GAUSSIAN MAPS, GIESEKER-PETRI LOCI AND LARGE THETA-CHARACTERISTICS

GAVRIL FARKAS

1. Introduction

For an integer g ≥ 1 we consider the moduli space S_{g} of smooth spin curves parametrizing pairs (C, L), where C is a smooth curve of genus g and L is a thetacharacteristic, that is, a line bundle on C such that L^{2} ≅ K_{C}. It has been known classically that the natural map π : S_{g} → M_{g} is finite of degree 2^{2g} and that S_{g} is a disjoint union of two components S_{g}^{even} and S_{g}^{odd} corresponding to even and odd theta-characteristics. A geometrically meaningful compactification \overline{S}_{g} of S_{g} has been constructed by Cornalba by means of stable spin curves of genus g (cf. [C]). The space \overline{S}_{g} and more generally the moduli spaces \overline{S}_{g,n}^{1/r} of stable n-pointed r-spin curves of genus g, parametrizing pointed curves with r-roots of the canonical bundle, have attracted a lot of attention in recent years, partly due to a conjecture of Witten relating intersection theory on \overline{S}_{g,n}^{1/r} to generalized KdV hierarchies (see e.g. [JKV]).

For each g, r ≥ 0 one can define the locus

S_{g}^{r} := {(C, L) ∈ S_{g} : h^{0}(L) ≥ r + 1 and h^{0}(L) ≡ r + 1 mod 2}.

We also set M_{g}^{r} := π(S_{g}^{r}). It has been proved by Harris that each component of S_{g}^{r} has dimension ≥ 3g − 3 − \binom{r+1}{2} (cf. [H]). This bound is known to be sharp when r is very small: it is a classical result that S_{g}^{1} is a divisor in S_{g}, while for r = 2, 3 we have that S_{g}^{r} has pure codimension r(r + 1)/2 in S_{g} for all g ≥ 8 (cf. [T1]). On the other hand clearly the bound is far from optimal when r is relatively large with respect to g in the sense that there are examples when S_{g}^{r} ≠ ∅ although 3g −3−\binom{r+1}{2} is very negative: the hyperelliptic locus H_{g} ⊂ M_{g} is contained in M_{g}^{[(g−1)/2]} and there are Castelnuovo extremal curves C ⊂ P^{r} of genus 3r such that K_{C} = O_{C}(2), which gives that S_{3r}^{r} ≠ ∅ for all r ≥ 3 (see e.g. [CdC]). It is thus natural to ask to what extent Harris’ bound is sharp. We give a partial answer to this question by proving the following:

Theorem 1.1. For 1 ≤ r ≤ 11, r ≠ 10, there exists an explicit integer g(r) such that for all g ≥ g(r) the moduli space S_{g}^{r} has at least one component of codimension \binom{r+1}{2} in S_{g}. The general point [C, L] of such a component corresponds to a smooth curve C ⊂ P^{r}, with L = O_{C}(1) and K_{C} = O_{C}(2).

Research partially supported by the NSF Grant DMS-0140520.

For a precise formula for g(r) we refer to Section 3. We conjecture the existence of a component of S_{g}^{r} of codimension \binom{r+1}{2} for any r ≥ 1 and g ≥ \binom{r+2}{2} and we indicate a way to construct such a component (see Conjecture 3.4). Theorem 1.1 is proved inductively using the following result:

Theorem 1.2. We fix integers r, g_{0} ≥ 1. If S_{g_{0}}^{r} has a component of codimension \binom{r+1}{2} in S_{g_{0}}, then for every g ≥ g_{0}, the space S_{g}^{r} has a component of codimension \binom{r+1}{2} in S_{g}.

To apply Theorem 1.2 however, one must have a starting case for the inductive argument. This is achieved by carrying out an infinitesimal study of the loci S_{g}^{r} which will relate theta-characteristics to Gaussian maps on curves. Recall that for a smooth curve C and a line bundle L on C, the Gaussian or Wahl map ψ_{L} : ∧^{2}H^{0}(L) → H^{0}(K_{C} ⊗ L^{2}) is defined essentially by

ψ_{L}(s ∧ t) := s dt − t ds.

The map ψ_{L} has attracted considerable interest being studied especially in the context of deformation theory (see [W1] and the references therein). Wahl proved the remarkable fact that if C sits on a K3 surface then ψ_{K_{C}} cannot be surjective, which should be contrasted with the result of Ciliberto, Harris and Miranda saying that ψ_{K_{C}} is surjective for the general curve C of genus g = 10 or g ≥ 12 (cf. [CHM]). In a completely different direction, in a previous work we made essential use of the Gaussian map ψ_{K_{C}} for g = 10 to construct a counterexample to the Harris-Morrison Slope Conjecture on effective divisors on \overline{M}_{g} (cf. [FP]).

There are several powerful criteria in the literature ensuring the surjectivity of ψ_{L} when L has large degree (see e.g. [Pa], Theorem G), but very little seems to be known about when is the map ψ_{L} injective, or more generally, what is the behaviour of ψ_{L} when the line bundle L is special (cf. Question 5.8.1 in [W1]). In Section 5 we go some way towards answering this question by showing the following:

Theorem 1.3. For the general curve C of genus g and for any line bundle L on C of degree d ≤ g + 2, the Gaussian map ψ_{L} is injective.

We refer to Theorem 5.4 for a more general statement that bounds the dimension of Ker(ψ_{L}) even when d > g + 2. In the case when L is a very ample line bundle giving an embedding C ⊂ P^{n}, Theorem 1.3 can be interpreted as saying that the associated curve C → P^{N} obtained by composing the Gauss map C → G(2, n + 1), C ∋ p ↦ 𝕋_{p}(C), with the Plücker embedding of the Grassmannian of lines, is nondegenerate. Alternatively one can read this result in terms of (absence of) certain self-correspondences on the general curve C (see Proposition 5.7).

In Section 4 we relate the Gieseker-Petri loci on \overline{M}_{g} to the moduli spaces S_{g,n}^{r} of npointed spin curves consisting of collections (C, p_{1}, …, p_{n}, L), where (C, p_{1}, … p_{n}) ∈ M_{g,n} and L is a degree k line bundle on C such that L^{2} ⊗ O_{C}(p_{1} + ⋯ + p_{n}) = K_{C} and h^{0}(L) ≥ r + 1. Here of course we assume that 2k + n = 2g − 2.

