices for our purposes.

\overline{S_{g}} is a normal, projective scheme which admits a proper morphism π onto \overline{M_{g}}

π : \overline{S_{g}} ⟶ \overline{M_{g}}.

As expected, the degree of π is 2^{2g}, and \overline{S_{g}} is a disjoint union of two irreducible components, \overline{S_{g}}^{+} and \overline{S_{g}}^{−}, corresponding, respectively, to even and odd theta-characteristics. We are mostly interested in odd theta characteristics; the degree of the restriction of π to \overline{S_{g}}^{−} is, of course, N_{g}.

The points of \overline{S_{g}} parametrize line bundles on certain quasistable curves of arithmetic genus g. Some more notation: let Y be a stable curve and let Σ be a set of nodes of Y, denote by Y_{Σ}^{ν} the normalization of Y at all nodes in Σ and by Y_{Σ} the quasistable curve obtained by “blowing-up” Y at all nodes in Σ. Thus Y_{Σ}^{ν} is the closure of what remains of Y_{Σ} after removing all of its destabilizing components.

A point in \overline{S_{g}} is (the isomorphism class of) a spin curve ξ, that is the following set of data. First, a subset Σ_{ξ} = Σ of nodes of Y and the quasistable curve Y_{Σ}. Second, a line bundle L_{ξ} = L on Y_{Σ} and a homomorphism α : L^{2} ⟶ ω_{Y_{Σ}} such that L has degree 1 on all destabilizing components, and the restriction of α to Y_{Σ}^{ν} is an isomorphism (see [Co] for details). We shall say that ξ is supported on Σ. As it will be made clear later in the paper (Section 3), not all subsets of nodes of Y are the support of a spin curve: those that are will be called “admissible” and will be studied extensively later.

The scheme structure on \overline{S_{g}} is obtained in [Co] after constructing the “universal deformation space” of a spin curve ξ. First (in Section 2) he describes its group of automorphisms Aut(ξ) and shows that there is an exact sequence of finite groups

1 ⟶ Aut_{0}(ξ) ⟶ Aut(ξ) ⟶ Aut(Y)

where Aut_{0}(ξ) is the so-called group of “inessential” automorphisms, (acting trivially on Y). Call B_{ξ} the base of the universal deformation space of ξ and B_{Y} the base of the universal deformation space of Y; B_{ξ} and B_{Y} will be viewed, as in [Co], as analytic spaces, that is, as 3g −3-dimensional discs. There is a commutative diagram of morphisms

B_{ξ} ⟶ B_{ξ}/Aut_{0}(ξ) ⟶ B_{ξ}/Aut(ξ) ↪ \overline{S_{g}}
δ↓  ^{π_{ξ}}↓  ↓^{π}
B_{Y} =  B_{Y}  ⟶^{ρ_{Y}} B_{Y} /Aut(Y) ↪ \overline{M_{g}}

with B_{ξ}/Aut_{0}(ξ) ⊂ B_{Y} ×\overline{M_{g}} \overline{S_{g}}; everything is shown to satisfy the necessary compatibility conditions. We recall how the covering δ is defined ([Co] section 5): choose coordinates u_{1}, ...., u_{3g−3} for B_{Y} so that the first #Σ correspond to the loci where the i-th node of Σ is preserved. Choose cordinates t_{1}, ...., t_{3g−3} on B_{ξ} and define δ as the base change u_{i} = δ^{∗}(t_{i}^{2}) for i ≤ #Σ and u_{i} = δ^{∗}(t_{i}) for i > #Σ.

Let Y ⟶ T be a family of generically stable curves. Since T is nonsingular and \overline{M_{g}} is projective, its moduli map T − \{t_{0}\} → \overline{M_{g}} extends to T and is denoted by

ϕ_{Y} : T ⟶ \overline{M_{g}}

The pull-back to T of \overline{S_{g}}^{−} is a curve over T denoted by

S_{Y} := ϕ_{Y}^{∗} \overline{S_{g}}^{−}.

Proposition 2.2.1. Let Y ⟶ T be a general one-parameter deformation of the stable curve Y. a ) The curve S_{Y} is smooth. b ) Let ξ be a spin curve on Y supported on the set Σ. Then the index of ramification of the finite covering

S_{Y} ⟶ T at the point corresponding to ξ is 2^{e_{Y} (Σ)} − 1 where e_{Y} (Σ) = #Σ − γ_{c}(Y_{Σ}^{ν}) + 1.

Remark. The same result holds for even spin curves, that is, if we replace S_{Y} by ϕ_{Y}^{∗} \overline{S_{g}}^{+}. The proof is the same.

Proof. It suffices to show that S_{Y} is non singular at the points lying over t_{0}. The moduli morphism ϕ_{Y} factors locally through the map ϕ_{Y} : O_{T,t_{0}} ⟶ B_{Y}; let \overline{T} := Imϕ_{Y}. We shall prove, more precisely, that if \overline{T} is transverse in B_{Y} to the loci where the nodes of Y are preserved, then S_{Y} is smooth.

It suffices to prove that ϕ_{Y}^{∗} ∘ ρ_{Y}^{∗} \overline{S_{g}}^{−} is smooth. By the above diagram, this is equivalent to show that the curve π_{ξ}^{−1}\overline{T} is smooth. Such a curve is the quotient via a finite group of the curve δ^{−1}\overline{T}; this last curve is smooth, because, by assumption, it is transverse to the branch locus of δ (look at the explicit description of δ given above). Hence its quotient by any finite group is smooth; this proves a).

For b) as well, most of the argument is already in Cornalba’s paper (Section 5), together with various explicit examples (from (5.3) to (5.7) ).

It suffices to look at how the covering π_{ξ}^{−1}(\overline{T}) ⟶ \overline{T} ramifies. The natural map δ : B_{ξ} ⟶ B is a finite covering of degree 2^{#Σ} totally ramified over the origin (as explained above); π_{ξ}^{−1}(\overline{T}) is the quotient of δ^{−1}(\overline{T}) via Aut_{0}(ξ). The structure of Aut_{0}(ξ) is described in [Co], Lemma 2.2 in terms of the graph associated to ξ (having verteces the connected components of Y_{Σ}^{ν} and edges the destabilizing components of Y_{Σ}); his result can be re-stated by saying that Aut_{0}(ξ) is a vector space over ℤ/2ℤ of dimension equal to the number of connected components of Y_{Σ}^{ν} minus 1:

dim_{ℤ/2ℤ} Aut_{0}(ξ) = γ_{c}(Y_{Σ}^{ν}) − 1.

Hence π_{ξ}^{−1}(\overline{T}) ⟶ \overline{T} ramifies at the point corresponding to ξ with index 2^{#Σ−γ_{c}(Y_{Σ}^{ν})−1} + 1.

2.3. We now introduce the scheme S_{Y} parametrizing odd spin curves having Y as stable model: S_{Y} is the (scheme-theoretic) special fiber of ρ_{Y}^{∗} \overline{S_{g}}^{−} = B_{Y} ×\overline{M_{g}} \overline{S_{g}}^{−} ⟶ B_{Y}.

Thus S_{Y} is a zero-dimensional scheme of length N_{g}. If Aut(Y) is trivial, then S_{Y} is the fiber of \overline{S_{g}}^{−} over the point Y ∈ \overline{M_{g}}. The following statement is an obvious consequence of 2.2.1.

Corollary 2.3.1. Let Y be a stable curve. L(S_{Y}) = {2^{n}, s.t. ∃ξ ∈ S_{Y} : e_{Y} (Σ_{ξ}) = n}.

The number e_{Y} (Σ), defined in the statement of 2.2.1, will be called the exponent of ξ and will play a crucial role in the sequel. Notice that, denoting by Z = Y_{Σ}^{ν} and by g_{Z} its arithmetic genus, we have that e_{Y} (Σ) = g − g_{Z}. The following simple observation will be used later.

Claim. Assume that ξ ∈ S_{Y} has exponent at least g − 3; then h^{0}(Z, L_{ξ} ⊗ O_{Z}) = 1.

In fact ξ is an odd spin curve, by definition; thus L_{ξ} ⊗ O_{Z} is an odd (hence effective) square root of ω_{Z}, that is, h^{0}(Z, L_{ξ} ⊗ O_{Z}) is odd (see [Co] section 6). Now Z is a curve of genus at most 3, hence of course h^{0}(Z, L_{ξ} ⊗ O_{Z}) ≤ 2.

Denote by S_{ξ} ⊂ S_{Y} the (unique) connected component of S_{Y} supported on ξ. Denote by Ŝ_{Y} the union of all components of S_{Y} corresponding to spin curves whose multiplicity is at least 2^{g−3}, that is

Ŝ_{Y} := ⋃_{ξ:e_{Y} (Σ_{ξ})≥g−3} S_{ξ}

Of course, Ŝ_{Y} can be empty, for example, it will be empty if Y is of compact type, in which case S_{Y} is reduced (see [Co] and also 3.3.1).

2.4. Relating the abstract and the projective pictures. The set up in this section is the following. Y is a stable curve and Y ⟶ T a one-parameter deformation of Y with smooth and theta-generic general fiber. A generically canonical, birational model of Y ⟶ T is the following: a projective family of curves: ℙ^{g−1} × T ⊃ W ⟶ T and a birational T -map ρ : Y − − > W which is an isomorphism away from t_{0} and such that for t ≠ t_{0}, the induced map Y_{t} ⟶ W_{t} is the canonical morphism.

Unless otherwise specified, no assumption will be made on the central fiber W of W, or on the restriction of ρ on the central fibers (notice that the locus of canonical curves is not closed in Hilb^{p(x)}[ℙ^{g−1}]).

To such a picture we add two, differently defined, families of odd theta-characteristics: to the abstract family Y ⟶ T we associate S_{Y} ⟶ T (defined in 2.2); to the projective family W ⟶ T we associate J_{W} ⟶ T (defined in 2.1). To make our notation more precise, consider θ_{W} : T ⟶ P_{N_{g}}, defined in 2.1. For t ≠ t_{0}, θ_{W}(t) represents the configuration of the N_{g} theta-hyperplanes of W_{t}. Since W_{t} ∈ V, this configuration does not depend on W and we have θ_{W}(t) = θ(W_{t}). For t_{0} the configuration θ_{W}(t_{0}) corresponds to a hypersurface of degree N_{g} in ℙ^{g−1} for which we shall use the notation

θ^{W}(W) := θ_{W}(t_{0})

(as usual, the above symbols will be abused to denote both points in P_{N_{g}} and hypersurfaces in ℙ^{g−1}). If W ∈ V we have, of course, θ^{W}(W) = θ(W) defined before. Consider now J_{W} ⊂ T × (ℙ^{g−1})^{∗}. If t ≠ t_{0}, the fiber of J_{W} over t only depends on W_{t} and will be denoted by J_{W_{t}}. The special fiber (J_{W})_{t_{0}} depends, a priori, on W. We shall denote

J_{W}^{W} := (J_{W})_{t_{0}}

and, if W ∈ V we shall simply write J_{W} := (J_{W})_{t_{0}}.

