ON MODULAR PROPERTIES OF ODD THETA-CHARACTERISTICS

Lucia Caporaso

Abstract. A general canonical curve X ⊂ ℙ^{g−1} determines a finite set θ(X) of hyperplanes that are tangent to X at g − 1 points. Such a set is in bijective correspondence with the set of odd theta-characteristics of X. We generalize the definition of θ(X) to certain singular curves (including Deligne-Mumford stable curves) in a way that is compatible with degenerations. We give the explicit description of θ(X), with its enumerative and local data. This is applied to show that some singular curves can be recovered from θ(X). This paper generalizes the preliminary steps used in [CS] to show that, if g = 3, θ(X) uniquely determines X.

1. Introduction. A theta-characteristic on a smooth, projective, complex curve X of genus g is a square root of the canonical line bundle K_{X}, that is, a divisor class D ∈ PicX (of degree g−1) such that 2D = K_{X}. Such a D is called “odd” (respectively “even”) if h^{0}(X, D) is odd (respectively even).

Assume X to be non-hyperelliptic, fix a canonical model X^{c} for X and let H be a hyperplane that is tangent to X^{c} at g − 1 (not necessarily distinct) points, that is, H ∩ X^{c} = ∑ 2p_{i} where p_{i} ∈ X^{c}; clearly the divisor H ∩ X^{c} is an effective theta-characterstic. Conversely, if D is an effective theta-characteristic, then there exists a hyperplane H ⊂ ℙ^{g−1} such that D = \frac{1}{2}H ∩ X^{c}, in particular, H is tangent to X^{c} at g −1 points. We shall call such hyperplanes the theta-hyperplanes of X^{c}.

It is well known that a smooth curve has exactly 2^{g−1}(2^{g} − 1) odd theta-characteristics. A simple consequence of the Petris’s conjecture (proved by Gieseker, see [ACGH] V.1.7) is that, if the curve is general in moduli, then its odd theta-characteristics coincide with the effective ones. In other words a general smooth curve X of genus g has exactly

N_{g} := 2^{g−1}(2^{g} − 1)

effective divisors D such that

2D = K_{X} and h^{0}(X, D) = 1.

Hence the canonical model of a general X has N_{g} distinct theta-hyperplanes.

Another intrinsic interpretation can be given using the Jacobian variety J(X) polarized by a theta divisor Θ_{X}: effective theta-characteristics are in natural one-to-one correspondence with the 2-torsion points of J(X) contained in Θ_{X}. If X is general, then the Riemann Singularity Theorem implies that Θ_{X} is smooth at such N_{g} points. As a consequence the images of the odd theta-characteristics under the Gauss map of Θ_{X} are, once again, put in bijective correspondence with the theta-hyperplanes of a canonical model of X.

Summarizing, the theta-characteristics of a curve X can be viewed abstractly as the 2-torsion points of J(X) lying on Θ_{X}. Or projectively, as the hyperplanes that are tangent to a canonical model of X in g − 1 points. This second point of view allows one to compare the odd thetacharacteristics of different curves. This paper is written in such a perspective.

Let X and Y be two abstract curves as above. If we are given for them canonical models X^{c} and Y^{c} in the same ℙ^{g−1}, we can ask: if X^{c} and Y^{c} have the same N_{g} theta-hyperplanes does it follow that X = Y?

We shall denote by θ(Z) the union of all theta-hyperplanes of a smooth canonical curve Z. Another version of the above question is: given two distinct canonical curves Z and Z′, is it

always true that θ(Z) ≠ θ(Z′)?

In [CS] the question above is answered affirmatively for curves of genus 3: it is shown that a general plane quartic can be recovered by its 28 bitangent lines. We conjectured that the same holds for any genus.

The method in [CS] is a degeneration technique, which can be summarized in three steps as follows.

The first step is to describe how the set of 28 bitangent lines degenerates when smooth curves degenerate to (certain) singular ones. This requires constructing a geometrically meaningful analogue for θ(X) which is compatible with deformations to smooth curves, that is to say, that does not depend on the choice of the family.

The second step is to show that some singular curves can in fact be recovered from their generalized set of bitangent lines (constructed in the previous step).

Finally one needs an argument to reduce the proof of the main theorem for smooth curves to the second step. This is done in [CS] using Geometric Invariant Theory and deformation theory for plane curves.

In this paper we generalize the first two stages of the process to curves of any genus.

For the first step, we concentrate on those singular curves that have been so far proved to be the “best”degenerations of smooth ones: stable curves in the sense of Deligne-Mumford. Denote by \overline{M_{g}} the moduli space of stable curves of genus g.

Maurizio Cornalba exhibited in [C] a geometrically meaningful compactification of the moduli space of pairs

(smooth curve X, theta-characteristic on X).

His space (called \overline{S_{g}}) is finite over \overline{M_{g}} and has good functorial properties. He furthermore shows that \overline{S_{g}} has two disjoint irreducible components, \overline{S_{g}}^{+} and \overline{S_{g}}^{−}, corresponding to the compactification of the loci of (respectively) even and odd theta-characteristics of smooth curves. We shall here give a closer look at \overline{S_{g}}^{−} (whose degree over \overline{M_{g}} is N_{g}) describing the enumerative and local data for its natural strata over the loci of \overline{M_{g}} parametrizing curves with a given number of nodes.