We recall that the Gieseker-Petri Theorem asserts that for a general curve C of genus g and for any line bundle L on C, the map μ_{0}(L) : H^{0}(L) ⊗ H^{0}(K_{C} ⊗ L^{−1}) → H^{0}(K_{C}) is injective (see e.g. [EH2]). It is straightforward to see that if μ_{0}(L) is not injective then h^{0}(L), h^{0}(K_{C} ⊗ L^{−1}) ≥ 2 and it is an old problem to describe the locus in M_{g} where the Gieseker-Petri Theorem fails, in particular to determine its components and their dimensions.

We fix integers r, d ≥ 1 such that ρ(g, r, d) = g − (r + 1)(g − d + r) ≥ 0. As usual, G_{d}^{r}(C) is the variety of linear systems 𝔤_{d}^{r} on C, and if (L, V) ∈ G_{d}^{r}(C), we denote by μ_{0}(V) : V ⊗ H^{0}(K_{C} ⊗ L^{−1}) → H^{0}(K_{C}) the multiplication map. We define the Gieseker-Petri locus of type (r, d)

GP_{g,d}^{r} := {[C] ∈ M_{g} : ∃ a base point free (L, V) ∈ G_{d}^{r}(C) with μ_{0}(V) not injective}.

There are only two instances when this locus in well understood. First, GP_{g,g−1}^{1} can be identified with the above introduced locus M_{g}^{1} of curves with a vanishing theta-null which is known to be an irreducible divisor (cf. [T3]). Then for even g ≥ 4, GP_{g,(g+2)/2}^{1} is a divisor on M_{g} which has an alternate description as the branch locus of the natural map H_{g,(g+2)/2} → M_{g} from the Hurwitz scheme of coverings of P^{1} of degree (g + 2)/2 with source curve of genus g. This last divisor played a crucial role in the proof that M_{g} is of general type for even g ≥ 24 (cf. [EH3]). It is natural to ask whether more generally, all loci GP_{g,d}^{r} are divisors and we give a partial affirmative answer to this question:

Theorem 1.4. For integers g ≥ 4 and (g + 2)/2 ≤ k ≤ g − 1, the Gieseker-Petri locus GP_{g,k}^{1} has a divisorial component.

As an easy consequence we mention the following:

Corollary 1.5. For g ≥ 4 and 0 ≤ n ≤ g − 4, the moduli space S_{g,n}^{1} has at least one component of dimension 3g − 4.

This last statement can be compared to Polishchuk’s recent result that the moduli space S_{g,n}^{0} is of pure dimension 3g − 3 + n/2 (cf. [Po], Theorem 1.1).

2. Limit theta-characteristics

In this section, after briefly recalling some basic facts about stable spin curves, we characterize limit theta-characteristics on certain stable curves of compact type after which we prove Theorem 1.2.

We review a few things about the moduli space \overline{S}_{g} (see [C] for more details). If X is a nodal curve, a smooth rational component R of X is called exceptional if #(R∩\overline{(X − R)}) = 2. The curve X is called quasistable if every two exceptional components are disjoint. Every quasistable curve is obtained by blowing-up some of the nodes of a stable curve.

A stable spin curve consists of a triple (X, L, α), where X is a quasistable curve with p_{a}(X) = g, L is a line bundle on X of degree g − 1 with L_{R} = O_{R}(1) for each exceptional component R and α : L^{2} → ω_{X} is a homomorphism such that α_{C} ≠ 0 for any non-exceptional component C of X. A family of stable spin curves is a triple (f : C → T, L, α), where f : C → T is a flat family of quasistable curves, L is a line bundle on C and α : L^{2} → ω_{f} is a homomorphism such that α_{C_{t}} gives a spin structure on each fibre C_{t} = f^{−1}(t).

The stack \overline{S}_{g} of stable spin curves of genus g has been constructed in [C] where it is also proved that there exists a finite map π : \overline{S}_{g} → \overline{M}_{g} whose fibre over [C] ∈ \overline{M}_{g} is the set of stable spin structures on quasistable curves stably equivalent to C.

Remark 2.1. Suppose C = C_{1} ∪_{p} C_{2} is a curve of compact type with C_{1} and C_{2} being smooth curves and g(C_{1}) = i, g(C_{2}) = g − i. Then it is easy to see that there are no spin structures on C itself. In fact, π^{−1}([C]) consists of spin structures on the quasistable curve X = C_{1} ∪_{q} R ∪_{r} C_{2} obtained from C by “blowing-up” C at the node p. Each such spin structure is given by a line bundle L on X such that L_{C_{1}}^{2} = K_{C_{1}}, L_{C_{2}}^{2} = K_{C_{2}} and L_{R} = O_{R}(1). More generally, a spin structure on any curve of compact type corresponds to a collection of theta-characteristics on the components.

Assume now that C = C_{1} ∪_{p} C_{2} is a curve of compact type where C_{1} and C_{2} are smooth curves of genus i and g − i respectively. We define an r-dimensional limit thetacharacteristic on C (in short, a limit θ_{g}^{r}), as being a pair of line bundles (L_{1}, L_{2}) with L_{i} ∈ Pic^{g−1}(C_{i}), together with (r + 1)-dimensional subspaces V_{i} ⊂ H^{0}(L_{i}) such that

(1) {l_{i} = (L_{i}, V_{i})}_{i=1,2} is a limit linear series 𝔤_{g−1}^{r} in the sense of [EH1].

(2) L_{1}^{2} = K_{C_{1}}(2(g − i)p) and L_{2}^{2} = K_{C_{2}}(2ip).

Using this terminology we now characterize singular curves in \overline{M}_{g}^{r}:

Lemma 2.2. Suppose [C = C_{1} ∪_{p} C_{2}] ∈ \overline{M}_{g}^{r}. Then C possesses a θ_{g}^{r}.