Let us consider the multiplicity sets; if t ≠ t_{0} we have, obviously, L(θ(W_{t})) = L(J_{W_{t}}) = {1}. For the special fibers we have, by construction, L(θ^{W}(W)) = L(J_{W}^{W}).

Lemma 2.4.1. Let Y ⟶ T be a general one-parameter deformation of the stable curve Y; let W ⟶ T be a generically canonical, birational image of Y ⟶ T. (a) There exists a natural, surjective, birational T -morphism

μ : S_{Y} ⟶ J_{W}.

(b) Assume furthermore that K_{Y} is very ample and that the central fiber W of W ⟶ T is a canonical image of Y, with W ∈ V. Then the morphism μ above induces an isomorphism on the central fibers. In particular, L(S_{Y}) = L(J_{W}).

Proof. By assumption, the generic fiber Y_{t} of Y is smooth and non-hyperelliptic and the generic fibers of W ⟶ T are in V. There is a natural T -birational map

μ : S_{Y} − −− > J_{W}

which is an isomorphism away from t_{0}. It associates to an odd theta-characteristic on Y_{t} the corresponding theta-hyperplane of its canonical model W_{t}. By part (a) of Proposition 2.2.1 the morphism μ extends to the whole of S_{Y}. This proves (a).

For (b), notice that by (a) it is enough to show that μ induces a set theoretic bijection on the central fibers, that is the set theoretic map

μ_{0} : (S_{Y})_{red} ⟶ (J_{W})_{red}

is a bijection. The proof is a bit long, but very simple and standard.

Throughout the rest of the argument, we shall identify Y with W by the canonical isomorphism given in the statement. Let ξ ∈ S_{Y} be a spin curve supported on the set of nodes Σ_{ξ} = Σ. To ξ there corresponds a line bundle L_{ξ} on the quasistable curve Y_{Σ} such that, if Z := Y_{Σ}^{ν} ⊂ Y_{Σ}, we have (L_{ξ} ⊗ O_{Z})^{2} ≅ ω_{Z} and such that, for every destabilizing component E of Y_{Σ}, the restriction of L_{ξ} to E is O_{E}(1). Since W ∈ V, there exists a unique effective divisor D_{ξ} ∈ PicZ such that

O_{Z}(D_{ξ}) = L_{ξ} ⊗ O_{Z}.

Moreover, D_{ξ} is reduced and suppD_{ξ} ⊂ (Y_{Σ})_{smooth}.

Consider the canonical morphism σ : Y_{Σ} ⟶ W ⊂ ℙ^{g−1} (which of course contracts all the destabilizing components); locally at every point of D_{ξ}, σ is an isomorphism. The set of points Σ ∪ σ(D_{ξ}) ⊂ W is a set of g − 1 points in general position (recall that W is theta-generic), which therefore spans a unique hyperplane H_{ξ} containing Σ. It is clear that H_{ξ} is tangent to W at every point of σ(D_{ξ}).

In conclusion, we have explicitely described μ_{0}(ξ); this has the advantage of proving that μ_{0} is injective. In fact μ_{0}(ξ) = μ_{0}(ξ′) ⇔ H_{ξ} = H_{ξ′}, this implies that Σ_{ξ} = Σ_{ξ′} and that D_{ξ} = D_{ξ′}. This is enough to ensure that ξ = ξ′ (see [Co]).

It remains to show that μ_{0} is surjective. Let H be a theta-hyperplane of W, so that (t_{0}, H) ∈ J_{W}, and let Σ = H ∩ W_{sing}. Then H is tangent to W at g − 1 − #Σ smooth points of W, let D be the divisor defined by H ∩ W_{smooth} = 2D. Denote by σ : Y_{Σ} ⟶ W the canonical morphism, let Z := Y_{Σ}^{ν} ⊂ Y_{Σ} and denote by σ_{Z} := σ_{|Z} the restriction to Z. Let Δ := σ_{Z}^{∗}(Σ) so that deg Δ = 2#Σ and

σ_{Z}^{∗}H = 2σ_{Z}^{∗}(D) + Δ.

On the other hand, W is a canonical image of Y_{Σ} hence

σ_{Z}^{∗}H = ω_{Y_{Σ}} ⊗ O_{Z} = ω_{Z} ⊗ O_{Z}(Δ).

Hence ω_{Z} = 2σ_{Z}^{∗}(D).

Finally, by gluing σ_{Z}^{∗}(D) to O_{E}(1) on every destabilizing component E of Y_{Σ}, we obtain a line bundle L which (regardless of the gluing data, which are proven to be irrelevant in [Co]) corresponds to a spin curve ξ on Y such that μ_{0}(ξ) = H. Proving that μ_{0} is surjective.

Let H ⊂ Hilb^{p(x)}[ℙ^{g−1}] be the projective scheme defined as the closure of V. Let P be a projective scheme and consider the space of morphisms from T to P

Hom[P] := {τ : T ⟶ P }.

Since T is a smooth curve, any rational map from T to P extends to a (uniquely defined) regular morphism \overline{T} ⟶ P, where \overline{T} is the smooth compactification of T. Thus Hom[P] has a natural scheme structure. We can apply this to the projective varieties \overline{M_{g}}, H and P_{N_{g}}. We shall thus consider the scheme of maps from T to \overline{M_{g}}: Hom[\overline{M_{g}}]; the scheme of maps from T to H: Hom[H], and the scheme Hom[P_{N_{g}}] as above.

There is a rational map ϕ : H − − > \overline{M_{g}} (regular at least on V), which is the moduli map associated to the universal family over H. Recall also that we have a rational map θ : H − − > P_{N_{g}} which is regular on V. Therefore we have two morphisms of schemes:

ϕ_{∗} : Hom[H] ⟶ Hom[\overline{M_{g}}]

given by composing with ϕ (that is, for ψ ∈ Hom[H], we define ϕ_{∗}(ψ) := ϕ ∘ ψ);

θ_{∗} : Hom[H] ⟶ Hom[P_{N_{g}}]

given by composing with θ (as above). We have

Lemma 2.4.2. Let Y ⟶ T be a one-parameter deformation of Y, with smooth and theta-generic general fiber. Let W ⟶ T be a generically canonical, birational image of Y. Then

L(S_{Y}) ≥ L(J_{W}^{W}).

Proof. First we consider the case in which Y ⟶ T is a general one parameter deformation of Y. By 2.2.1, we have that S_{Y} is a smooth curve and, by 2.4.1(a), there is a surjective T -morphism

S_{Y} ⟶ J_{W}.

By comparing the fibers over t_{0} we obtain L(S_{Y}) ≥ L(J_{W}^{W}) which is what we wanted.

Consider now a one-parameter deformation Y ⟶ T of Y and a generically canonical model W ⊂ T × ℙ^{g−1} as stated. As usual, denote by ψ_{W} ∈ Hom[H] and ϕ_{Y} ∈ Hom[\overline{M_{g}}] the corresponding moduli morphisms; of course, ϕ_{Y} = ϕ ∘ ψ_{W}. Pick now a local curve U^{M} ⊂ Hom[\overline{M_{g}}] whose special point is ϕ_{Y}. More precisely, let R be a discrete valuation ring and let

U^{M} := Im\{ϕ_{R} : SpecR ⟶ Hom[\overline{M_{g}}]\}

be such that the image of the generic point, ϕ_{R}(η) = ϕ_{η}, is a general one-parameter deformation of Y, and the image of the special point is the given ϕ_{R}(s) = ϕ_{Y}.

Let U^{H} ⊂ Hom[H] be a lifting of U^{M} passing through ψ_{W}; that is,

U^{H} := Im{ψ_{R} : SpecR ⟶ Hom[H]}

such that the image of the generic point, ψ_{R}(η) = ψ_{η} is such that ϕ ∘ ψ_{η} = ϕ_{η} while the image of the special point is the original ψ_{W}. Denote by W^{η} ⊂ T × ℙ^{g−1} the family of curves corresponding to ψ_{η} (that is, W^{η} is the pull-back, via ψ_{η}, of the universal family over H and ψ_{η} = ψ_{W^{η}}).

Lastly, let U^{P} ⊂ Hom[P_{N_{g}}] be defined as U^{P} := θ_{∗}(U^{H}). Thus

U^{P} = Im{θ_{R} : SpecR ⟶ Hom[P_{N_{g}}]}

such that, denoting, as usual, θ_{η} and θ_{s} the images of the generic and special point (respectively) of SpecR, we have θ_{η} = θ ∘ ψ_{W^{η}} = θ_{W^{η}} and θ_{s} = θ_{W}. Now we shall construct a universal family of theta-hypersurfaces over U = SpecR using the morphism defined above

θ_{R} : U ⟶ Hom[P_{N_{g}}] ↪ Hilb(T × P_{N_{g}})

(where the immersion on the right is the structural one, used to give a scheme structure to Hom[P_{N_{g}}]). To do that, we shall proceed in a standard way, using the various universal families over the Hilbert schemes in our set-up.

There is a universal family F ⟶ Hilb(T × P_{N_{g}}) which we can pull back to U via θ_{R}. Denote by G := F ×_{Hilb(T ×P_{Ng})} U ⟶ U this pull-back. All of its fibers are thus isomorphic to T. Since U = SpecR, we get that this fibration is in fact trivial, so that G ≅ T × U; let us fix one of such isomorphisms from now on.

Consider also the universal family of (reducible) hypersurfaces over P_{N_{g}}, denoted by P_{N_{g}} × ℙ^{g−1} ⊃ I ⟶ P_{N_{g}}. We have a diagram

G ⟶  F  ↪ Hilb(T × P_{N_{g}}) × T × P_{N_{g}} ⟶ P_{N_{g}}
↓  ↓
U ⟶ Hilb(T × P_{N_{g}})

so that we can pull back I ⟶ P_{N_{g}} to G. Denote Θ̃ := G ×_{P_{Ng}} I such a pull back. Finally

Θ̃ ⟶ G ≅ T × U ⟶ U

is the family of theta-hypersurfaces that we wanted. Let us now consider the restriction Θ̃^{0} of Θ̃ to {t_{0}} × U, i.e. Θ̃^{0} ⟶ {t_{0}} × U ≅ SpecR. This is a family of schemes of pure dimension g − 2, for which we get L((Θ̃^{0})_{η}) ≥ L((Θ̃^{0})_{s}). By construction, the special fiber is the original θ^{W}(W) = (Θ̃^{0})_{s}. For the generic fiber, of course, (Θ̃^{0})_{η} = θ^{W^{η}}(W^{η}) (where, as usual, W^{η} denotes the central fiber of W^{η}). Thus the above inequality translates into L(θ^{W^{η}}(W^{η})) ≥ L(θ^{W}(W)). Recall now that W^{η} is a birational, canonical image of a general one-parameter deformation of Y. Therefore we can apply to W^{η} the result of the first part of the proof, that is: L(S_{Y}) ≥ L(J_{W^{η}}^{W^{η}}); since L(J_{W^{η}}^{W^{η}}) = L(θ^{W^{η}}(W^{η})), we conclude that L(S_{Y}) ≥ L(θ^{W}(W)) = L(J_{W}^{W}).