The second step will be carried out combining the previous analysis with the results of [CS]. We prove (in Theorem 5) that a general, nodal canonical curve X ⊂ ℙ^{g−1} which is the union of two rational normal curves meeting at g + 1 points is uniquely determined by its generalized θ(X) (X is a so called “split” curve, see below).

Finally we show that singular, irreducible, general curves of (arithmetic) genus 4 can be recovered by their odd theta-characteristics in the following sense: if X and Y are singular, irreducible curves in \overline{M_{4}} such that there exist suitable canonical models X^{c} and Y^{c} having the same thetahyperplanes, then X = Y (this is Theorem 6 ).

2. Enumerative results. Fix g ≥ 3, let r = g − 1 and d = 2g − 2. Denote by H_{d,g}^{r} the Hilbert scheme of curves in ℙ^{r} having Hilbert polynomial p(x) = dx−g +1. Throughout X will be a curve in ℙ^{r} satisfying the following set of assumptions:

X is reduced of degree d and arithmetic genus g, X has at most nodes as singularities, no irreducible component of X is contained in a hyperplane, X is embedded in ℙ^{r} by the (complete) linear series |ω_{X}| (where ω_{X} denotes the dualizing line bundle of X).

A curve satisfying the above requirements will be called throughout a canonical curve. We shall denote by V′ the open subset of H_{d,g}^{r} parametrizing curves satisfying the above properties.

Let δ be the number of nodes of X; then either X is irreducible and 0 ≤ δ ≤ g, or X is reducible, in which case it is easy to see that X must be the union of 2 rational normal curves (of degree g − 1 of course) meeting transversally at g + 1 points.

Definition. Let H ⊂ ℙ^{r} be a hyperplane; we shall say that H is a theta-hyperplane of X if H ∩ X is everywhere non-reduced and if H meets X in g − 1 distinct points. Let i be an integer with 0 ≤ i. We shall say that H is a theta-hyperplane of type i for X if H contains exactly i singular points of X. We shall denote by θ_{i}(X) the set of such hyperplanes and by t_{i}(X) its cardinality.

As already noticed in the introduction, a general, smooth X has a finite number of distinct theta-hyperplanes, and this number is equal to N_{g} = 2^{g−1}(2^{g} − 1); these are the the hyperplanes that are tangent to X at g − 1 points.

We shall say that a smooth curve X as above is theta-generic if it has N_{g} theta-hyperplanes. We can define a rational map

θ : V′ · · · · · · · > Sym^{N_{g}}(ℙ^{r})^{∗}

by the rule θ(X) = (H_{1}, ...., H_{N_{g}}) with X smooth, theta-generic, H_{i} a theta hyperplane of X and H_{i} ≠ H_{j}.

It is not hard to see (arguing like in [CS] 2.3.1 or using [C]) that θ can be defined on an open subset of the Hilbert scheme larger than V′, including some singular curves. More precisely, if X ⊂ ℙ^{r} is a reduced, nodal, canonical curve with δ nodes, having no component contained in a hyperplane, and general among all curves with δ nodes, X has a finite set of distinct thetahyperplanes. Thus one can define θ(X) which will be a hypersurface of degree N_{g} all of whose irreducible components are theta-hyperplanes of X, which will now appear with a certain multiplicity in θ(X).

We shall call such a curve theta-generic and we shall denote by V the set of all such curves. Of course θ can be defined over V.

To describe the hypersurface θ(X) we start by computing t_{i}(X).

The multiplicity with which each hyperplane occurs in θ(X) will be computed later.

Proposition 1. Let X ∈ V be irreducible with δ nodes. Then

t_{0}(X) = 2^{2(g−δ)}2^{δ−1} = 2^{2g−δ−1} if δ ≥ 1;
t_{i}(X) = 2^{2(g−δ)}2^{δ−i−1}\binom{δ}{i} = \binom{δ}{i}2^{2g−δ−i−1} if i < δ < g − 1;
t_{δ}(X) = 2^{g−δ−1}(2^{g−δ} − 1) if δ < g − 1;

t_{g−1} = \{ \begin{matrix}1, & if δ = g − 1; \\ g, & if δ = g. \end{matrix}

Remark. We stated the result in such a redundant fashion, with double equalities, for the reader’s convenience: the first formula highlights the inductive pattern, while the second is more compact.

Proof. We start by proving that, if δ ≥ 1, then there are 2^{2(g−δ)}2^{δ−1} theta-hyperplanes of type 0. Let H be such a hyperplane, then

H ∩ X = 2 ∑_{i=1}^{g−1} P_{i}

where P_{i} is a smooth point of X. Let ν : X^{ν} ⟶ X be the normalization of X and let

E = ∑_{i=1}^{g−1} ν^{−1}(P_{i}).

Now denote by D the class of ν^{∗}H in the Picard group of X^{ν}, so that D = K_{X^{ν}} + G where G is an effective divisor of degree 2δ and deg D = 2g − 2. Of course E is an effective divisor such that 2E ∼ D; conversely any divisor class E′, such that 2E′ ∼ D, is effective, since the genus of X^{ν} is g − δ wich is at most equal to deg E′.