Proof. We may assume that there exists a 1-dimensional family of curves f : C → B with smooth general fibre C_{b} and central fibre C_{0} = f^{−1}(0) stably equivalent to C, together with a line bundle L on C − C_{0} and a rank (r + 1) subvector bundle V ⊂ f_{∗}(L) over B^{∗} := B − {0} such that L_{C_{b}}^{2} ≡ ω_{C_{b}} for all b ∈ B^{∗}. Then for i = 1, 2 there are unique line bundles L_{i} on C for extending L and such that deg_{Y} (L_{i}) = 0 for every component Y of C_{0} different from C_{i}. If we denote by L_{i} := L_{i}|_{C_{i}} and V_{i} ⊂ H^{0}(L_{i}) the (r + 1)-dimensional subspace of sections that are limits in L_{i} of sections in V, then by Theorem 2.6 of [EH1] we know that {(L_{i}, V_{i})}_{i=1,2} is a limit 𝔤_{g−1}^{r}. Finally, since L_{i}^{2} and K_{C_{i}} are isomorphic off p they must differ by a divisor supported at p which accounts for condition (2) in the definition of a θ_{g}^{r}. □

We describe explicitly the points in \overline{M}_{g}^{r} ∩ Δ_{1}, where Δ_{1} is the divisor of curves with an elliptic tail:

Proposition 2.3. Let [C = C_{1} ∪_{p} E] be a stable curve with C_{1} smooth of genus g − 1 and E an elliptic curve. If [C] ∈ \overline{M}_{g}^{r} then either (1) [C_{1}] ∈ M_{g−1}^{r}, or (2) there exists a line bundle L_{1} on C_{1} such that (C_{1}, L_{1}) ∈ S_{g−1}^{r−1} and p ∈ Bs|L_{1}|. If moreover p ∈ C_{1} is a general point, then possibility (2) does not occur hence [C_{1}] ∈ M_{g−1}^{r}.

Proof. We know that C carries a limit θ_{g}^{r}, say l = {l_{C_{1}}, l_{E}}. By the compatibility relation between l_{C_{1}} and l_{E}, the vanishing sequence a^{l_{C_{1}}} (p) of l_{C_{1}} at p is ≥ (0, 2, …, r + 1). If l_{C_{1}} has a base point at p then if we set L := L_{C_{1}}(−p) we see that (C_{1}, L) ∈ S_{g−1}^{r} and we are in case (1). Otherwise we set M := L_{C_{1}}(−2p) and then h^{0}(C_{1}, M) = r, M^{2} = K_{C_{1}}(−2p) and |M + p| is a theta-characteristic on C_{1} having p as a base point.

For the the last statement, we note that a curve has finitely many positive dimensional theta-characteristics each of them having only a finite number of base points, so possibility (2) occurs for at most finitely many points p ∈ C_{1}. □

We can now prove Theorem 1.2. More precisely we have the following result:

Proposition 2.4. Fix r, g ≥ 1. If S_{g−1}^{r} has a component of codimension \binom{r+1}{2} in S_{g−1}, then S_{g}^{r} has a component of codimension \binom{r+1}{2} in S_{g}.

Proof. Suppose [C_{1}, L_{1}] ∈ S_{g−1}^{r} is a point for which there exists a component Z ∋ [C_{1}, L_{1}] of S_{g−1}^{r} with codim(Z, S_{g−1}) = \binom{r+1}{2}. We fix a general point p ∈ C_{1} and set C := C_{1} ∪_{p} E, where (E, p) is a general elliptic curve. We denote by X := C_{1}∪_{q} R∪_{s}E the curve obtained from C by blowing-up p, and we construct a spin structure on X given by a line bundle L on X with L_{C_{1}} = L_{1}, L_{R} = O_{R}(1) and L_{E} = O_{E}(t − s), where t − s is a non-zero torsion point of order 2. Clearly h^{0}(X, L) = h^{0}(C_{1}, L_{1}) ≥ r + 1. We first claim that (X, L) is a smoothable spin structure which would show that [X, L] ∈ \overline{S}_{g}^{r}.

To see this we denote by (f : X → B, L, α : L^{2} → ω_{f}) the versal deformation space of (X, L), so that if B_{1} denotes the versal deformation space of the stable model C of X, there is a commutative diagram:

B ⟶^{σ} B/Aut(X, L) ↪ \overline{S}_{g}
↓^{ϕ} ↓^{π}
B_{1} ⟶ B_{1}/Aut(C) ↪ \overline{M}_{g}

We define B^{r} := {b ∈ B : h^{0}(X_{b}, L_{b}) ≥ r+1, h^{0}(X_{b}, L_{b}) ≡ r+1 mod 2} and Theorem 1.10 from [H] gives that every component of B^{r} has dimension ≥ dim(B) − r(r + 1)/2. We also consider the divisor Δ ⊂ B corresponding to singular spin curves. To conclude that (X, L) is smoothable we show that there exists a component W ∋ 0 of B^{r} not contained in Δ (here 0 ∈ B is the point corresponding to (X, L)).

Assume that on the contrary, every component of B^{r} containing 0 sits inside Δ. It is straightforward to describe B^{r} ∩ Δ: if (X_{b} = C_{b} ∪ R_{b} ∪ E_{b}, L_{b}) where b ∈ B, g(C_{b}) = g − 1, g(E_{b}) = 1, is a spin curve with h^{0}(X_{b}, L_{b}) ≥ r + 1, then either (1) h^{0}(C_{b}, L_{b|C_{b}}) ≥ r + 1 or (2) h^{0}(C_{b}, L_{b|C_{b}}) = r and L_{b|E_{b}} = O_{E_{b}} (put it differently, L_{b|E_{b}} is the only odd theta characteristic on E_{b}). Since even and odd theta characteristics do not mix, it follows that any component 0 ∈ W ⊂ B^{r} will consist entirely of elements b for which h^{0}(C_{b}, L_{b|C_{b}}) ≥ r + 1. Moreover, there is a 1:1 correspondence between such components of B^{r} and components of S_{g−1}^{r} through [C_{1}, L_{1}]. But then the locus

Z_{1} := {b ∈ Δ : [C_{b}, L_{b|C_{b}}] ∈ Z, h^{0}(E_{b}, L_{b|E_{b}}) = 0}

is a component of B^{r} containing 0 and dim(Z_{1}) = dim(Z) + 2 = 3g − 4 − \binom{r+1}{2}, which contradicts the estimate on dim(B^{r}).