3. Combinatorics of stable curves

This section is essentially independent of the rest of the paper; its goal is 3.4.2, which will play a crucial role later on. We have seen that, for a stable curve Y, the multiplicity set L(S_{Y}) only depends on the combinatorial data of Y (2.2.1 and 2.3.1). Here we shall make this more precise by re-defining it in purely combinatorial terms and thus simplifying its computation. The key fact will be that L(S_{Y}) is a fine invariant for certain types of curves: namely, it completely determines the topological and combinatorial structure of Y (among all stable curves) if Y is of compact type (this is easy: 3.3.1) and if Y is split (this is more interesting: 3.4.1, 3.4.2).

3.1. Let Y be a nodal curve. Γ_{Y} is the dual graph of Y, that is, the verteces of Γ_{Y} correspond to the irreducible components of Y, the edges joining two verteces correspond to the nodes contained in the two corresponding components.

Recall that Y_{sing} is the set of all nodes of Y and, if Σ ⊂ Y_{sing}, the curve Y_{Σ}^{ν} is the normalization of Y at the nodes contained in Σ. Assume that Y has arithmetic genus g, δ nodes and γ irreducible components; let Y = ⋃_{i=1}^{γ} C_{i} be the decomposition of Y into irreducible components, and let g_{i} be the geometric genus of C_{i}. Recall the genus formula: g = ∑ g_{i} + δ − γ + γ_{c}(Y).

Let N be a node of Y. We say that N is internal if N is contained in a unique irreducible component of Y, otherwise we say that N is external. A (external) node N is called separating if γ_{c}(Y) = γ_{c}(Y − N) − 1. If all nodes of Y are separating, Y is called “of compact type”.

Definition. Exponent of a set of nodes. Let Σ be a set of nodes of Y. The exponent of Σ is the number e_{Y} (Σ) below

e_{Y} (Σ) = p_{a}(Y) − p_{a}(Y_{Σ}^{ν})

As we already know (see 2.3), an equivalent definition is e_{Y} (Σ) := #Σ − γ_{c}(Y_{Σ}^{ν}) + γ_{c}(Y).

The reason for the name “exponent” comes from Proposition 2.2.1. Keeping the same notation, we list a couple of straightforward consequences of the definition.

Property A. If Σ ⊂ Σ′ then e_{Y} (Σ) ≤ e_{Y} (Σ′). Moreover, for every Σ,

0 = e_{Y} (∅) ≤ e_{Y} (Σ) ≤ e_{Y} (Y_{sing}) = g − ∑_{i=1}^{γ} g_{i}

Property B. If N is a node of Y not contained in Σ, then

e_{Y} (Σ ∪ \{N\}) = \{ \begin{matrix}e_{Y} (Σ) + 1 & if N is not separating for Y_{Σ}^{ν}; \\ e_{Y} (Σ) & if N is separating for Y_{Σ}^{ν}. \end{matrix}

We shall need to compute the exponent of a distinguished type of sets of nodes:

Definition. Admissible sets of nodes. A set Σ ⊂ Y_{sing} is said to be admissible if for every subcurve W of Y, the number of nodes in the intersection W ∩ W^{c} which are not in Σ is even. Equivalently, Σ is admissible if for every i = 1, ...., γ the number #[(C_{i} ∩ C_{i}^{c}) ∩ (Y_{sing} − Σ)] is even.

The equivalence of the two definitions above is an elementary verification. We denote by A_{Y} the set of all admissible sets of nodes of Y:

A_{Y} := {Σ ⊆ Y_{sing} s.t. Σ is admissible}

The previous definition is implicit in Cornalba’s paper; it is motivated by the fact that Σ is admissible if and only if the dualizing sheaf ω_{Y_{Σ}^{ν}} has even degree on every subcurve. It is important to stress that a set Σ is the support of a spin curve on Y if and only if Σ ∈ A_{Y} (see [Co]).

Some simple examples; the set of all nodes Σ = Y_{sing} is certainly admissible. Similarly the set of all external nodes of Y is admissible, in fact the above definition ignores the internal nodes. The empty set is not always admissible, consider for example a singular curve of compact type Y = C_{1} ∪ C_{2} with #C_{1} ∩ C_{2} = 1.

Here are a few immediate properties:

Property C. If Σ ∈ A_{Y}, then Σ contains all the separating nodes of Y.

Property D. Let Σ ⊂ Y_{sing}, denote Z = Y_{Σ}^{ν} and let Σ_{1} ⊂ Z_{sing}. Then Σ_{1} ∈ A_{Z} if and only if Σ_{1} ∪Σ ∈ A_{Y}. In particular ∅ ∈ A_{Z} if and only if Σ ∈ A_{Y}.

3.2. Cyclic sets of nodes. Given our nodal curve Y, consider its dual graph Γ_{Y}. To any subset, Σ, of nodes of Y, we can associate a graph Γ_{Σ}, which is the subgraph of Γ_{Y} generated by all edges representing the nodes contained in Σ. We shall say that a non-empty Σ is a cyclic set of nodes if its graph Γ_{Σ} is a closed polygon.

An equivalent, graph-free, definition is the following. Up to reordering the irreducible components of Y, we can write a cyclic set of nodes as

Σ = {N_{1,2}, N_{2,3}, ..., N_{h−1,h}, N_{h,1}}

with 1 ≤ h ≤ γ, meaning that N_{i,j} ∈ C_{i} ∩C_{j}. For example, if N is an internal node of Y, then {N} is a cyclic set. Observe that, if Y is not of compact type, then Y always admits some cyclic set of nodes. Conversely, a curve of compact type does not admit any cyclic set of nodes.

Lemma 3.2.1. Let Σ be the complement in Y_{sing} of a cyclic set of nodes. Then Σ is admissible and e_{Y} (Σ) = g − ∑ g_{i} − 1. More generally, let Σ be the complement in Y_{sing} of a disjoint union of n cyclic sets of nodes. Then Σ is admissible and e_{Y} (Σ) ≤ g − ∑ g_{i} − n.

Proof. If Σ_{1}....., Σ_{n} are cyclic sets of nodes of Y such that Σ_{i} ∩ Σ_{j} = ∅ for all i ≠ j, then their union contains an even number of external nodes on every irreducible component of Y. Thus Σ = Y_{sing} −∪_{1}^{n}Σ_{i} is admissible by definition. If n = 1 so that Σ is the complement of Σ_{1} we easily compute

e_{Y} (Σ) = (δ − #Σ_{1}) − (γ − #Σ_{1} + 1) + γ_{c}(Y) = δ − γ + γ_{c}(Y) − 1 = g − ∑ g_{i} − 1.

If n is arbitrary, the inequality in the statement is straightforward.

Definition. Exponent set. We shall call the finite set of integers, that occur as exponents of admissible sets of nodes, the exponent set of Y, and we will denote it by E_{Y}:

E_{Y} := {e, such that there exists Σ ∈ A_{Y} with e_{Y} (Σ) = e}

Remark. It is clear that E_{Y} depends only on the combinatorial data of Y. More precisely, if two curves Y and Y′ have the same genus and the same dual graph, then E_{Y} = E_{Y′}.

Lemma 3.2.2.

a ) If Y is not of compact type, then {g − ∑ g_{i} − 1, g − ∑ g_{i}} ⊂ E_{Y}.

b ) If Σ is admissible and e_{Y} (Σ) = g − ∑ g_{i}, then Σ = Y_{sing}.

c ) If N ∈ Y is an internal node, then either g − ∑ g_{i} − 2 ∈ E_{Y} or Y_{N}^{ν} is of compact type and E_{Y} = {0, 1}.

Proof. Since Y_{sing} is admissible and e_{Y} (Y_{sing}) = g − ∑ g_{i}, we have that g − ∑ g_{i} ∈ E_{Y}. To see that g − ∑ g_{i} − 1 ∈ E_{Y}, observe that, since Y is not of compact type, Y contains a cyclic set of nodes, whose complement is admissible and has exponent precisely g − ∑ g_{i} − 1, by 3.2.1. This proves a).

For b) we have to show that if Σ is admissible and its exponent is maximum (that is e_{Y} (Σ) = e(Y_{sing}) = g − ∑ g_{i}) then Σ contains every node of Y. By contradiction, let Σ ≠ Y_{sing} and let Z = Y_{Σ}^{ν}; the curve Z is singular. By Property D, ∅ ∈ A_{Z} hence Z is free from separating nodes, thus Z is not of compact type and it contains a cyclic set of nodes Σ_{0}. We have thus a cyclic set of nodes Σ_{0} ⊂ Y_{sing} such that Σ ⊂ Σ_{0}^{c}. Hence we have, by Property A,

e_{Y} (Σ) ≤ e_{Y} (Σ_{0}^{c}) = g − ∑ g_{i} − 1

which is a contradiction. To prove c) let Z = Y_{N}^{ν}; since N is internal, we have that

A_{Y} = {Σ, Σ ∪ N, ∀Σ ∈ A_{Z}};

By Property B, e_{Y} (Σ ∪ N) = e_{Y} (Σ) + 1 = e_{Z}(Σ) + 1 hence

E_{Y} = {n, n + 1, ∀n ∈ E_{Z}}.

If Z is of compact type, E_{Z} = {0} hence E_{Y} = {0, 1}. Otherwise, part (a) applied to Z says that, E_{Z} contains the number n = (g − 1) − ∑ g_{i} − 1 (the arithmetic genus of Z is, of course, g − 1). Hence E_{Y} also contains n = g − ∑ g_{i} − 2 and we are done.

3.3. Let Y be stable. There is a relation between E_{Y} and S_{Y}:

L(S_{Y}) = \{ \begin{matrix}\{2^{n}, ∀n ∈ E_{Y} , n ≠ g − ∑ g_{i}\} & if g_{i} = 0 ∀i \\ \{2^{n}, ∀n ∈ E_{Y} \} & otherwise \end{matrix}

In fact, L(S_{Y}) is described in 2.3.1, and recall that for every ξ ∈ S_{Y}, the support Σ_{ξ} of ξ is admissible. Now, (see [Co] section 6) if Σ is admissible and Σ ≠ Y_{sing}, then there exist odd (and even) spin curves supported on Σ (an equal number of odd and even, in fact). By 3.2.2, e_{Y} (Σ) = g − ∑ g_{i} iff Σ = Y_{sing}. A spin curve ξ supported on Y_{sing} is given by a line bundle L on Y^{ν} which restricts to a theta-characteristic L_{i} on every irreducible component C_{i}^{ν} of Y^{ν}. For ξ to be odd it is necessary and sufficient that L_{i} be an odd theta-characteristic on C_{i}^{ν} for an odd number of components of Y^{ν}. Therefore there exist odd spin curves supported on Y_{sing}, unless all components of Y^{ν} are rational; in fact there are obviously no odd theta-characteristics on ℙ^{1}. In other words, if g_{i} = 0 for every i, a spin curve of exponent g is necessarily even.