Let W ⊂ ℙ(H^{0}(X^{ν}, D)^{∗}) be the r-dimensional linear subspace such that our ambient space ℙ^{r} is equal to W. Given a square root E of D, the number of theta-hyperplanes of type 0 of X coming from the linear system |E| is thus equal to the number of divisors in |E| whose square is in the sublinear system W. Call x(δ) such a number (we will see that it does not depend on the choice of E). Since on X^{ν} there are exactly 2^{2(g−δ)} distinct divisor classes whose square is equal to D, we get that

t_{0}(X) = 2^{2(g−δ)}x(δ).

We now compute x(δ). There is a natural map

α : |E| ⟶ |D|

sending E to 2E. Now, by Riemann-Roch, dim |D| = g +δ −2 and dim |E| = δ −1 (our assumption that X is theta-generic implies that the divisor E on X^{ν} is non-special). Notice that if δ = 1 our result follows immediately.

The map α is the composition of a Veronese mapping of degree 2 from ℙ^{δ−1} = |E|, with a general linear projection; thus the degree of Imα is equal to 2^{δ−1}. Since W is a linear subspace of |D| of codimension δ − 1 (transverse to Imα since X is theta-generic), we conclude that

x(δ) = deg(W ∩ Imα) = 2^{δ−1}

which is what we wanted.

For the second formula we start with the case i = 1 (and δ ≥ 2) for simplicity. If i = 1 let N be a fixed node of X and let us compute the number of theta hyperplanes of type 1 containing N. Project X from N to ℙ^{r−1}, the image is a nodal irreducible curve Y having δ − 1 nodes; Y is also canonical, not contained in a hyperplane and theta-generic. The hyperplanes of type 0 of Y are in bijective correspondence with the hyperplanes of type 1 of X containing N. By the previous formula, we obtain

t_{0}(Y) = 2^{2(g−1−δ+1)}2^{δ−2} = 2^{2(g−δ)}2^{δ−2}

and, since there are δ nodes, t_{1}(X) = δ · t_{0}(Y); we are done in this case.

Let now 1 < i < g − 1; we proceed in a similar fashion. Pick a set S of i nodes and consider first the case i < g − 2. Project X from the i − 1-dimensional linear space spanned by S (if S spans a set of smaller dimension, pick a general space of dimension i − 1 containing it; this is not going to happen for the general X); the image is a canonical curve Y ⊂ ℙ^{g−1−i} of genus g − i and having δ − i nodes. The theta-hyperplanes of type i containing S are in bijective correspondence with the set of theta-hyperplanes of type 0 of Y. Since there are \binom{δ}{i} choices for S we get, using the first formula

t_{i}(X) = \binom{δ}{i}t_{0}(Y) = \binom{δ}{i}2^{2(g−i−δ+i)}2^{δ−i−1}

and we are done. Now assume i = g − 2. Let π : X ⟶ X′ ⊂ ℙ^{2} be the projection from g − 3 nodes in S and let N′ ∈ X′ be the image of the remaining node in S (the one not contained in the center of the projection). The curve X′ is an integral nodal curve of degree 4 having δ − g + 3 nodes. The set of theta-hyperplanes of type g − 2 of X containing S is in bijective correspondence with the set θ_{N′}(X′) of theta-lines of type 1 of X′ containing N′. By induction

#θ_{N′}(X′) = \frac{1}{δ − g + 3}t_{1}(X′) = 2^{g−δ}2^{δ−g+1}.

Since there \binom{δ}{g−2} choices for S we get that

t_{g−2}(X) = \binom{δ}{g − 2}#θ_{N′}(X′) = \binom{δ}{g − 2}2^{g−δ}2^{δ−(g−2)−1}

which is what we wanted.

The formula for t_{δ}(X) is obtained again projecting from the nodes of X and getting a smooth curve Y. The number of theta-hyperplanes of type δ of X is equal to the number of effective theta characteristics of the theta-generic, smooth, canonical curve Y.

The last two equalities are clear: just notice that g − 1 nodes on a curve of degree 2g − 2 must span a hyperplane (or the curve is contained in a hyperplane).

Definition. A curve X in V is called a split curve if X = C_{1} ∪ C_{2} is the union of two rational normal curves meeting transversally at g + 1 points.

Remark. Split curves are the only reducible curves in V, and they are characterized by the property of having g + 1 nodes.

Proposition 2. Let X be a split curve of (arithmetic) genus g. Then for 0 ≤ j ≤ g − 1 we have:

t_{j}(X) = \{ \begin{matrix}\binom{g+1}{j}2^{g−j−1} & if j ≢ g (mod 2) \\ 0 & if j ≡ g (mod 2) \end{matrix}

Remark. In particular, if g is odd, then t_{1}(X) = t_{3}(X) = .... = t_{g−2}(X) = 0.

If g is even then t_{0}(X) = t_{2}(X) = .... = t_{g−2}(X) = 0.

Notice also that for every g, a split curve has \binom{g+1}{2} theta-hyperplanes of type g − 1.

Proof. Start with t_{0}(X). If g is even, that is, r is odd, then t_{0}(X) must vanish: let X = C_{1} ∪ C_{2} and let H be any theta-hyperplane of type 0. Then H is tangent to X at r distinct smooth points, since r is odd then H must be tangent to (say) C_{1} in at least \frac{r+1}{2} points, so that deg H ∩C_{1} ≥ r +2 which is obviously impossible.