Thus (X, L) is smoothable. We now show that at least one component of \overline{S}_{g}^{r} passing through [C, L_{C}] has codimension \binom{r+1}{2}. Suppose this is not the case. Then each component of \overline{S}_{g}^{r} ∩ σ(Δ) through [C, L_{C}] has codimension ≤ \binom{r+1}{2} − 1 in σ(Δ). Recalling that p ∈ C_{1} was general, Proposition 2.3 says that any such component corresponds to curves C′_{1} ∩ E′ where E′ is elliptic and [C′_{1}] ∈ M_{g−1}^{r}. But then σ(Z_{1}) is such a component and we have already seen that codim(Z_{1}, Δ) = \binom{r+1}{2}, which yields the desired contradiction. □

Remark 2.5. Retaining the notation from the proof of Theorem 2.3, if [C_{1}, L_{1}] ∈ S_{g−1}^{r} is such that L_{1} is very ample, then a smoothing [C′, L_{C′}] ∈ S_{g}^{r} of [C = C_{1} ∪_{p} E, L_{C}] corresponds to a very ample L_{C′}. Indeed, assuming by contradiction that there exist points x, y ∈ C′ such that h^{0}(L_{C′}(−x − y)) ≥ h^{0}(L_{C′}) − 1, we have three possibilities depending on the position of the points r, s ∈ C to which x and y specialize. The case x, y ∈ E can be ruled out immediately, while x, y ∈ C_{1} would contradict the assumption that L_{1} is a very ample line bundle. Finally, if x ∈ C_{1} and y ∈ E, one obtains that {x, p} fails to impose independent conditions on |L_{1}|, a contradiction. Thus L_{C′} is very ample.

3. Gaussian maps and theta-characteristics

It may be helpful to review a few things about Gaussian maps on curves and to explain the connection between Gaussians and theta-characteristics. This will enable us to construct components of S_{g}^{r} of dimension achieving the Harris bound.

For a smooth projective variety X and a line bundle L, we denote by R(L) the kernel of the multiplication map H^{0}(L) ⊗ H^{0}(L) → H^{0}(L^{2}). Following J. Wahl (see e.g. [W1]), we consider the Gaussian map Φ_{L} = Φ_{X,L} : R(L) → H^{0}(Ω_{X}^{1} ⊗ L^{2}), defined locally by

s ⊗ t ↦ s dt − t ds.

Since R(L) = ∧^{2}H^{0}(L) ⊕ S_{2}(L), with S_{2}(L) = Ker{Sym^{2}H^{0}(L) ⟶^{μ_{L}} H^{0}(L^{2})}, it is clear that Φ_{L} vanishes on symmetric tensors and it makes sense to look at he restriction

ψ_{L} = ψ_{X,L} := Φ_{L|∧^{2}H^{0}(L)} : ∧^{2}H^{0}(L) → H^{0}(Ω_{X}^{1} ⊗ L^{2}).

If X ⊂ P^{r} is an embedded variety with L = O_{X}(1), one has the following interpretation for Φ_{L}: we pull back the Euler sequence to X to obtain that R(L) = H^{0}(Ω_{P^{r}|X}^{1} ⊗ L^{2}) and then Φ_{L} can be thought of as the map obtained by passing to global sections in the morphism Ω_{P^{r}|X}^{1} ⊗ L^{2} → Ω_{X}^{1} ⊗ L^{2}. Furthermore, if N_{X} is the normal bundle of X in P^{r}, tensoring the exact sequence

(1)  0 ⟶ N_{X}^{∨} ⟶ Ω_{P^{r}|X}^{1} ⟶ Ω_{X}^{1} ⟶ 0

by O_{X}(2), we obtain that Ker(Φ_{L}) = Ker(ψ_{L}) ⊕ S_{2}(L) = H^{0}(N_{X}^{∨}(2)). If X is projectively normal, from the exact sequence 0 → I_{X}^{2} → I_{X} → N_{X}^{∨} → 0 it is straightforward to check that Ker(ψ_{L}) = H^{1}(P^{r}, I_{X}^{2}(2)).

The map ψ_{L} has been extensively studied especially when X is a curve, in the context of the deformation theory of the cone over X (cf. e.g. [W1]). The connection between Gaussian maps and spin curves is given by the following tangent space computation due to Nagaraj (cf. [N], Theorem 1): for (C, L) ∈ S_{g}^{r}, if we make the standard identifications T_{[C,L]}(S_{g}) = T_{[C]}(M_{g}) = H^{1}(C, T_{C}) = H^{0}(C, K_{C}^{2})^{∨}, then

T_{[C,L]}(S_{g}^{r}) = (Im(ψ_{L}) : ∧^{2}H^{0}(L) → H^{0}(K_{C}^{2}))^{⊥}.

In other words, to show that a component Z of S_{g}^{r} has codimension \binom{r+1}{2} in S_{g}, it suffices to exhibit a spin curve [C, L] ∈ Z such that h^{0}(L) = r+1 and ψ_{L} is injective. We construct such curves as sections of certain homogeneous spaces having injective Gaussians and then we apply Theorem 1.2 to increase the range of (g, r) for which we have a component of S_{g}^{r} of codimension \binom{r+1}{2}. We will use repeatedly the following result of Wahl relating the Gaussian map of a variety to that of one of its sections (cf. [W2], Propositions 3.2 and 3.6):

Proposition 3.1. 1. Suppose X ⊂ P^{r} is a smooth, projectively normal variety such that ψ_{X,O_{X}(1)} is injective. If Y ⊂ X is a subvariety with ideal sheaf I satisfying the conditions

H^{1}(X, I(1)) = 0, H^{1}(X, I^{2}(2)) = 0, H^{1}(X, N_{X}^{∨}(2) ⊗ I) = 0,

then the Gaussian ψ_{Y,O_{Y} (1)} is injective too.

2. Let X ⊂ P^{r} be a smooth, projectively normal, arithmetically Cohen-Macaulay variety and Y = X ∩ P^{r−n} ⊂ P^{r−n} a smooth codimension n linear section, where n < r. If H^{i}(X, N_{X}^{∨}(2 − i)) = 0 for 1 ≤ i ≤ n and ψ_{X,O_{X}(1)} is injective, then Y is projectively normal and the Gaussian ψ_{Y,O_{Y} (1)} is also injective.