Corollary 3.3.1. A curve Y is of compact type if and only if E_{Y} = {0}.

Remark. By 3.3, if Y is stable, this is equivalent to saying that Y is of compact type if and only if S_{Y} is reduced. Proof. If Y is of compact type, the only admissible set is Y_{sing} (by Property C), whose exponent is 0. The converse follows from Lemma 3.2.2.

Lemma 3.3.2. Let Y be a connected, nodal curve and let X be its stable model. Then E_{X} = E_{Y}.

Proof. Let σ : Y ⟶ X be the natural map, contracting all smooth rational components of Y meeting their complementary curve in less than 3 points. There is a natural bijection β between A_{Y} and A_{X}

\begin{aligned}β : & A_{Y} & ⟶ & A_{X} \\ & Σ & ↦ & σ(Σ) ∩ X_{sing} \end{aligned}

whose inverse is, denoting by Y_{sep} the subset of all separating nodes of Y,

\begin{aligned}β^{−1} : & A_{X} & ⟶ & A_{Y} \\ & Σ′ & ↦ & [σ^{−1}(Σ′) ∩ Y_{sing}] ∪ Y_{sep} \end{aligned}

It is a trivial verification to show that the two above maps are each other inverse and that they preserve the exponents, that is e_{Y} (Σ) = e_{X}(β(Σ)).

3.4. We now describe E_{X} for a split curve X. Of course e(X_{sing}) = g. Let Σ be a subset of nodes such that Σ ≠ X_{sing}; then e_{X}(Σ) = #Σ and Σ is admissible if and only if #Σ ≢ g mod 2. Thus if g is odd, the exponents appearing in E_{X} are 0, 2, 4, ...., g − 1, g (in particular, the only odd exponent is g). Symmetrically for g even. Summarizing:

Let X be a split curve of genus g. Then

E_{X} = \{ \begin{matrix}\{0, 2, 4, ...., g − 3, g − 1, g\} & if g is odd; \\ \{1, 3, 5, ...., g − 3, g − 1, g\} & if g is even. \end{matrix}

We shall prove that the converse is also true, in other words split curves are identified in \overline{M_{g}} by their exponent set. Something more precise is true.

Theorem 3.4.1. Let Y be a DM stable curve of genus g. Assume that g ∈ E_{Y} and that g − 2 ∉ E_{Y}. Then either Y is a split curve, or Y is the “polygonal” curve of genus 3.

Where recall that the polygonal curve of genus 3 is the nodal curve made of four copies of ℙ^{1} meeting pairwise in one point (having a total of six nodes). Its canonical model is the union of four general lines in ℙ^{2}, whence the name “polygonal”.

We shall apply the Theorem above to obtain the following crucial result:

Corollary 3.4.2. Let X and Y be stable curves of genus g; assume that X is split. (a) If E_{X} = E_{Y}, then Y is split. (b) If L(S_{X}) = L(S_{Y}) then Y is split.

Proof of the Corollary. We can exclude that Y is the polygonal curve of genus 3, in fact, in that case E_{Y} = {2, 3} whereas the exponent set of a split curve X of genus 3 is {0, 2, 3}. Similarly, L(S_{Y}) does not contain 1, whereas L(S_{X}) does.

Assume now that E_{X} = E_{Y} = E. If X is split, then E satisfies the assumptions of Theorem 3.4.1(see above), hence Y is split.

This proves (a). Let us show that (b) implies (a). For any stable curve Z, the relation between E_{Z} and L(S_{Z}) has been explained at the beginning of 3.3. This yields that E_{X} and E_{Y} contain precisely the same integers n such that n ≤ g − 1. It remains to show that g ∈ E_{Y}. Notice that g − 2 ∉ E_{Y} and g − 1 ∈ E_{Y}; therefore, if g ∉ E_{Y}, we get a contradiction to 3.2.2(a) (Y is not of compact type by 3.3.1).

Proof of the Theorem. Recall the basic notation: Y = ∪_{i=1}^{γ}C_{i}. The proof is in three steps.

Step 1: Y is a union of smooth rational components. The fact that g ∈ E_{Y} implies that e_{Y} (Y_{sing}) = g (see 3.2.2) and that all irreducible components of Y have geometric genus 0. If Y contained an internal node, by 3.2.2, part c), E_{Y} would contain g − 2, which is not the case.

Step 2: It suffices to assume that Y is free from separating nodes. Given our stable curve Y, denote by Y^{∗} a stable curve of genus g obtained by smoothing out all the separating nodes of Y. This is to say that if N is a separating node, and N = C_{1} ∩ C_{2}, the curve Y^{∗} is obtained by replacing C_{1} ∪ C_{2} (which is a curve of arithmetic genus 0 by the Step 1) with a smooth rational curve D.

Notice that, unless Y is free from separating nodes, Y^{∗} contains a separating component: D (that is D^{c} is disconnected).

By Property C there is a natural bijection between the admissible sets of Y and those of Y^{∗}: to an admissible set Σ of Y (which must contain all the separating nodes of Y) we associate the admissible set Σ^{∗} = Σ − {N : N is separating} on Y^{∗}. By Property B we have e_{Y} (Σ) = e(Σ^{∗}), hence E_{Y} = E_{Y^{∗}}.

If we show that Y^{∗} is a split curve, then Y^{∗} has no separating components, hence Y = Y^{∗} and we are done. From now on we shall assume that Y is free from separating nodes.

Step 3. We can assume that Y is free from internal and separating nodes. We proceed by induction on g. The case g = 3 is the first case to be treated. A straightforward case by case analysis (there are only 3 cases) yields the result. By the previous steps, there are only three cases. If Y is not split, then Y has more than two components. If Y has three components, then its dual graph is uniquely determined and so is E_{Y} (compare with the Remark in 3.2); Y has 5 nodes and E_{Y} = {1, 2, 3}. If Y has four components, then Y is the polygonal curve and E_{Y} = {2, 3}. There are no other cases.

Assume g ≥ 4. Let N be any node of Y and let Z = Y_{N}^{ν} be the normalization of Y at N. Since N is not an internal node of Y, to complete the proof of the Theorem it suffices to prove that Z is a split curve.

Notice that Z is connected (Y is free from separating nodes), nodal, and its arithmetic genus is g − 1. Moreover, Z is a union of smooth, rational components (because Y is) and hence g − 1 ∈ E_{Z}.

Let now Σ ∈ A_{Z}, then clearly Σ∪N ∈ A_{Y} and e_{Z}(Σ) = e_{Y} (Σ∪N)−1. We conclude that E_{Z} ⊂ E_{Y} −1, that is, if n ∈ E_{Z} then n + 1 ∈ E_{Y}. In particular, g − 3 ∉ E_{Z}, because, by hypothesis, g − 2 ∉ E_{Y}.

Let now X be the stable model of Z (possibly equal to Z). By Lemma 3.3.2, E_{X} = E_{Z}. Therefore X satisfies the assumptions of the Theorem. By induction we obtain that X is a split curve of genus g − 1, unless X is the polygonal curve of genus 3.

The rest of the proof consists in showing that Z is stable (i.e. X = Z) and that Z is not the polygonal curve of genus 3.

Assume first that X is split.

By contradiction, assume that Z is not stable; then Z has one or two destabilizing components (since Z is the partial normalization of the stable curve Y at a unique node N). Hence Z, and likewise Y, has three or four irreducible components. We need to distinguish the components of Y from those of Z; let us denote by C_{i}^{Z} the irreducible component of Z which naturally (via the normalization at N) corresponds to C_{i} in Y.

We can write X = D_{1} ∪ D_{2} with #D_{1} ∩ D_{2} = g. Let C_{1} and C_{2} be the two components of Y that correspond to D_{1} and D_{2}. Then #C_{1} ∩ C_{2} ≥ 2; pick two nodes in C_{1} ∩ C_{2} and denote them by N_{1,2}, N_{2,1}. Obviously the set {N_{1,2}, N_{2,1}} is a cyclic set of nodes of Y.

The fact that X is split implies that every destabilizing component of Z meets both C_{1}^{Z} and C_{2}^{Z} in one point, thus (on Y) we have that, for all i ≥ 3 (so that C_{i}^{Z} is destabilizing in Z), C_{i} ∩ C_{1} ≠ ∅ and C_{i} ∩ C_{2} ≠ ∅. There are two possible cases, according to the number of destabilizing components of Z, which is 1 or 2.

First case: Z has one destabilizing component, C_{3}^{Z}. In this case, (up to switching C_{1} and C_{2}) N ∈ C_{2}∩C_{3} so let us rename N = N_{2,3}. Since C_{2}^{Z} ∩ C_{3}^{Z} ≠ ∅ there must be another node in C_{2} ∩ C_{3}, let it denote by N_{3,2}. The set {N_{2,3}, N_{3,2}} is a cyclic set of nodes of Y.

Let now Σ be the complement of the two cyclic sets we constructed above, that is

Σ := Y_{sing} − ({N_{1,2}, N_{2,1}} ∪ {N_{2,3}, N_{3,2}}).

By Lemma 3.2.1, Σ ∈ A_{Y} and its exponent is easily seen to be equal to g − 2. Then g − 2 ∈ E_{Y}, a contradiction.

In the second case Z has two destabilizing components, C_{3}^{Z} and C_{4}^{Z}. Now N ∈ C_{3} ∩C_{4}, so we rename it: N = N_{3,4}. Since C_{3}^{Z} and C_{4}^{Z} both meet C_{1}^{Z} and C_{2}^{Z}, we find (in Y) nodes N_{2,3} ∈ C_{2} ∩C_{3} and N_{4,2} ∈ C_{2} ∩C_{4}. The set of nodes {N_{2,3}, N_{3,4}, N_{4,2}} is cyclic. Continuing as in the previous case, we let Σ be the set

Σ := Y_{sing} − ({N_{1,2}, N_{2,1}} ∪ {N_{2,3}, N_{3,4}, N_{4,2}}).

which is admissible, by 3.2.1, and whose exponent is g − 2. A contradiction.

We conclude that Z is stable, which is what we wanted.

To complete the proof we must show that X is not the polygonal curve of genus 3. This is done just as above, showing, by a trivial case by case analysis, that Y contains two disjoint cyclic sets of nodes whose complement has exponent g − 2.

4. Configurations of theta-hyperplanes of split curves

4.1. The configuration of theta-hyperplanes of a projective split curve is described in details in [C], where it is proved (Theorem 5) that split curves are uniquely determined by it, among all curves in V. In this section, we shall apply Geometric Invariant Theory ([GIT]) and give a sharper version of such a result (4.4.5), to be used later.

It is worth pointing out a nice feature of split curves: the projection of a split curve in ℙ^{g−1} from every subset of i ≤ g − 3 of its nodes is a split curve in ℙ^{g−1−i}.