Now consider g odd and write r = 2h. Let C be a rational normal curve in ℙ^{r} and let T_{C} ⊂ (ℙ^{r})^{∗} be the closure of the locus of hyperplanes in ℙ^{r} that are tangent to C at h distinct points. Then we claim that T_{C} is an irreducible subvariety in (ℙ^{r})^{∗} of dimension h and degree 2^{h}.

The first part of the statement is immediate: just notice that through any h points P_{1}, ...., P_{h} of C there exists a unique hyperplane H_{P_{1},....,P_{h}} that is tangent to C at those h points. In fact, let L_{i} be the line tangent to C at P_{i}; then L_{1}, ...., L_{h} must be in general position, since deg C = 2h, hence they span a unique hyperplane. We get that T_{C} can be constructed as the closure of the image of the map

Sym^{h}(C) \ diagonals ⟶ (ℙ^{r})^{∗}
(P_{1}, ...., P_{h})  ↦ H_{P_{1},....,P_{h}}

Now to compute the degree, let J ≅ ℙ^{h} be a general h-dimensional linear subspace of ℙ^{r}. The degree of T_{C} is equal to the number of hyperplanes containing J and tangent to C at h points. Let π : C ⟶ ℙ^{h} be the projection from J. Having chosen J general, the image π(C) is a nodal, rational curve D of degree 2h in ℙ^{h} having h + 1 nodes. The theta-hyperplanes of type 0 of D are in bijective correspondence with the hyperplanes in ℙ^{r} containing J and tangent to C at h points. By Proposition 1, t_{0}(D) = 2^{h} = 2^{g−1}; we conclude that deg T_{C} = 2^{h}.

A consequence of this analysis is that if X = C_{1} ∪ C_{2} is a split curve of odd genus, the number of its theta-hyperplanes of type 0 is obtained by intersecting T_{C_{1}} with T_{C_{2}}: this is the intersection of two subvarieties of dimension h and degree 2^{h} in a projective space of dimension r = 2h. Transversality is ensured by the condition that X is theta-generic, so that we get

deg T_{C_{1}} ∩ T_{C_{2}} = 2^{2h} = 2^{g−1}

and we are done. The rest of the proof of the proposition is by iterating projections of X onto a lower degree split curve. Let us show the first case, the others are obtained in the same fashion.

To see that a split curve X of odd genus g has no hyperplanes of type 1, project from a node N onto a split curve Y of (even) genus g − 1. Theta hyperplanes of type 1 of X containing N correspond to theta-hyperplanes of type 0 of Y, which do not exist by the previous argument. Hence X has no theta-hyperplanes of type 1.

Consider now a split curve X of even genus and let N be a node. Repeat the above construction: now Y has 2^{g−2} theta-hyperplanes of type 0, by the previous part of the proof, hence X has 2^{g−2} theta-hyperplanes of type 1 through N. Since X has g + 1 nodes, we get

t_{1}(X) = (g + 1)2^{g−2}

and we are done.

The rest of the proof is absolutely identical, and we leave it to the reader.

Finally, we have the following

Lemma 3. Let X ∈ V; then a theta-hyperplane of type i appears with multiplicity 2^{i} as an irreducible component of θ(X).

Proof. This follows easily from [C], proof of (5.2), as explained in example (5.4).

Remark. The previous results imply that if X and Y are curves in V such that θ(X) = θ(Y), then they have the same number of nodes.

The reader can at this point check that deg θ(X) = ∑ 2^{i}t_{i}(X) = 2^{g−1}(2^{g} − 1).

We conclude this section with a somewhat parenthetic discussion about cuspidal curves, from which the rest of the paper is independent. The goal is to describe the enumerative data of the configuration of theta-hyperplanes for such curves.

Let Z ⊂ ℙ^{g−1} be an integral curve of degree 2g − 2, having γ ordinary cusps, that is, Z has γ double points q_{1}, ...., q_{γ} such that if ν : Z^{ν} ⟶ Z is the normalization of Z, then ν^{−1}(q_{i}) is one point p_{i}. Notice that the genus of Z^{ν} is h = g − γ and the arithmetic genus of Z is g. Let K be the canonical class of Z^{ν}; the projective model Z is obtained by a (g − 1)-dimensional linear series in the linear system |K + 2 ∑_{1}^{γ} p_{i}| ≅ ℙ^{g+γ−2}.

The discussion at the beginning of Section 2 can be extended to such cuspidal curves; in particular we have:

(1) The definition of theta-hyperplanes, and that of theta-hyperplanes of type i (0 ≤ i ≤ γ) remains unchanged for such cuspidal curves.

(2) If Z is generic, then Z has a finite number of theta-hyperplanes.

(3) One can extend the definition of θ(Z) to generic cuspidal curves, so that θ(Z) will be a nonreduced hypersurface of degree N_{g}, all of whose irreducible components are theta-hyperplanes.

The details for g = 3 are carried out in [CS]. Here we are only interested in the enumerative aspects. The theta-hyperplanes of Z can be stratified according to the number of cusps that they contain. Recall that θ_{i}(Z) denotes the set of theta-hyperplanes containing exactly i cusps, and that t_{i}(Z) := #θ_{i}(Z).