We will apply Proposition 3.1 in the case of the Grassmannian X = G(2, n) of 2dimensional quotients of ℂ^{n} and for the line bundle L = O_{G(2,n)}(1) which gives the Plücker embedding. In this case ψ_{O_{G(2,n)}(1)} is bijective (cf. [W2], Theorem 2.11).

We need to compute the cohomology of several vector bundles on G(2, n) and we do this using Bott’s theorem (see [FH] for a standard reference). Recall that G(2, n) = SL_{n}(ℂ)/P, where the reductive part of the parabolic subgroup P consists of matrices of type diag(A, B) ∈ SL_{n}(ℂ) where A ∈ GL_{2}(ℂ) and B ∈ GL_{n−2}(ℂ). We denote by Q the universal rank 2 quotient bundle defined by the tautological sequence

0 ⟶ U ⟶ O_{G(2,n)}^{⊕n} ⟶ Q ⟶ 0.

Every irreducible vector bundle over G(2, n) comes from a representation of the reductive part of P. If e_{1}, …, e_{n} is an orthonormal basis of ℝ^{n}, the positive roots of SL_{n}(ℂ) are {e_{i} −e_{j}}_{i<j} and we use the notation E(a_{1}, …, a_{n}) for the vector bundle corresponding to the representation with highest weight a_{1}e_{1}+⋯+a_{n}e_{n}. We then have the identifications Q = E(1, 0, …, 0), O_{G(2,n)}(1) = det(Q) = E(1, 1, 0, …, 0) and U = E(0, 0, 1, 0, …, 0). The cotangent bundle Ω_{G(2,n)}^{1} = Q^{∨} ⊗ U is irreducible and corresponds to the highest weight (0, −1, 1, 0, …, 0). Bott’s theorem can be interpreted as saying that the cohomology group H^{i}(G(2, n), E(a_{1}, …, a_{n})) does not vanish if and only if i is the number of strict inversions in the sequence (n+a_{1}, n−1+a_{2}, …, 1+a_{n}) and all the entries of this sequence are distinct.

First we establish the following vanishing result:

Proposition 3.2. Let 𝔾 = G(2, n) ⊂ P^{N} with N = \binom{n}{2} − 1, be the Grassmannian of lines in its Plücker embedding. We have the following vanishing statements:

(1) H^{i}(N_{𝔾}^{∨}(2 − i)) = 0 for all 1 ≤ i ≤ 2n − 5, i ≠ 2 and for i = 2 and n ≤ 6.

(2) H^{i}(Ω_{𝔾}^{1} ⊗ Q(−i)) = 0 for 0 ≤ i ≤ 2n − 7.

(3) H^{i+1}(N_{𝔾}^{∨} ⊗ Q(−i)) = 0 for 1 ≤ i ≤ min(n, 2n − 7).

(4) H^{∗}(Q(−i)) = 0 for 1 ≤ i ≤ n.

(5) H^{i+1}(N_{𝔾}^{∨}(−i)) = 0 for 0 ≤ i ≤ n − 1.

Proof. (1). We start with the case i ≥ 3. From the exact sequence (1) it suffices to show that (a) H^{i−1}(𝔾, Ω_{𝔾}^{1}(2 − i)) = 0 and that (b) H^{i}(𝔾, Ω_{P^{N} |𝔾}^{1}(2 − i)) = 0. From the Euler sequence (b) at its turn is implied by the vanishings H^{i−1}(O_{𝔾}(2−i)) = H^{i}(O_{𝔾}(1−i)) = 0 which are obvious, while (a) is a consequence of Bott’s theorem (or of Kodaira - Nakano vanishing). When i = 1, one checks that H^{0}(𝔾, Ω_{𝔾}^{1}(1)) = 0 (Bott again), and that H^{1}(𝔾, Ω_{P^{N} |𝔾}^{1}(1)) = 0 (Euler sequence). The remaining case i = 2 is handled differently and we employ the Griffiths vanishing theorem: since 𝔾 is scheme theoretically cut out by quadrics, the vector bundle E = N_{𝔾}^{∨}(2) is globally generated. From the exact sequence (1) one finds that det(E) = O_{𝔾}((n−3)(n−4)/2) and we can write N_{𝔾}^{∨} = K_{𝔾}⊗E ⊗det(E)⊗L, with L an ample line bundle, precisely when n ≤ 6.

Part (2) is a consequence of Le Potier vanishing (cf. [LP]), while (4) follows from Bott vanishing since Q(−i) = E(1 − i, −i, 0, …, 0). To prove (3) we tensor the exact sequence (1) by Q(−i) and we have to show that H^{i}(Ω_{𝔾}^{1} ⊗ Q(−i)) = H^{i+1}(Ω_{P^{N} |𝔾}^{1} ⊗ Q(−i)) = 0 which we already treated in parts (2) and (4). Finally, (5) is handled similarly to (1) and we omit the details. □

For certain r we construct half-canonical curves C ⊂ P^{r} of genus g(r) with injective Gaussian. This combined with Theorem 1.2 proves Theorem 1.1.

Proposition 3.3. For 3 ≤ r ≤ 11, r ≠ 10, there exists a smooth half-canonical curve C ⊂ P^{r} of genus g(r) (to be specified in the proof), such that the Gaussian map ψ_{O_{C}(1)} is injective. It follows that S_{g(r)}^{r} is smooth of codimension r(r + 1)/2 at the point [C, O_{C}(1)].