Let H be a theta-hyperplane of type i of a projective split curve X, thus H contains exactly i nodes of X (0 ≤ i ≤ g − 1) and it is tangent to X at g − 1 − i smooth points, equally distributed among the two irreducible components of X. The set of all hyperplanes of type i is denoted by Θ_{i}(X) and its cardinality by t_{i}(X) = #Θ_{i}(X). Clearly, Θ_{i}(X) is empty for all i having the same parity of g. We need the following

Claim. For every projective split curve X, we have t_{g−3}(X) = 4\binom{g+1}{g−3} and t_{g−1}(X) = \binom{g+1}{g−1}.

For the first formula, it suffices to check that for any subset Σ of g − 3 nodes of X, there are 4 thetahyperplanes of type g − 3 containing Σ, and containing no other node of X. If g = 3, we are simply saying that two plane conics meeting transversely have exactly 4 distinct tangent lines in common (which is clear by looking at the duals) and that these 4 tangent lines do not go through the common points (also easy, see [CS]). The general case is done by projecting X from Σ to a split curve of genus 3 in ℙ^{2}.

The second formula is obvious: the nodes of X are in general linear position, therefore there is a unique theta-hyperplane of type g − 1 for every set of g − 1 nodes of X.

4.2. The dualizing sheaf of a split curve Y might fail to be very ample: if this is the case, the canonical image W of Y is a double rational normal curve in ℙ^{g−1}. In other words, Y has a degree 2 morphism onto ℙ^{1} and it is thus in the closure, in \overline{M_{g}}, of the locus of smooth, hyperelliptic curves (see [HM] 3.159 and 3.160). A split curve of this type will be called a hyperelliptic split curve. Of course in such a case, the canonical model W is not theta-generic and its theta-hyperplanes are not consistently defined.

We shall leave the projective point of view, for a moment, and consider an abstract split curve Y and the scheme parametrizing its odd stable spin curves S_{Y}. We are interested in the spin curves parametrized by Ŝ_{Y}, whose exponent is at least g − 3 and whose space of global sections has dimension 1 (see 2.3).

Let ξ ∈ Ŝ_{Y} have exponent g − 3; then ξ is supported on a set of g − 3 nodes of Y. Moreover, every set Σ ⊂ Y_{sing} such that #Σ = g − 3 is the support of exactly 4 odd, distinct spin curves (this follows from [Co]). Therefore the number of spin curves of Y having exponent g − 3 is 4\binom{g+1}{g−3}.

There are no spin curves having exponent g − 2. The set of ξ ∈ S_{Y} having exponent g − 1 is easily seen to have cardinality equal to the number of subsets of g − 1 nodes of Y, that is \binom{g+1}{g−1}.

Summarizing, the zero-dimensional scheme Ŝ_{Y} has 4\binom{g+1}{g−3} + \binom{g+1}{g−1} irreducible components, its length is equal to 2^{g−3} · 4\binom{g+1}{g−3} + 2^{g−1}\binom{g+1}{g−1} and L(Ŝ_{Y}) = \{2^{g−3}, 2^{g−1}\}.

Let σ : Y ⟶ W ⊂ ℙ^{g−1} be a canonical model of the split curve Y, thus W is either a split curve or σ is a two-to-one morphism onto a rational normal curve C. Let

{N_{1}, ...., N_{g+1}} := σ(Y_{sing})

so that the points {N_{1}, ...., N_{g+1}} are all distinct and, of course, in general linear position, since they lie on a rational normal curve.

Let Y ⟶ T be a general one-parameter deformation of Y to smooth, theta-generic curves, and let Y ⟶ W ⊂ ℙ^{g−1} × T be a canonical image inducing σ on the central fibers. Then θ^{W}(W) is defined as the limit of the theta-hypersurfaces of the generic fibers (see 2.4).

We shall now describe a configuration of hyperplanes Θ̂(Y, W) ⊂ θ^{W}(W) which corresponds to Ŝ_{Y} and is independent of the choice of Y and W. We start by writing explicitely

Θ̂(Y, W) := {2^{g−3}Θ_{g−3}(Y, W), 2^{g−1}Θ_{g−1}(Y, W)}

where Θ_{g−3}(Y, W) will correspond to spin curves having exponent g − 3 and Θ_{g−1}(Y, W) to spin curves having exponent g − 1.

We have two cases: either σ is an isomorphism, or it is not. In the first case, W is a projective split curve; we shall simplify the notation, writing Θ̂(Y, W) = Θ̂(W), and we have

Θ̂(Y, W) = Θ̂(W) := {2^{g−3}Θ_{g−3}(W), 2^{g−1}Θ_{g−1}(W)}

defined at the beginning of the section. The hyperplanes in Θ_{g−3}(W) correspond naturally to spin curves of exponent g − 3 and similarly, theta-hyperplanes in Θ_{g−1}(W) correspond to spin curves of exponent g − 1.

If K_{Y} is not very ample and the image of σ is a rational normal curve C, we define

Θ_{g−3}(Y, W) := {< N_{i_{1}}, ...., N_{i_{g−3}}, T_{N_{i_{g−2}}} C >, ∀i_{1}, ...., i_{g−2} s.t. i_{j} ≠ i_{h}}

where by T_{N}C we denote the tangent line to C at N. The other piece of Θ̂(Y, W) is defined by

Θ_{g−1}(Y, W) = {< N_{i_{1}} + ... + N_{i_{g−1}} > s.t. 1 ≤ i_{1} < .... < i_{g−1} ≤ g + 1}

Lemma 4.2.1. Let Y be a split curve and let Y ⟶ W be a canonical model. The configuration Θ̂(Y, W) defined above has the following naturality property. For every one-parameter deformation Y ⟶ T of Y to theta-generic curves, and for any choice of a canonical model Y ⟶ W ⊂ ℙ^{g−1} × T restricting to the above Y ⟶ W, we have that

Θ̂(Y, W) ⊂ θ^{W}(W).

Moreover, there is a natural bijection between the irreducible components of Θ̂(Y, W) and the irreducible components of Ŝ_{Y}.

Proof. Consider first a general one-parameter deformation Y of Y. Then, by 2.4.1(a), we have a natural, birational T -morphism μ : S_{Y} ⟶ J_{W}; denote by μ̂ : Ŝ_{Y} ⟶ J_{W} its restriction to Ŝ_{Y}. We are stating that its image μ̂(Ŝ_{Y}) is independent of the choice of the family, and it corresponds to Θ̂(Y, W).

If W is a split curve, this follows from the discussion at the beginning of the section.

Otherwise Y is hyperelliptic, thus, as we mentioned above, Y can be expressed as the specialization of a family of smooth hyperelliptic curves Y_{η}. We shall use this to describe Ŝ_{Y}.

The Weierstrass points W_{1}, ....W_{2g+2} of the generic, smooth fiber Y_{η}, specialize in pairs to the g + 1 nodes of Y. Let us fix the notation so that, for i = 1, ...., g + 1, the pair (W_{i}, W_{i+g+1}) specializes to the node that maps to N_{i}. Recall now that for smooth, hyperelliptic curves, a complete description of the set of theta-characteristics is given in [Mu2], Proposition 6.1. The hyperplanes of Θ_{g−3}(Y, W) correspond to the specialization of odd theta-characteristics of type O_{Y_{η}}(W_{i_{1}} + W_{i_{2}} + ... + W_{i_{g−3}} + W_{i_{g−2}} + W_{i_{g−2}+g+1}), with 1 ≤ i_{1} < .... < i_{g−2} ≤ g + 1.

By 2.3, there is no ambiguity for the choice of the hyperplane corresponding to a given ξ ∈ Ŝ_{Y}.

In a completely similar fashion we deal with spin curves of exponent g−1. This is in some sense easier, in fact such spin curves are supported on sets of g−1 nodes (all of such sets will occur). Moreover, for a given set Σ of g −1 nodes of Y there exists a unique, odd spin curve supported on Σ (by [Co]). As before, viewing Y as specialization of smooth hyperelliptic curves, the hyperplanes of Θ_{g−1}(Y, W) correspond to the specializations of theta-characteristics of type O_{Y_{η}}(W_{i_{1}} + W_{i_{2}} + ... + W_{i_{g−2}} + W_{i_{g−1}}) with 1 ≤ i_{1} < .... < i_{g−1} ≤ g + 1.

So our statement holds for a general Y. Now an arbitrary one-parameter deformation of Y (satisfying the assumptions) can be obtained as a specialization of general ones, for which the result holds. A standard specialization argument (as in 2.4.2) gives the result in general.

4.3. We need to study the behaviour of the above defined configurations of hyperplanes, with respect to the natural action of G = P GL(g) on Sym^{m}(ℙ^{g−1})^{∗}. The spaces of configurations of hyperplanes are fully understood from the Geometric Invariant Theory point of view. In [GIT] a criterion for stability is proved, which we need to recall. For any Ω ∈ Sym^{m}(ℙ^{g−1})^{∗}, define, for every h = 0, ...., g−2, μ_{h}(Ω) as the maximum multiplicity of an h-dimensional linear subspace Λ ⊂ ℙ^{g−1} as a subscheme of Ω (viewed now as a degree m-hypersurface in ℙ^{g−1}). Thus, for example, μ_{0}(Ω) is the maximum order of a singular point of Ω and μ_{g−2}(Ω) is the maximum multiplicity of a component (that is, a hyperplane) of Ω. The criterion for stability is the following ([GIT] Proposition 4.3):

GIT Stability Criterion. Ω is stable if and only if for every h = 0, ...., g − 2 we have μ_{h}(Ω) < Max_{h}(m), where Max_{h}(m) := m\frac{g−1−h}{g}.

We are going to apply it to the next result.

Lemma 4.3.1. Let C ⊂ ℙ^{g−1} be a rational normal curve and let N_{1}, ...., N_{g+1} be distinct points on C. Let Ω ∈ Sym^{m}(ℙ^{g−1})^{∗} be a configuration of hyperplanes of type a), b) or c) below. Then Ω is GIT-stable with respect to the natural action of G on Sym^{m}(ℙ^{g−1})^{∗}.

a ) m = 4\binom{g+1}{g−3} and Ω = Θ_{g−3}(X) where X ⊂ ℙ^{g−1} is a projective split curve.

b ) m = 4\binom{g+1}{g−3} and Ω = \{< N_{i_{1}}, ...., N_{i_{g−3}}, T_{N_{i_{g−2}}} C >, ∀i_{1}, ...., i_{g−2} s.t. i_{j} ≠ i_{h}\} c ) m = \binom{g+1}{g−1} and Ω = \{< N_{i_{1}}, ...., N_{i_{g−1}} >, 1 ≤ i_{1} < .... < i_{g−1} ≤ g + 1\}

Remark. If Ω is as in b), we have, of course, that Ω = Θ_{g−3}(Y, W). Similarly, if Ω is as in c), then Ω = Θ_{g−1}(X) = Θ_{g−1}(Y, W). Proof. To prove the result for cases a) and b), notice that for every h we have

μ_{h}(Θ_{g−3}(X)) ≤ μ_{h}(Θ_{g−3}(Y, W));

therefore, if Θ_{g−3}(Y, W) satisfies the criterion, Θ_{g−3}(X) also does. It is thus enough to deal with case b). Let us compute μ_{h}(Θ_{g−3}(Y, W)). The h-dimensional linear subspaces contained in Θ_{g−3}(Y, W), and having the highest multiplicity are, clearly, those spanned by h + 1 among the generating points N_{i}. Of course, the definition of Θ_{g−3}(Y, W) being symmetric with respect to the N_{i}, varying the (h + 1)-t-uple does not change the multiplicity. Therefore we can pick a specific set, say {N_{1}, ...., N_{h+1}}, of h + 1 nodes and have

μ_{h}(Θ_{g−3}(Y, W)) = #{H ⊂ Θ_{g−3}(Y, W) : {N_{1}, ...., N_{h+1}} ⊂ H}.