To better state and discuss the next result we need a new piece of notation. A smooth curve of genus h has 2^{2h} theta-characterstics. We shall denote by N_{h}^{+} := 2^{h−1}(2^{h} + 1) the number of even theta-characteristics (recall that N_{h} := 2^{h−1}(2^{h} − 1) is the number of odd ones).

Proposition 4. Let Z ⊂ ℙ^{g−1} be a general, integral curve of degree 2g − 2 having γ ordinary cusps and no other singularities. Then if i < g − 1

t_{i}(Z) = \{ \begin{matrix}\binom{γ}{i}N_{h} & if i ≡ γ (mod 2) \\ \binom{γ}{i}N_{h}^{+} & if i ≢ γ (mod 2) \end{matrix}

If i = g − 1

t_{g−1}(Z) = \{ \begin{matrix}1, & if γ = g − 1; \\ g, & if γ = g. \end{matrix}

Proof. Let S(Z^{ν}) ⊂ Pic^{h−1}Z^{ν} be the set of theta-characteristics of Z^{ν} (notation as above). Denote by Θ(Z) the support of θ(Z), in other words: Θ(Z) := ∪_{i=0}^{γ}θ_{i}(Z) and, of course, #Θ(Z) = ∑_{1}^{γ} t_{i}(Z).

In the first part of the proof we show that there is a surjective map of sets

τ : Θ(Z) ⟶ S(Z^{ν})

all of whose fibers have cardinality 2^{γ−1}. In particular, we get #Θ(Z) = 2^{g−2γ−1}.

To define τ, let H ∈ Θ(Z), then there exists a unique effective divisor D_{H} on Z^{ν} such that ν^{∗}(H ∩Z) = 2D_{H}. We have that 2D_{H} ∼ K +2 ∑ p_{i} and hence D_{H} −∑ p_{i} is a theta-characteristic on Z^{ν}. We define

τ(H) = D_{H} − ∑_{1}^{γ} p_{i}.

To show that τ is surjective, pick T ∈ S(Z^{ν}) and consider the divisor D_{T} := T + ∑ p_{i}. By genericity of Z we have that h^{0}(Z^{ν}, D_{T}) = γ; there is a natural morphism

α : ℙ^{γ−1} ≅ |D_{T} | ⟶ |2D_{T} | = |K + 2 ∑ p_{i}| ≅ ℙ^{g+γ−2}

defined by α(D) = 2D (compare with the proof of Proposition 1). Let now W ⊂ |2D_{T} | be the g −1-dimensional linear subpace such that our ambient space ℙ^{g−1} is W. The fiber τ^{−1}(T) is given exactly by the points in Imα ∩ W.

Now, α is the composition of the degree 2 Veronese map with a general linear projection. Thus dim Imα = γ − 1 and deg Imα = 2^{γ−1}. We obtain that Imα ∩ W is non-empty (by dimension count) hence τ is surjective. More precisely, our genericity assumption implies tha the intersection above is transverse, that is, that Imα ∩ W is made of 2^{γ−1} distinct points. Thus ∀T ∈ S(Z^{ν}) we have #τ^{−1}(T) = 2^{γ−1}.

In the second part of the proof we work by induction on γ. If γ = 1 the description in the first part becomes particularly simple.

If T is odd, that is, T is effective (Z^{ν} is generic), the divisor 2(T + p_{1}) is cut out by a unique theta-hyperplane of Z, passing through the cusp.

If T is even, that is, T is not effective, we still have that T + p_{1} is effective (by Riemann-Roch) and that |T + p_{1}| = {D_{T} }, where D_{T} is an effective divisor not containing p_{1} in its support. Hence 2D_{T} is cut out by a theta-hyperplane of type 0 of Z. Hence the case γ = 1 of the Proposition is proved.

Let now γ be arbitrary and g − 2 > i > 0. Fix a set G ⊂ {q_{1}, ...., q_{γ}} made of i cusps of Z and let π : Z ⟶ Z^{G} ⊂ ℙ^{g−i−1} be the projection from the linear space spanned by G, so that Z^{G} is an integral curve of degree 2(g − i − 1) having γ − i cusps, to which we can apply the induction hypothesis. In particular

t_{0}(Z^{G}) = \{ \begin{matrix}N_{h} & iff γ − i ≡ 0 (mod 2) ⟺ γ ≡ i (mod 2) \\ N_{h}^{+} & iff γ − i ≢ 0 (mod 2) ⟺ γ ≢ i (mod 2) \end{matrix}

Moreover, Z has precisely t_{0}(Z^{G}) theta hyperplanes of type i containing G; since there are \binom{γ}{i} choices for G we get

t_{i}(Z) = \binom{γ}{i}t_{0}(Z^{G})

and we are done if 0 < i < g − 2 .

Just like in the proof of Proposition 1, the case i = g − 2 needs to be treated separately. Let π : Z ⟶ Z′ ⊂ ℙ^{2} be the projection form a subset of G containing g − 3 cusps and let Q′ ∈ Z′ be the image of the remaining cusp. Z′ is thus an integral curve having γ − g + 3 ordinary cusps (one of which is Q′) and no other singularities. The set of theta-hyperplanes of type g − 2 of Z containing G is in one-to-one correspondence with the set θ_{Q′}(Z′) of theta-lines of type 1 of Z′ containing Q′. By induction

#θ_{Q′}(Z′) = \{ \begin{matrix}N_{h} & iff γ − g + 3 ≡ 1 (mod 2) ⟺ γ ≡ g − 2 (mod 2) \\ N_{h}^{+} & iff γ − g + 3 ≢ 1 (mod 2) ⟺ γ ≢ g − 2 (mod 2) \end{matrix}

Since of course t_{g−2}(Z) = \binom{γ}{g−2}#θ_{Q′}(Z′) we are done in this case.