Proof. Each case will require a different construction. We treat every situation separately in increasing order of difficulty. r=3. We let C be a (3, 3) complete intersection in P^{3}, hence g(C) = g(3) = 10 and K_{C} = O_{C}(2). Clearly N_{C} = O_{C}(3) ⊕ O_{C}(3), so trivially H^{1}(N_{C}^{∨}(2)) = 0 which proves that ψ_{O_{C}(1)} is injective.

r=4. Now C is a complete intersection of type (2, 2, 3) in P^{4}. Then g(C) = g(4) = 13 and N_{C} = O_{C}(2)^{2}⊕O_{C}(3). Using that C is projectively normal we get that H^{1}(P^{4}, I_{C}^{2}(2)) = 0, hence ψ_{O_{C}(1)} is injective again.

r=5. This is the last case when C can be a complete intersection: C is of type (2, 2, 2, 2) in P^{5}, thus g(C) = g(5) = 17 and like in the r = 4 case we check that H^{1}(P^{5}, I_{C}^{2}(2)) = 0.

r=8. We choose the Grassmannian G(2, 6) ⊂ P^{14}. A general codimension 6 linear section of G(2, 6) is a K3 surface S ⊂ P^{8} with deg(S) = 14 and we let C := S ∩ Q ⊂ P^{8} be a quadric section of S. Then C is half-canonical and g(C) = g(8) = 29. We claim that ψ_{S,O_{S}(1)} is injective, which follows from Proposition 3.1 since H^{i}(N_{G(2,6)}^{∨}(2 − i)) = 0 for 1 ≤ i ≤ 6. To obtain that H^{1}(P^{8}, I_{C}^{2}(2)) = 0, by Proposition 3.1 we have to check that H^{1}(S, O_{S}(−1)) = H^{1}(S, O_{S}(−2)) = 0 (Kodaira vanishing), and that H^{1}(N_{S}^{∨}) = 0 ⇔ H^{1}(N_{S}) = 0. Note that S is a general K3 surface of genus 8 having ρ(S) = 1 and since by transcendental theory, the Hilbert scheme of such K3 surfaces is irreducible, it will suffice to exhibit a single K3 surface of genus 8 having this property: we let S degenerate to a union R_{1} ∪_{B} R_{2} of two rational scrolls of degree 7 in P^{8} joined along an elliptic curve B ∈ | − K_{R_{i}}| for i = 1, 2. Then R_{1} ∪_{B} R_{2} is a limit of smooth K3 surfaces X ⊂ P^{8} of degree 14 and H^{1}(R_{1} ∪_{B} R_{2}, N_{R_{1}∪_{B}R_{2}}) = 0 (see [CLM], Theorem 1.2 for more details on this degeneration). It follows that H^{1}(X, N_{X}) = 0, for a general prime K3 surface X ⊂ P^{8} of degree 14 and then H^{1}(S, N_{S}) = 0 as well.

r=7. In this situation we choose the 10-dimensional spinor variety X ⊂ P^{15} corresponding to a half-spin representation of Spin(10) (see [M] for a description of the projective geometry of X). One has that X is a homogeneous space for SO(10), K_{X} = O_{X}(−8) and deg(X) = 12. A general codimension 8 linear section of X is a K3 surface S ⊂ P^{7} of degree 12. Take now C to be a quadric section of S and then K_{C} = O_{C}(2) and g(C) = g(7) = 25. Since N_{X}^{∨} is irreducible (cf. e.g. [W2], Theorem 2.14), we obtain that the Gaussian map ψ_{X,O_{X}(1)} is injective.

To show that ψ_{S,O_{S}(1)} is injective we verify that H^{i}(N_{X}^{∨}(2 − i)) = 0 for 1 ≤ i ≤ 8. For 3 ≤ i ≤ 8 this follows from Kodaira-Nakano vanishing for the twists of sheaves of holomorphic forms on X in a way similar to the proof of Proposition 3.2, while the i = 1 it is a consequence of Bott vanishing. For i = 2 we use Griffiths vanishing: since X is cut out by quadrics (see e.g. [M], Proposition 1.9), the vector bundle E := N_{X}^{∨}(2) is globally generated, det(E) = O_{X}(2) and one can write N_{X}^{∨} = K_{X} ⊗ E ⊗ det(E) ⊗ O_{X}(4). In this way we obtain that H^{2}(N_{X}^{∨}) = 0. Thus ψ_{S,O_{S}(1)} is injective, and to have the same conclusion for the Gaussian of C, the only non-trivial thing to check is that H^{1}(N_{S}) = 0, which can be seen by letting S degenerate again to a union of two rational scrolls like in the case r = 8.

r=6. We consider the Grassmannian 𝔾 = G(2, 5) ⊂ P^{9} and we denote by X ⊂ P^{6} a general codimension 3 linear section of 𝔾, by S := X ∩ Q a general quadric section of X and by C := S ∩ Q′ a general quadric section of S. Then S is a K3 surface of genus 6, K_{C} = O_{C}(2) and g(C) = g(6) = 21. Using Propositions 3.1 and 3.2 we see easily that ψ_{X,O_{X}(1)} is injective. We claim that ψ_{S,O_{S}(1)} is injective as well which would follow from H^{1}(X, N_{X}^{∨}) = 0. Since N_{X/𝔾}^{∨} = O_{X}(−1)^{⊕3}, the vanishing of H^{1}(X, N_{X}^{∨}) is implied by that of H^{1}(N_{𝔾}^{∨} ⊗ O_{X}) which in its turn is implied by H^{i+1}(N_{𝔾}^{∨}(−i)) = 0 for 0 ≤ i ≤ 3 (use the Koszul resolution). These last vanishing statements are contained in Proposition 3.2 and in this way we obtain that ψ_{S,O_{S}(1)} is injective. We finally descend to C. To conclude that ψ_{C,O_{C}(1)} is injective it is enough to verify that H^{1}(N_{S}) = 0. We could check this again via the Koszul complex, but it is more economical to use that S is a general K3 surface of genus 6 and to invoke once more [CLM], Theorem 1.2, like in the previous cases.

r=11. We start with the Grassmannian X = G(2, 7) ⊂ P^{20} for which K_{X} = O_{X}(−7) and we let C be a general codimension 9 linear section of X. Then C ⊂ P^{11} is a smooth half-canonical curve of genus g(C) = g(11) = 43. To conclude that ψ_{C,O_{C}(1)} is injective we apply directly the second part of Proposition 3.1: the vanishing H^{i}(N_{G(2,7)}^{∨}(2 − i)) = 0 is guaranteed by Proposition 3.2 for all 1 ≤ i ≤ 9, i ≠ 2. For i = 2 we can no longer employ Griffiths vanishing so we proceed differently: we use (1) together with the vanishing H^{2}(X, Ω_{P^{20}|X}^{1}) = 0 coming from the Euler sequence, to write down the exact sequence