Now there are two types of hyperplanes H contributing to μ_{h}(Θ_{g−3}(Y, W)): either H does not contain the tangent line T_{N_{i}}C for any i = 1, ...., h + 1, or it does. The hyperplanes H of the first type are

\binom{g + 1 − (h + 1)}{g − 3 − (h + 1)}4 = \binom{g − h}{4}4

where the binomial coefficient is the number of (g − 3)-uples containing the fixed set {N_{1}, ...., N_{h+1}}, and the coefficient 4 is there because for every chosen g − 3 as above, we have a choice of 4 points where the hyperplane is tangent to C.

The hyperplanes of the second type contain the tangent line to C at one of the N_{i}, for i ≤ h + 1. The total number of them is

\binom{g + 1 − (h + 1)}{g − 3 − h}(h + 1) = \binom{g − h}{3}(h + 1)

where the binomial coefficient is the number of g − 3-uples containing a fixed subset of h elements in the set {N_{1}, ...., N_{h+1}}; the factor (h + 1) represents the choice of the N_{i} such that H contains the tangent line to C at N_{i}. In conclusion

μ_{h}(Θ_{g−3}(Y, W)) = \binom{g − h}{4}4 + \binom{g − h}{3}(h + 1) = \frac{(g − h)(g − h − 1)(g − h − 2)(g − 2)}{3!}

On the other hand, the strict upper bound allowed for stability by the criterion of GIT is

Max_{h}(m) = m\frac{g − 1 − h}{g} = 4\binom{g + 1}{g − 3}\frac{g − 1 − h}{g} = \frac{(g + 1)(g − 1)(g − h − 1)(g − 2)}{3!}.

Since 0 ≤ h, it is clear that for every h

μ_{h}(Θ_{g−3}(Y, W)) < Max_{h}(m)

and therefore Θ_{g−3}(Y, W) and Θ_{g−3}(X) are GIT-stable.

Now we treat case c), which is simpler. We have for every h

μ_{h}(Ω) = \binom{g + 1 − h − 1}{g − 1 − h + 1} = \binom{g − h}{2} = \frac{(g − h)(g − h − 1)}{2}.

On the other hand, the (strict) upper bound given by the criterion above is, for every h

Max_{h}(m) = m\frac{g − 1 − h}{g} = \binom{g + 1}{2}\frac{g − 1 − h}{g} = \frac{(g + 1)(g − 1 − h)}{2}

it is thus evident that, for every h ≥ 0, μ_{h}(Ω) < Max_{h}(m), and hence Ω is GIT-stable.

Remark. The interested reader can generalize the above computation to see that such stability results are special cases of a more general phenomenon, about the stability of configurations of hyperplanes spanned by sets of points and tangent lines of a rational normal curve.

Corollary 4.3.2. Let X be a projective split curve; let Y be an abstract split curve and Y ⟶ W a canonical model. Consider the configurations Θ̂(X) and Θ̂(Y, W) with respect to the natural action of G.

(a) They are GIT-stable. (b) If they are in the same G-orbit, then W is a projective split curve.

Proof. The proof of (a) is a straightforward application of the criterion already used for the proof of the previous result. Let m_{1} = 4\binom{g+1}{g−3}, m_{2} = \binom{g+1}{g−1} and m = 2^{g−3}m_{1} + 2^{g−1}m_{2}. By definition,

Max_{h}(m) = 2^{g−3}Max_{h}(m_{1}) + 2^{g−1}Max_{h}(m_{2})

where 0 ≤ h ≤ g − 2. Let now Ω ∈ Sym^{m}(ℙ^{g−1})^{∗} be one of the two configurations in the statement. Then we can write Ω = {2^{g−3}Ω_{1}, 2^{g−1}Ω_{2}} where Ω_{1} is either Θ_{g−3}(X) or Θ_{g−3}(Y, W), and hence a configuration of m_{1} hyperplanes of type a) or b) in the previous Lemma 4.3.1 a); similarly, Ω_{2} is a configuration of m_{2} hyperplanes of type c) in 4.3.1. We have

μ_{h}(Ω) = 2^{g−3}μ_{h}(Ω_{1}) + 2^{g−1}μ_{h}(Ω_{2}) < 2^{g−3}Max_{h}(m_{1}) + 2^{g−1}Max_{h}(m + 2) = Max_{h}(m)

(where the inequality comes from 4.3.2), hence we are done.

For (b) we must prove that the G-orbits of Θ̂(X) and Θ̂(Y, W) are different, if W is not a split curve. This follows from the fact that the two, regarded as hypersurfaces, have different singularities. More precisely, by looking at the points N_{i}, one sees that

μ_{0}(Θ_{g−3}(Y, W)) > μ_{0}(Θ_{g−3}(X))

(recall in fact that if H ∈ Θ_{g−3}(X), then H is tangent to X at two smooth points). Since, as already noticed, μ_{h}(Θ_{g−1}(Y, W)) = μ_{h}(Θ_{g−1}(X)) for every h ≥ 0, we conclude that

μ_{0}(Θ̂(Y, W)) > μ_{0}(Θ̂(X))

and hence the two configurations cannot possibly be projectively equivalent.

4.4. We conclude with a strengthening of Theorem 5 of [C] which will be used later. The improvement consists essentially in the fact that to recover the curves it is enough to consider theta-hyperplanes of type g − 3 and g − 1, rather than all of them. The proof is very similar.

Proposition 4.4.5. Let X and X′ be two split curves in ℙ^{g−1}. If Θ̂(X) = Θ̂(X′), then X = X′.

Proof. The argument of Theorem 5 in [C] is divided into two parts; the first part (of combinatorial type) shows that, if the two curves have the same theta-hyperplanes of type g −1, then they have the same singular points. We can use that part, since, by definition, Θ̂(X) = Θ̂(X′) implies that X and X′ have the same theta-hyperplanes of type g − 1 (which are, of course, those hyperplanes having multiplicity 2^{g−1}).

Thus, denote X_{sing} = X′_{sing} = {N_{1}, ...., N_{g+1}}, X = C_{1} ∪ C_{2} and X′ = C′_{1} ∪ C′_{2}, with C_{i} and C′_{i} rational normal curves.

Now let, for j = 1, 2

Λ_{j} =< N_{j}, N_{3}, N_{4}, ...., N_{g−3}, N_{g−2} >

thus Λ_{1} and Λ_{2} are two linear subspaces of dimension g−4 intersecting in < N_{3}, ...., N_{g−2} >. Let, for j = 1, 2, π_{j} : ℙ^{g−1} − − > ℙ^{2} be the projection from Λ_{j} onto ℙ^{2} and let X_{j} = π_{j}(X) and X′_{j} = π_{j}(X′). X_{j} and X′_{j} are two plane quartics with the same singularities, moreover X_{j} is split, because X is split, notice in fact that X_{j} (respectively, X′_{j}) is the normalization of X (respectively, of X′) at the nodes N_{j}, N_{3}, ...., N_{g−2}. Since X_{sing} = X′_{sing} we have that X_{j} and X′_{j} have the same singularities. There is a natural bijection between the theta-lines of type 0 of X_{j} (respectively, of X′_{j}) and the theta-hyperplanes of type g − 3 of X (respectively, of X′) that contain N_{j}, N_{3}, ...., N_{g−2}. By assumption, X and X′ have the same theta-hyperplanes of type g − 3, therefore X_{j} and X′_{j} have the same theta-lines of type 0. We conclude (see [CS]) that X_{j} = X′_{j} for j = 1, 2.

We have thus proved that projecting X and X′ to ℙ^{2} from the same g − 4-dimensional linear subspace spanned by g − 3 of their (common) nodes, one obtains the same split curve in ℙ^{2}. It is clear that, up to re-naming the nodes, we can chose Λ_{1} and Λ_{2} so that for j = 1, 2 we have that π_{j}(C_{1}) = π_{j}(C′_{1}) and, of course, π_{j}(C_{2}) = π_{j}(C′_{2}).

Let S_{j} be the cone over C_{1} with vertex Λ_{j}, that is

S_{j} := ⋃_{P ∈C_{1}} < Λ_{j}, P >,

then the fact that π_{j}(C_{1}) = π_{j}(C′_{1}) implies that C′_{1} ⊂ S_{j} for both j = 1, 2. We shall now show that this implies that C_{1} = C′_{1}. It suffices to show that that

S_{1} ∩ S_{2} ⊂ C_{1}∪ < Λ_{1}, Λ_{2} > .

By contradiction; suppose that there is a point Q such that Q ∈ S_{1} ∩ S_{2} but Q ∉ C_{1}∪ < Λ_{1}, Λ_{2} >. Then the linear space < Q, Λ_{j} > must intersect C_{1} in a point P_{j}; we thus find two new points P_{1} and P_{2} lying in the intersection of C_{1} with the hyperplane H :=< Q, Λ_{1}, Λ_{2} >. Then

{N_{1}, N_{2}, ...., N_{g−2}, P_{1}, P_{2}} ⊂ C_{1} ∩ H

that is, deg C_{1} ∩ H ≥ g, which is not possible. Therefore C_{1} = C′_{1}.

Repeating the argument for C_{2} and C′_{2} we conclude that X = X′.

5. The local picture near split curves

The purpose of this section is to analyze the differential of θ : V → Sym^{N_{g}}(ℙ^{g−1})^{∗} at a point parametrizing a split curve, and to prove (in 5.2.2) that it is injective. 5.1. Vector bundles on ℙ^{1}: elementary transformations. We recall a few well known facts concerning vector bundles on rational curves. Denote by C = ℙ^{1} and by λ the line bundle of degree 1 on C. Let E be a vector bundle on C. An elementary transformation of E is a vector bundle E′ such that there is an exact sequence

(1).  0 → E′ → E → ℂ_{x} → 0

where ℂ_{x} is a torsion sheaf supported on a point x ∈ C with fiber ℂ. We have

rk(E′) = rk(E); deg(E′) = deg(E) − 1

Let us denote by ℙ(E) the projective bundle over C associated to E. The exact sequence (1) corresponds to a point of ℙ(E), contained in the fiber ℙ(E)_{x}, called the center of the elementary transformation E′.