The formula for i = g − 1 is clear (the cusps are in general linear position, by our genericity assumption).

It remains to compute t_{0}(Z). Let us do it for γ odd, for simplicity. It is obvious how to modify the computation that follows if γ is even. By the first part of the proof, every theta-characteristic of Z^{ν} determines 2^{γ−1} theta -hyperplanes of Z. So far, we have found that the total number of theta-hyperplanes corresponding to odd theta-characteristics of Z^{ν} is equal to

t_{γ}(Z) + t_{γ−2}(Z) + .... + t_{3}(Z) + t_{1}(Z) = N_{h} · ∑_{i=1}^{\frac{γ+1}{2}} \binom{γ}{2i − 1} = N_{h} · 2^{γ−1}

where the last equality follows from observing that ∑i=1^{\frac{γ+1}{2}} \binom{γ}{2i−1} is equal to the number of subsets

having odd cardinality of a set of γ elements; thus ∑i=1^{\frac{γ+1}{2}} \binom{γ}{2i−1} = \frac{2^{γ}}{2} .

We conclude that the odd theta-characteristics of Z^{ν} determine only theta-hyperplanes of type i > 0. In fact, an analogous computation gives that the total number of theta-hyperplanes of type i > 0 corresponding to even theta-characteristics of Z^{ν} is equal to

t_{γ−1}(Z) + t_{γ−3}(Z) + .... + t_{4}(Z) + t_{2}(Z) = N_{h}^{+} · ∑_{i=1}^{\frac{γ−1}{2}} \binom{γ}{2i} = N_{h}^{+} · (2^{γ−1} − 1)

since ∑i=1^{\frac{γ−1}{2}} \binom{γ}{2i} is the number of non-empty subsets of even cardinality in a set with γ elements. By the first part of the proof we know that

#Θ(Z) = 2^{γ−1}(N_{h}^{+} + N_{h})

and we just showed that

∑_{i=1}^{γ} t_{i}(Z) = 2^{γ−1}(N_{h}^{+} + N_{h}) − N_{h}^{+}.

We conclude that t_{0}(Z) = N_{h}^{+}.

3. Recovering curves from their theta-hyperplanes. The first result here states that split curves are uniquely determined, among all curves in V, by the weighted configuration of their theta-hyperplanes. Using the terminology of [CS], we could restate the Theorem below by saying that split curves have the theta-property.

Theorem 5. Let X ∈ V be a split curve, and let Y ∈ V be such that θ(X) = θ(Y). Then X = Y.

Proof. Let X = C_{1}∪C_{2} be a split curve and let Y ∈ V be such that θ(X) = θ(Y); denote Θ = θ(X). Then Y is also a split curve, since split curves can be characterized among all curves in V as those having either no hyperplanes of even type (if g is even) or no hyperplanes of odd type (if g is odd). This property is clearly reflected on the structure of Θ, whose irreducible components have all multiplicity an odd power of 2 if g is even, or an even power of 2 if g is odd.

Part 1: X and Y have the same nodes. Recall first that the nodes of X, being points on a rational normal curve, are in general linear position, that is, any h < g of them span a linear space of dimension h − 1. We shall now exhibit an algorithm to determine the nodes of X and Y as the g + 1 “cluster” points of the weighted configuration of hyperplanes given by Θ. More precisely, we shall prove that such nodes can be recovered from the configuration of hyperplanes that they span. This is really a special case of a more general phenomenon; see the discussion immediately following the proof of the Theorem.

For the first step of the algorithm, pick a hyperplane H ⊂ Θ having multiplicity 2^{g−1}, so that H contains exactly g − 1 nodes of X; call such nodes N_{1}, N_{2}, ...., N_{g−1} (which we will now recover) and call the remaining nodes of X (not lying on H) M_{1}, M_{2}. There are \binom{g+1}{g−1}−1 remaining theta-hyperplanes of type g − 1; let H′ be one of them. Of course

dim H ∩ H′ = g − 3

and there are two possibilities:

(a) H′ =< N_{i_{1}}, ...., N_{i_{g−2}}, M_{j} >, hence

J := H ∩ H′ =< N_{i_{1}}, ...., N_{i_{g−2}} > .

Observe also that such a J can be obtained by intersecting H with exactly 2 different H′, that is

J = H∩ < N_{i_{1}}, ...., N_{i_{g−2}}, M_{1} >= H∩ < N_{i_{1}}, ...., N_{i_{g−2}}, M_{2} > .

(b) H′ =< N_{i_{1}}, ...., N_{i_{g−3}}, M_{1}, M_{2} >, then K := H ∩ H′ contains only g − 3 nodes N_{i}’s and all such intersections are distinct, since for every g − 3-uple of nodes N_{i}s there is a unique H′ of type (b) containing it.

We can distinguish intersections H ∩ H′ of type (a) (the J’s) from those of type (b) (the K’s) using their multiplicity, since every J is obtained from two different H′’s, while every K is obtained in only one way.