(2)  0 ⟶ H^{1}(N_{X}^{∨}) ⟶ H^{1}(Ω_{P^{20}|X}^{1}) ⟶ H^{1}(Ω_{X}^{1}) ⟶ H^{2}(N_{X}^{∨}) ⟶ 0,

where H^{1}(Ω_{P^{20}|X}^{1}) ≅ H^{0}(O_{X}) ≅ ℂ and H^{1}(Ω_{X}^{1}) ≅ ℂ. From Bott’s theorem at most one of the cohomology groups of the irreducible bundle N_{X}^{∨} are ≠ 0, hence either H^{2}(N_{X}^{∨}) = 0 and then we are done, or else, if H^{2}(N_{X}^{∨}) ≠ 0 then H^{1}(N_{X}^{∨}) = 0, and the map in the middle of the sequence (2) is bijective which yields a contradiction. r=9. This is the most involved case. We look at the ample vector bundle F := Q(1) on 𝔾 = G(2, 6) ⊂ P^{14} and choose a general section s ∈ H^{0}(𝔾, F). We denote by Z the zero locus of s, by I = I_{Z/𝔾} the ideal of Z inside 𝔾, and by I_{Z} and I_{𝔾} the ideals of Z and 𝔾 in P^{14} respectively. By adjunction, we have that I/I^{2} = Q^{∨}(−1) ⊗ O_{Z} and the Koszul complex gives a resolution for Z:

0 ⟶ O_{𝔾}(−3) ⟶ Q^{∨}(−1) ⟶ I ⟶ 0.

We first claim that Z ⊂ P^{14} is nondegenerate and projectively normal. This will follow if we show that H^{0}(𝔾, I(1)) = 0 and H^{1}(𝔾, I(r)) = 0 for r ≥ 1. Using the Koszul resolution, the first vanishing is implied by H^{0}(Q^{∨}) = H^{1}(O_{𝔾}(−2)) = 0 which is clear. For the second vanishing we have to check that H^{1}(Q^{∨}(r − 1)) = H^{2}(O_{𝔾}(r − 3)) = 0 for r ≥ 1. Since Q^{∨}(r − 1) = E(r − 1, r − 2, 0, 0, 0, 0) and O_{𝔾}(r − 3) = E(r − 3, r − 3, 0, 0, 0, 0) this can be checked instantly using Bott’s theorem.

Next we claim that the ψ_{Z,O_{Z}(1)} is injective. By Proposition 3.1, we have to verify that (1) H^{1}(𝔾, I^{2}(2)) = 0 and that (2) H^{1}(𝔾, N_{𝔾}^{∨}(2) ⊗ I) = 0. We start with (1). From the exact sequence

0 ⟶ I^{2}(2) ⟶ I(2) ⟶ Q^{∨}(1) ⊗ O_{Z} ⟶ 0,

using that Z is projectively normal, (1) is implied by the bijectivity of the map H^{0}(I(2)) → H^{0}(Q^{∨}(1) ⊗ O_{Z}). This is a consequence of the isomorphism Q^{∨}(1) ≅ Q and of the Koszul resolution giving that H^{0}(Z, Q^{∨}(1) ⊗ O_{Z}) = H^{0}(𝔾, Q^{∨}(1)) = H^{0}(𝔾, I(2)), where for the first isomorphism one uses that H^{0}(𝔾, I⊗Q) = H^{1}(𝔾, I⊗Q) = 0, which is straightforward to check via Bott’s theorem.

We turn to (2). The cohomology of I ⊗N_{𝔾}^{∨}(2) is computed from the Koszul complex of I, which yields an isomorphism H^{1}(N_{𝔾}^{∨} ⊗ I(2)) = H^{1}(N_{𝔾}^{∨} ⊗ Q^{∨}(1)) (because we have H^{i}(N_{𝔾}^{∨}(−1)) = 0 for i = 1, 2- this being checked via the sequence (1)). Next we write the cohomology sequence associated to the exact sequence

0 ⟶ N_{𝔾}^{∨} ⊗ Q^{∨}(1) ⟶ Ω_{P^{14}|𝔾}^{1} ⊗ Q^{∨}(1) ⟶ Ω_{𝔾}^{1} ⊗ Q^{∨}(1) ⟶ 0.

The map H^{1}(Ω_{P^{14}|𝔾}^{1} ⊗ Q^{∨}(1)) → H^{1}(Ω_{𝔾}^{1} ⊗ Q^{∨}(1)) is an isomorphism: from the Euler sequence one obtains that H^{1}(Ω_{P^{14}|𝔾}^{1} ⊗ Q^{∨}(1)) = H^{0}(Q^{∨}(1)), while tensoring by Ω_{𝔾}^{1}(1) the dual of the tautological sequence, one gets that H^{1}(Ω_{𝔾}^{1}(1) ⊗ Q^{∨}) = H^{0}(U^{∨} ⊗ Ω_{𝔾}^{1}(1)) = H^{0}(Q^{∨}(1)) (or alternatively, use for this [LP], Corrolaire 2). Moreover H^{0}(Ω_{𝔾}^{1} ⊗ Q^{∨}(1)) injects into H^{0}(Ω_{𝔾}^{1}(1))^{⊕6} which is zero by Bott’s theorem. Hence H^{1}(N_{𝔾}^{∨} ⊗ Q^{∨}(1)) = 0 and this proves that ψ_{Z,O_{Z}(1)} is injective.

We now take a general codimension 5 linear section of Z which is a curve C ⊂ P^{9} with K_{C} = O_{C}(2). A routine calculation gives that deg(C) = 3deg(𝔾) = 42, hence g(C) = g(9) = 43. We claim that ψ_{C,O_{C}(1)} is injective. Since ψ_{Z,O_{Z}(1)} is injective, by Proposition 3.1 we are left with checking that Z is ACM (this amounts to H^{i}(O_{Z}(j)) = 0 for i ≠ 0, 6 = dim(Z), which easily follows from the Koszul complex) and that H^{i}(Z, N_{Z}^{∨}(2 − i)) = 0 for 1 ≤ i ≤ 5 (here N_{Z} = (I_{Z}/I_{Z})^{∨} is the normal bundle of Z in P^{14}). We employ the exact sequence

0 ⟶ N_{𝔾}^{∨} ⊗ O_{Z} ⟶ N_{Z}^{∨} ⟶ I/I^{2} ⟶ 0,

from which it will suffice to show that (a) H^{i}(Z, I/I^{2}(2 − i)) = H^{i}(Q^{∨}(1 − i) ⊗ O_{Z}) = 0 for 1 ≤ i ≤ 5 and that (b) H^{i}(N_{𝔾}^{∨}(2 − i) ⊗ O_{Z}) = 0, which in turn is a consequence of H^{i}(N_{𝔾}^{∨}(2 − i)) = H^{i+1}(N_{𝔾}^{∨}(−1 − i)) = 0 and of the vanishing H^{i+1}(N_{𝔾}^{∨} ⊗ Q^{∨}(1 − i)) = 0 (for all these use Proposition 3.2).