It is well known that the projective bundle ℙ(E′) is obtained from ℙ(E) by blowing up the center and then blowing down the proper transform F̃ of the fiber containing it (see [Ma]); ℙ(E) is obtained from ℙ(E′) by the analogous “inverse” process with center the image of F̃. For example, if E has rank 2, we write ℙ(E) ≅ 𝔽_{n} and ℙ(E′) ≅ 𝔽_{n′}, with the usual notation 𝔽_{n} := ℙ(λ^{0} ⊕ λ^{n}). Of course, in our situation, |n − n′| = 1. More precisely, we have that n = n′ − 1 if and only if the center p ∈ ℙ(E) belongs to the (−n)- curve of ℙ(E), if and only if the center q ∈ ℙ(E′) of the inverse process does not belong to the (−n−1)-curve of ℙ(E′).

An exact sequence of vector bundles

0 → F → E → L → 0

with rkL = 1, defines a section σ of ℙ(E); σ contains the center of E′ if and only if F ⊂ E′. In this case we have an exact sequence:

0 → F → E′ → Lλ^{−1} → 0

In particular, if E = F ⊕ L then E′ = F ⊕ Lλ^{−1}.

Given an exact sequence

0 → E_{n} → E → F → 0

where F = ℂ_{x_{1}} ⊕ ⋯ ⊕ ℂ_{x_{n}} is a torsion sheaf supported on n points of C, we say that E_{n} is obtained from E by a sequence of n elementary transformations, centered at n points of ℙ(E).

Conversely, a set Γ of n points of ℙ(E), no two on the same fiber, defines a vector bundle E_{Γ} endowed with an exact sequence

0 → E_{Γ} → E → F → 0

where F is as above. Of course, E_{Γ} is obtained applying the elementary transformations corresponding to the points of Γ.

We need to consider the following special case. Let r = rkE and let Γ′ be a set of r points, all contained in a fiber of ℙ(E), and in general position in such a fiber. Then one can consistently define E_{Γ′} = E ⊗ λ^{−1}, so that ℙ(E) ≅ ℙ(E_{Γ′}). Similarly, if n = rq and Γ_{0} is a union of disjoint r-uples of points, as the Γ′ above, then E_{Γ_{0}} = E ⊗ λ^{−q}.

We shall need the following very simple Lemma.

Lemma 5.1.1. Let E ⟶ T be a family of vector bundles over ℙ^{1}. Let E_{t} = ⊕λ^{n_{t}^{i}} be the generic fiber and let E_{0} = ⊕λ^{n_{0}^{i}} be the special fiber. Then

a ) max_{i}{n_{t}^{i}} ≤ max_{i}{n_{0}^{i}} and min_{i}{n_{t}^{i}} ≥ min_{i}{n_{0}^{i}}.

b )If E_{0} is balanced (i.e. n_{0}^{i} = n_{0}^{j} for all i, j) then E_{t} is balanced and E_{t} ≅ E_{0}.

Proof. Let M_{t} := max_{i}{n_{t}^{i}} and M_{0} := max_{i}{n_{0}^{i}}. By contradiction; if M_{t} > M_{0}, we can consider the family E′ ⟶ T of vector bundles, obtained by tensoring the given family E by λ^{−M_{t}}. Then E′_{0} has no global sections, being a sum of line bundles of negative degree. On the other hand E′_{t} has one summand equal to λ^{0}, thus it has non-zero sections. This is a contradiction. The statement about the minimum is obtained in the same way, working on the sequence of dual vector bundles. Part b) follows immediately fom a).

5.2. Normal bundles of split curves. Let now g ≥ 3 and let C ⊂ ℙ^{g−1} be a rational normal curve. Let us consider the normal bundle of C:

N_{C} := N_{C/ℙ^{g−1}} ≅ ⨁^{g−2} λ^{g+1}

We have degN_{C} = (g − 2)(g + 1), h^{0}(N_{C}) = (g − 2)(g + 2) and ℙ(N_{C}) ≅ C × ℙ^{g−3}. This isomorphism can be interpreted in terms of the identifications:

ℙ(N_{C}) ≅ ℙ(N_{C}(−1)) = {(x, H) : x ∈ C, T_{x}C ⊂ H} ⊂ C × (ℙ^{g−1})^{∗}

(T_{x}C is the tangent line to C at x) coming from the surjections:

⊕^{g}O_{C} → T_{ℙ|C}(−1) → N_{C}(−1)

More precisely, we have: ℙ(N_{C}) ≅ C × |λ^{g−3}|. Thus a fiber ℙ(N_{C})_{x}, x ∈ C, can be identified with the complete linear system |λ^{g−3}| cut out by all hyperplanes containing T_{x}C.

Let X = C_{1} ∪ C_{2} ⊂ ℙ^{g−1} be a split curve of genus g and let C_{1} ∩ C_{2} = {N_{1}, …, N_{g+1}} = X_{sing}. Let Ω = Θ_{g−3}(X) be the set of all theta-hyperplanes of X passing through exactly g − 3 nodes of X (described in the previous section). We write Ω = Ω_{1} ∪ … ∪ Ω\binom{g+1}{g−3}, where Ω_{i} is the set of four theta-hyperplanes containing a fixed subset Σ_{i} of g − 3 nodes. We denote by H_{i,j}, j = 1, …, 4, the four hyperplanes of Ω_{i}.

Consider the everywhere nonreduced zero-dimensional scheme

Z := [ ⋃_{H∈Ω} H] ∩ X_{reg} ⊂ X

and let, for k = 1, 2, Z_{k} = Z ∩ C_{k}. Then Z_{k} ⊂ C_{k} is supported at the 4\binom{g+1}{g−3} points below

\{p_{i,1}^{k}, …, p_{i,4}^{k}, i = 1, …, 4\binom{g + 1}{g − 3}\}

For a general split curve, such points are distinct and p_{i,j}^{k} ≠ N_{h} for all choices of indeces. We have natural surjections N_{X} → T_{Z}^{1} and N_{C_{k}} → T_{Z_{k}}^{1}, where T_{−}^{1} denotes the first cotangent sheaf of −, and we define

N′_{X} := ker{N_{X} → T_{Z}^{1}}
N′_{C_{k}} = ker{N_{C_{k}} → T_{Z_{k}}^{1}}

We have an exact and commutative diagram for k = 1, 2:

(3)  \begin{aligned}& & 0 & & 0 \\ & & ↓ & & ↓ \\ 0 & → & N′_{C_{k}} & → & N_{C_{k}} & → & T_{Z_{k}}^{1} & → & 0 \\ & & ↓ & & ↓ & & ∥ \\ 0 & → & (N′_{X})_{|C} & → & (N_{X})_{|C_{k}} & → & T_{Z_{k}}^{1} & → & 0 \\ & & ↓ & & ↓ \\ & & T & = & T \\ & & ↓ & & ↓ \\ & & 0 & & 0 \end{aligned}

where T = ℂ_{N_{1}} ⊕ ⋯ ⊕ ℂ_{N_{g+1}} is a torsion sheaf.

The proof of Theorem 5.2.2 is based on the following key result

Propositon 5.2.1. With the above notation, H^{0}(X, N′_{X}) = 0 for all g ≥ 4.

Proof. We have an exact sequence:

(2)  0 → N′_{X} → (N′_{X})_{|C_{1}} ⊕ (N′_{X})_{|C_{2}} → Q → 0

where Q = ℂ_{N_{1}}^{g−2} ⊕ ⋯ ⊕ ℂ_{N_{g+1}}^{g−2}, it therefore suffices to prove that, for k = 1, 2,

H^{0}((N′_{X})_{|C_{k}}) = 0.

To prove this, we will consider the above diagram (3). From its first row, we see that N′_{C_{k}} is obtained from N_{C_{k}} applying a sequence of 4\binom{g+1}{g−3} elementary transformations, centered at the points of the set

Γ := \{(p_{i,j}^{k}, Σ_{i}), i = 1, ...., \binom{g + 1}{g − 3}, j = 1, ...., 4\}

Let us fix now k = 1 and drop the index k for simplicity, denoting C_{1} = C. Let E := N_{C} = ⊕^{g−2}λ^{g+1}. Let n = 4\binom{g+1}{g−3}.

We can specialize our curve X = C ∪ C_{2} to a hyperelliptic split curve (see 4.2), whose canonical model X_{0} is supported on C. We can do that by keeping the component C fixed (i.e. C_{2} specializes to C) and by mantaining the nodes of every fiber at the points N_{1}, ...., N_{g+1}. Call X_{t} the generic fiber and X_{0} the special fiber. Let Γ_{t} be the n-uple of points in ℙ(E) defined as the centers of the n elementary transformation of the first row of diagram (3), which we can now re-write for X_{t}

0 ⟶ E_{Γ_{t}} ⟶ E ⟶ T_{t}^{1} ⟶ 0

As X_{t} specializes to X_{0}, the configuration of theta-hyperplanes Θ_{g−3}(X) specializes to a configuration that has been defined and studied in the previous section. Namely, call Y_{0} the abstract, hyperelliptic split curve whose canonical model is X_{0}. Then the limit configuration is Θ_{g−3}(Y_{0}, X_{0}) defined in 4.2, where W = X_{0}. Therefore the set Γ_{t} specializes to a set Γ_{0} ⊂ ℙ(E) which has implicitely been described in 4.2:

Γ_{0} = {(N_{j}, Σ_{i}) : #Σ_{i} = g − 3, N_{j} ∉ Σ_{i}}

In particular, Γ_{0} is entirely contained in the g + 1 fibers over N_{1}, ...., N_{g+1}; each such a fiber contains exactly \binom{g}{g−3} points of Γ_{0}, and such points are in general position in every fiber (because C is a rational normal curve).

At this point, we need to distinguish three cases; first: g ≢ 2 (mod 3) and g ≠ 4; second: g = 3x + 2 and third: g = 4,

• Suppose that g ≢ 2 (mod 3).

Then we can factor n = (g − 2)(g + 1)\frac{g(g−1)}{2}, so that Γ_{0} is a union of disjoint g − 2-uples contained in a fiber of ℙ(E) ⟶ C. We are thus in the situation described in 5.1; we obtain that

E_{Γ_{0}} = E ⊗ λ^{−(g+1) \frac{g(g−1)}{2}} = ⨁^{g−2} λ^{a}

where a = g + 1 − (g + 1)\frac{g(g−1)}{2}. Thus E_{t} specializes to the “balanced” vector bundle E_{0} = ⊕λ^{a}. By 5.1.1, E_{t} is also balanced and isomorphic to E_{0}.

We have therefore proved that N′_{C} = ⊕^{g−2}λ^{a}.

A straightforward computation shows that, if g ≥ 5, then a < −g − 1

The exact sequence in the first column of (3) expresses N′_{C} as obtained from (N′_{X})_{|C} by a sequence of g + 1 elementary transformations. In particular,

deg(N′_{X})_{|C} = deg N′_{C} + g + 1.

If g ≥ 5, we immediately deduce that each line bundle summand in the splitting of (N′_{X})_{|C} has negative degree, therefore H^{0}((N′_{X})_{|C}) = 0 and we are done.