For the second step, forget the K’s and, for a fixed i with 1 ≤ i ≤ g − 1, denote J^{i} ⊂ H

J^{i} =< N_{1}, ...., N_{i−1}, N_{i+1}, ....N_{g−1} >

the g − 3 dimensional subspace spanned by all N_{j}’s with the exception of N_{i}. Of course J^{i} is one of the intersections of type (a) above.

Now it is clear that the intersection of J^{i} and J^{h} is the g − 4 linear space spanned by all N_{j}’s with the exception of N_{i} and N_{h}. Moreover, for any subset of distinct integers {i_{1}, i_{2}, ...., i_{g−2}} ⊂ {1, 2, ...., g − 1} we get

J^{i_{1}} ∩ J^{i_{2}} ∩ .... ∩ J^{i_{g−2}} = N_{h}, h ∉ {i_{1}, i_{2}, ...., i_{g−2}}.

In this way we can recover all the nodes N_{1}, ...., N_{g−1} of X and Y.

To determine the remaining two nodes of X it is enough to repeat the argument starting from some different H. Or, go on with the proof of the Theorem, which only requires X and Y to have three nodes in common.

Part 2: Project from the nodes to show that X = Y. From now on, denote N_{1}, ....N_{g+1} the nodes of X and Y, and let Y = D_{1} ∪ D_{2}.

We use induction on r. The case r = 2 is 4.1.1 in [CS].

Let π_{i} be the projection from N_{i} to ℙ^{r−1}. Denote by X_{i} and Y_{i} the images of X and Y via π_{i}. Thus X_{i} and Y_{i} are split curves in ℙ^{r−1} and θ(X_{i}) = θ(Y_{i}). By induction X_{i} = Y_{i}. We can choose two nodes (say N_{1} and N_{2}) such that

π_{1}(C_{1}) = π_{1}(D_{1}) = E_{1}

and

π_{2}(C_{1}) = π_{2}(D_{1}) = E_{2}

(and also, of course π_{1}(C_{2}) = π_{1}(D_{2}), π_{2}(C_{2}) = π_{2}(D_{2})). Let us show that C_{1} = D_{1}. Denote by S_{i} ⊂ ℙ^{r} the cone over E_{i} with vertex N_{i} (that is

S_{i} = ∪_{P ∈X} \overline{N_{i}P} = ∪_{Q∈Y} \overline{N_{i}Q} )

then

C_{1} ∪ D_{1} ⊂ S_{1} ∩ S_{2}.

We shall now see that this implies C_{1} = D_{1}. To do that, it is enough to show that the intersection S_{1} ∩ S_{2} is a scheme entirely supported on C_{1} ∩ \overline{N_{1}N_{2}} (and thus it cannot contain a rational normal curve different from C_{1}). By contradiction, assume that there is a point Q ∈ S_{1} ∩ S_{2} such that Q does not belong to C_{1} or to the line joining N_{1} with N_{2}. Then the line \overline{N_{i}Q} must intersect C_{1} in another point P_{i}. Thus we find 2 points, P_{1} and P_{2} such that the plane Π spanned by N_{1}, N_{2}, Q satisfies

P_{1} + P_{2} + N_{1} + N_{2} ⊂ Π ∩ C_{1}.

Now this is not possible, in fact pick r −3 other points on C_{1} and let H be the hyperplane spanned by Π and those points; then

deg H ∩ C_{1} ≥ deg Π ∩ C_{1} + r − 3 ≥ r + 1

while deg C_{1} = r and C_{1} is not contained in a hyperplane. (Notice that we did not need the points P_{1}, P_{2}, N_{1}, N_{2} to be distinct)

We conclude that C_{1} = D_{1}. In the same way we prove that C_{2} = D_{2} and hence Y = X, a contradiction.

Remark. As suggested by the referee, what proved in Part 1 is a special case of the following more general result. Let S = {P_{1}, ...., P_{t}} be a set of points in general linear position in ℙ^{r}, with t ≥ r+1. Let G be the configuration of \binom{t}{r} hyperplanes that they span, that is: G = \{H =< P_{i_{1}}, ...., P_{i_{r}} > with 1 ≤ i_{1} < .... < i_{r} ≤ t}. Then G uniquely determines S.

Notice that the case t = r + 1 is clear, and the next case t = r + 2 is precisely Part 1 of the proof of the Theorem. The general statement can be proved using an analogous argument. Let H ∈ G and rename the points in S so that S = {N_{1}, ...., N_{r}, M_{1}, ...., M_{s}} with H =< N_{1}, ...., N_{r} > and s + r = t. We shall show how to determine N_{1}, ...., N_{r}. Let H′ ∈ G and let us consider H ∩ H′. There are two posiibilities:

(a) J := H ∩ H′ is contained in s + 1 elements of G, that is, there exists H_{1}, ...., H_{s} ∈ G such that

J = H ∩ H_{1} = H ∩ H_{2} = .... = H ∩ H_{s}

(b) H ∩ H′ is contained in less than s + 1 elements of G.

Now, (a) occurs for H′ containing at most one M_{i} among its generators, that is, for example

J = H∩ < N_{1}, ...., N_{r−1}, M_{1} >= .... = H∩ < N_{1}, ...., N_{r−1}, M_{s} >=< N_{1}, ...., N_{r−1} >

On the other hand, (b) occurs for H′ containing at least 2 of the M_{i}. In fact, suppose that (say) H′ =< N_{1}, ...., N_{k}, M_{1}, ...., M_{h} > with 2 ≤ h ≤ r. If K = H ∩ H′ = H ∩ H′′ then H′′ =< N_{1}, ...., N_{k}, M_{i_{1}}, ...., M_{i_{h}} >, with {M_{1}, ...., M_{h}} ∩ {M_{i_{1}}, ...., M_{i_{h}}} = ∅. In fact if (say) M_{1} ∈ H′′, then M_{1} ∈ H′ ∩ H′′. Thus M_{1} ∈ H since H′ ∩ H′′ ⊂ H ∩ H′ ∩ H′, and this is impossible. Thus there exist at most \frac{s}{h} different hyperplanes H′ ∈ G such that K = H ∩ H′. Now we argue exactly as in the second step of the algorithm used to prove Part 1: we recover N_{1}, ...., N_{r} by intersecting linear spaces of codimension 2 of type (a), which are distinguished, being the only ones having multiplicity s + 1.

We conclude with a result for curves of genus 4.

Theorem 6. Fix g = 4. Let X ∈ V be a general singular curve. If Y ∈ V is a curve such that there exists a quadric surface Q containing X and Y and if θ(X) = θ(Y), then X = Y.

Remark. From a moduli theoretic point of view the condition that X and Y are contained in the same quadric does not appear so restrictive, since all quadrics of the same rank are projectively equivalent, and every canonical curve in ℙ^{3} is contained in a (unique) quadric.

Proof. First we show that X and Y have the same nodes. The fact that they have the same number of nodes is an immediate consequence of Lemma 3. Let Θ = θ(X). Let us show that the nodes of X are uniquely determined as the cluster points of θ(X).

By Proposition 1, if X has a unique node N, then Θ contains 28 planes H_{1}, ...., H_{28} of multiplicity 2, passing through N and tangent to X at smooth points (by Proposition 1). The rest of Θ consists in 64 distinct planes not containing N. The node N is determined as the unique point where all the H_{i}’s intersect. In fact, if the intersection of the H_{i} contained another point M, let J be the line through N and M and let ϕ : X^{ν} ⟶ ℙ^{1} be the degree 4 map given by all planes through J (X^{ν} is the normalization of X). Then the ramification locus of ϕ contains the divisors cut on X − N by the 28 planes H_{i}, contradicting the Riemann-Hurwitz formula (X^{ν} has genus 3).

If X has 2 nodes, applying Proposition 1we get that Θ contains 6 planes of multiplicity 4, which are the theta-planes of type 2. Intersecting any two of them gives the line J through the two nodes. The intersection of J with any plane of type 1 (i.e., any plane appearing in Θ with multiplicity 2) determines one of the two nodes. Thus we recover the two nodes of X which must therefore coincide with the two nodes of Y.

If X has 3 nodes, let H be the plane containing all of them (the only component having multiplicity 8 in Θ). Intersecting H with all the planes of multiplicity 4, each of which contains a pair of nodes of X (there are 12 such planes, 4 through each pair of nodes, by Proposition 1) gives 3 lines of H forming a triangle of vertices the three nodes of X.

If X has 4 nodes, then Θ contains 4 planes of multiplicity 8, each of which is spanned by the three nodes of X. The intersection of any three of them is thus a node of X (and of Y).

If X has more than 4 nodes, then X is necessarily a split curve, which has already been treated in Theorem 5.

We now argue by contradiction, assuming that X and Y are distinct, irreducible (by Theorem 5) with δ nodes.

Pick a node N of X and Y and let π : ℙ^{3} ⟶ ℙ^{2} be the projection from N. Let C = π(X) and D = π(Y), so that C and D are irreducible plane quartic having the same δ − 1 nodes. We have

θ(C) = θ(D)

in fact the bitangent lines of C and D correspond to the theta-hyperplanes of type 1 of X and Y containing N. By [CS] (4.2.1, 5.1.1, 5.1.2, 5.2.1) we obtain that C = D. Let now S be the cone over C of vertex N; by the above analysis, S contains both X and Y. But then

X ∪ Y ⊂ S ∩ Q

which contradicts Bézout’s Theorem, since deg X ∪ Y = 12, while deg S ∩ Q = 8.

We believe that the assumption that X is singular can be removed from the statement of the Theorem, and that a suitable analogue holds for curves of any genus.

Aknowledgments. It is a pleasure to thank Edoardo Sernesi for introducing me to these problems and for the conversations we had about them. Special thanks to Emma Previato for her enthusiastic encouragement and to the referee for important comments. Part of this paper was written while I was visiting the University of Strasbourg: my sincere gratitude to Olivier Debarre and to the Department of Mathematics for the invitation and for the wonderful working conditions provided.

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

[C] M. Cornalba: Moduli of curves and theta-characteristics. Lectures on Riemann surfaces (Trieste, 1987), 560–589, World Sci. Publishing, Teaneck, NJ, 1989 Mathematics Subject Classification: 14Nxx, 14Hxx. L.Caporaso: Università degli Studi del Sannio, Benevento, Italy

and Massachusetts Institute of Technology, Cambridge, MA, USA caporaso@math.mit.edu