• Let g = 3x + 2. Here the problem is that for every h = 1, ...., g + 1 the number f of points of Γ_{0} contained in the fiber over N_{h} is not a multiple of g − 2. More precisely we have

f := \binom{g}{g − 3} = x + (g − 2)\frac{3x(x + 1)}{2}

Pick on each of these g + 1 fibers a subset of f − x points of Γ_{0}, and call Γ′_{0} the union of these g + 1 subsets. Thus Γ′_{0} ⊂ Γ_{0},

#Γ′_{0} = (g + 1)(g − 2)\frac{3x(x + 1)}{2}

and, as before,

E_{Γ′_{0}} = E ⊗ λ^{−(g+1) \frac{3x(x+1)}{2}} = ⨁^{g−2} λ^{a′}

where a′ = g + 1 − (g + 1)\frac{3x(x+1)}{2}. As in the previous case, we conclude that N′_{C} ⊂ ⊕^{g−2}λ^{a′}.

One easily sees that, if x ≥ 1, then a′ < −g − 1. So we conclude by looking at the first column of diagram (3), exactly as we did before.

• If g = 4 the argument of the first case yields N′_{C} = λ^{−5} ⊕ λ^{−5}, which is not enough; we use a slightly different strategy. We can write (N′_{X})_{|C} = λ^{u} ⊕ λ^{v} with u ≤ v and u + v = −5. To show that (N′_{X})_{|C} has no sections it suffices to show that v − u < 5.

Consider the second column of diagram (3)

0 → λ^{5} ⊕ λ^{5} → (N_{X})_{|C} = λ^{c} ⊕ λ^{d} → T → 0

with c + d = 15 and c ≤ d, For i = 1, …, 5 let H_{i} be the plane spanned by the tangent lines at N_{i} to C_{1} = C and C_{2}:

H_{i} :=< T_{N_{i}}C_{1}, T_{N_{i}}C_{2} > .

It is a local computation to check that T_{N_{i}} is defined by the plane H_{i}.

Therefore the 5 centers of the elementary transformation above are the points (q_{i}, N_{i}) where q_{i} ∈ C ∩ H_{i} − {N_{i}} is the residual point of intersection of H_{i} with C_{i}. It is easy to see that p_{i} ≠ p_{j}. We deduce that such centers are not all contained in the same section. In particular, ℙ((N_{X})_{|C}) ≠ 𝔽_{5}, hence d − c < 5.

Now we look at the second row of diagram (3). By the same argument as in the previous part of the proof, using 5.1.1a), we can conclude that v − u ≤ d − c < 5; hence we are done.

We can now prove the

Theorem 5.2.2. Let X ⊂ ℙ^{g−1} be a general, projective split curve. Then θ is an immersion at (the point parametrizing) X.

Proof. Let X = C_{1} ∪ C_{2} ⊂ ℙ^{g−1} be a general split curve. To prove that θ is an immersion at X amounts to showing that T_{X}θ^{−1}(θ(X)) = (0). Note that θ^{−1}(θ(X)) is a closed subfamily of V consisting of canonical curves which have θ(X) as configuration of theta-hyperplanes. Consider the subsheaf N′_{X} ⊂ N_{X} studied above. Its sections define a subspace of H^{0}(X, N_{X}) = T_{X}V consisting of first order deformations of X which remain tangent to the hyperplanes of Ω at the points of Z. Therefore

T_{X}θ^{−1}(θ(X)) ⊂ H^{0}(X, N′_{X})

But H^{0}(X, N′_{X}) = 0 by 5.2.1, thus our statement is proved.

6. Characterizing canonical curves by their theta-hyperplanes

For the reader’s convenience, we restate our main Theorem, before proving it.

Theorem 6.1.1. Let X and X′ be general canonical curves in ℙ^{g−1} having genus g ≥ 4. If θ(X) = θ(X′) then X = X′.

Proof. By contradiction: suppose the statement false. Then we can find two families of canonical curves as follows. The first family

X ↪ T × ℙ^{g−1}
↓
T

is a general one-parameter deformation of a general split curve X, with X_{t} smooth for t ≠ t_{0} (notation as usual, see 1.1). For every t ∈ T, we assume that ψ_{X} (t) ∈ V.

The second family

X′ ↪ T × ℙ^{g−1}
↓
T

is such that the generic fiber X′_{t} is a smooth, theta-generic canonical curve, such that θ(X_{t}) = θ(X′_{t}) for every t ≠ t_{0}.

We cannot, for the moment, say much about the special fiber X′ of the second family, we shall use the existence of stable reduction to analyze it. Modulo replacing T by a finite covering, we can assume that X′ ⟶ T admits stable reduction over T; let it denote by Y ⟶ T. We have that, for t ≠ t_{0}, X′_{t} is a canonical (isomorphic) model of Y_{t}. The central fiber Y of Y ⟶ T is a stable curve, and we have a birational map Y − − > X′ which is an isomorphism away from the central fibers.

• We claim that Y is a split curve.

To prove that, consider first J_{X} ⟶ T defined at the end of 2.1, and the family of odd spin curves S_{X} ⟶ T defined in 2.2 (abusing notation denoting by the same symbol X the abstract family and its canonical model). We can compare the two: by our genericity assumption, S_{X} is a smooth curve (by 2.2.1) dominating J_{X} (by 2.4.1(a)). Notice also that, by construction, J_{X} = J_{X′}; in conclusion, we have a birational morphism of T -schemes

S_{X} ⟶ J_{X} = J_{X′}

and by 2.4.1(b),

L(S_{X}) = L(J_{X})  (1)

Consider now the pull-back to T of the space of odd spin curves of the family Y ⟶ T, as usual denoted by S_{Y}. Away from their special fibers, S_{Y} and J_{X′} are isomorphic over T; by what we have observed above, we have a birational T -morphism (recall that S_{X} is a smooth curve)

α : S_{X} ⟶ S_{Y}.

Looking at the central fibers we get that

L(S_{X}) ≥ L(S_{Y}),

equivalently, by (1)

L(J_{X}) ≥ L(S_{Y})  (2)

Now we apply 2.4.2, with W = X′. We get that

L(S_{Y}) ≥ L(J_{X′}^{X′})

but J_{X′} = J_{X} and hence, of course, J_{X} = J_{X′}^{X′} so that we get

L(S_{Y}) ≥ L(J_{X})  (3)

Combining (2) and (3) (see 1.2) and using (1), we conclude that

L(S_{X}) = L(S_{Y}).

Finally, X being a split curve, we use 3.4.2(b) to conclude that Y is a split curve. The claim is thus proved. Let now Y ⟶ W be a canonical map. Let

Y ⟶ W ↪ T × ℙ^{g−1}

be a canonical image of Y ⟶ T, whose restriction to the central fibers is the above Y ⟶ W. If t ≠ t_{0}, W_{t} is an isomorphic model of Y_{t} and it is thus projectively equivalent to X′_{t}. That is, there exists a morphism γ : T − {t_{0}} ⟶ G such that

X′_{t} = W_{t}^{γ(t)}.

Hence

θ(X′_{t}) = θ(X_{t}) = θ(W_{t})^{γ(t)}.

By 4.2.1, we can consider the distinguished subconfiguration of theta-hyperplanes Θ̂(X) ⊂ θ(X); by definition, Θ̂(X) is the subconfiguration (with multiplicities) of all components of θ(X) having multiplicity at least 2^{g−3}. Therefore it is the specialization of a well defined subconfiguration of hyperplanes Θ̂_{t} ⊂ θ(X_{t}). Similarly, consider the distinguished configuration Θ̂(Y, W) ⊂ θ^{W}(W). By construction, we have that Θ̂(Y, W) is the specialization of the family of configurations (Θ̂_{t})^{γ(t)}. Recall that, by 4.3.2, Θ̂(X) and Θ̂(Y, W) are GIT-stable points in Sym^{m}(ℙ^{g−1})^{∗}. We have just seen that they are specializations of two G-conjugate families. Therefore (since they are GIT-stable) they are themselves G-conjugate, that is, there exists an element g_{0} ∈ G such that

Θ̂(X) = (Θ̂(Y, W))^{g_{0}}.

Now this implies that W is a split curve, by 4.3.2, (b). Hence Θ̂(Y, W) = Θ̂(W) and

Θ̂(X) = Θ̂(W)^{g_{0}} = Θ̂(W^{g_{0}}).

Summarizing, X and W^{g_{0}} are split curves such that Θ̂(X) = Θ̂(W^{g_{0}}). We are thus in the position of applying 4.4.5, to conclude that W^{g_{0}} = X. Now W^{g_{0}} is the central fiber of the family over T whose generic fiber is W_{t}^{γ(t)}. By Theorem 5.2.2, we get that W_{t}^{γ(t)} = X_{t}, and we are done, since by construction W_{t}^{γ(t)} = X′_{t}.

References

[ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris: Geometry of Algebraic Curves 1 Grundlehren der Mathematischen Wissenschaften, 267. Springer, New York-Berlin, 1985

[CS] L. Caporaso, E. Sernesi: Recovering plane curves from their bitangents. Preprint Alg-Geom AG/0008239 To appear in Journ. of Alg. Geom.

[C] L. Caporaso: On modular properties of odd theta characteristics Advances in mathematics motivated by physics, pp.101-114, Contemporary Mathematics series AMS

[Co] M. Cornalba: Moduli of curves and theta-characteristics. Lectures on Riemann surfaces (Trieste, 1987), 560–589, World Sci. Publishing, Teaneck, NJ, 1989

[DM] P.Deligne, D.Mumford: The irreducibility of the space of curves of given genus. Publ Math. IHES 36 (1969) 75-120.

[HM] J. Harris, I. Morrison: Moduli of curves. Graduate texts in Math. 187, Springer (1998)

[L] D. Lehavi: Any smooth plane quartic can be reconstructed from its bitangents. Preprint AG/0111017 [Ma] M. Maruyama : Elementary transformations in the theory of algebraic vector bundles. Algebraic geometry (La Rabida, 1981), 241-266, Lecture Notes in Math. 961, Springer 1982.

[Mu1] D. Mumford: Curves and thir Jacobians. The University of Michigan Press, Ann Arbor, 1975.

[Mu2] D. Mumford: Tata lectures on theta II. Progress in mathematics 43, Birkhäuser 1984.

[Mu3] D. Mumford: Tata lectures on theta III. Progress in mathematics 97, Birkhäuser 1991.

[GIT] D. Mumford, J. Fogarty, F. Kirwan: Geometric Invariant Theory. E.M.G. 34 Springer 1994.

[S] R. Salvati Manni: On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure. Math. Zeitshrift 221 (1996), 231-241.

Lucia Caporaso Dipartimento di Matematica, Università di Roma Tre Largo S. L. Murialdo 1 I-00146 Roma - ITALY caporaso@mat.uniroma3.it

Edoardo Sernesi Dipartimento di Matematica, Università di Roma Tre Largo S. L. Murialdo 1 I-00146 Roma - ITALY sernesi@mat.uniroma3.it