We are left with (a) which is a consequence of H^{i}(Q^{∨}(1 − i)) = 0 (again, use Proposition 3.2), of H^{i+2}(Q^{∨}(−2 − i)) = 0, and of H^{i+1}(Q ⊗ Q(−2 − i)) = 0. For this last statement use that Q ⊗ Q = S^{2}Q ⊕ det(Q) and each summand being irreducible the vanishing can be easily verified via Bott’s theorem. □

We believe that there should be a uniform way of constructing half-canonical curves C ⊂ P^{r} for any r ≥ 3 of high genus g >> r and having injective Gaussian maps (though no longer as sections of homogeneous varieties). Together with Theorem 1.2 this prompts us to make the following:

Conjecture 3.4. For any r ≥ 3 and g ≥ \binom{r+2}{2}, there exists a component of S_{g}^{r} of codimension \binom{r+1}{2} inside S_{g}.

The bound g ≥ \binom{r+2}{2} is obtained by comparing the expected dimension 3g−3−\binom{r+1}{2} of S_{g}^{r} with the expected dimension of the Hilbert scheme Hilb_{g−1,g,r} of curves C ⊂ P^{r} of genus g and degree g − 1. We believe that there exists a component of Hilb_{g−1,g,r} consisting entirely of half-canonically embedded curves. To prove the Conjecture it would suffice to construct a smooth half-canonical curve C ⊂ P^{r} of genus g = \binom{r+2}{2} such that H^{1}(C, N_{C/P^{r}}) = 0, that is, Hilb_{g−1,g,r} is smooth at the point [C] and has expected dimension h^{0}(C, N_{C/P^{r}}) = 4(g − 1). Note that for such C, the map Ψ_{C,O_{C}(1)} would be injective, in particular C would not sit on any quadrics. This gives the necessary inequality g ≥ \binom{r+2}{2}. The main difficulty in proving Conjecture 3.4 lies in the fact that the degeneration techniques one normally uses to construct “regular” components of Hilbert schemes of curves, seem to be at odds with the requirement that C be half-canonical.

4. Gieseker-Petri loci

In this section we construct divisorial components of the loci GP_{g,k}^{1}. The method we use is inductive and close in spirit to the one employed in Section 2 to construct components of S_{g}^{r} of expected dimension. We begin by describing a setup that enables us to analyze the following situation: if {L_{b}}_{b∈B^{∗}} and {M_{b}}_{b∈B^{∗}} are two families of line bundles over a 1-dimensional family of smooth curves {X_{b}}_{b∈B^{∗}}, where B^{∗} = B − {b_{0}} with b_{0} ∈ B, we want to describe what happens to the multiplication map

μ_{b} = μ_{b}(L_{b}, M_{b}) : H^{0}(X_{b}, L_{b}) ⊗ H^{0}(X_{b}, M_{b}) → H^{0}(X_{b}, L_{b} ⊗ M_{b})

as X_{b} degenerates to a singular curve of compact type X_{0}.

Suppose first that C is a smooth curve and p ∈ C. We recall that if l = (L, V) is a linear series of type 𝔤_{d}^{r} with L ∈ Pic^{d}(C) and V ⊂ H^{0}(L), the vanishing sequence of l at p

a^{l}(p) : 0 ≤ a_{0}^{l}(p) < … < a_{r}^{l}(p) ≤ d,

is obtained by ordering the set {ord_{p}(σ)}_{σ∈V}. If L and M are line bundles on C and ρ ∈ H^{0}(L) ⊗ H^{0}(M) we write that ord_{p}(ρ) ≥ k, if ρ lies in the span of elements of the form σ ⊗ τ, where σ ∈ H^{0}(L) and τ ∈ H^{0}(M) are such that ord_{p}(σ) + ord_{p}(τ) ≥ k. Let μ_{L,M} : H^{0}(L) ⊗ H^{0}(M) → H^{0}(L ⊗ M) be the multiplication map. We shall use the following observation: suppose {σ_{i}} ⊂ H^{0}(L) and {τ_{j}} ⊂ H^{0}(M) are bases of global sections adapted to the point p ∈ C in the sense that ord_{p}(σ_{i}) = a_{i}^{L}(p) and ord_{p}(τ_{j}) = a_{j}^{M}(p) for all i and j. Then if ρ ∈ Ker(μ_{L,M}) then there must exist distinct pairs of integers (i_{1}, j_{1}) ≠ (i_{2}, j_{2}) such that

ord_{p}(ρ) = ord_{p}(σ_{i_{1}}) + ord_{p}(τ_{j_{1}}) = ord_{p}(σ_{i_{2}}) + ord_{p}(τ_{j_{2}}).

Suppose now that π : X → B is a family of genus g curves over B = Spec(R), with R being a complete DVR with local parameter t, and let 0, η denote the special and the generic point of B respectively. Assume furthermore that X_{η} is smooth and that X_{0} is singular but of compact type. If L_{η} is a line bundle on X_{η} then, as explained in [EH1], there is a canonical way to associate to each component Y of X_{0} a line bundle L^{Y} on X such that deg_{Z}(L_{|_{Z}}^{Y}) = 0 for every component Z of X_{0} different from Y. We set L_{Y} := L_{|_{Y}}^{Y} which is a line bundle on the smooth curve Y.

We fix σ ∈ π_{∗}L_{η} a section on the generic fibre. We denote by α the smallest integer such that t^{α}σ ∈ π_{∗}L^{Y}, that is, t^{α}σ ∈ π_{∗}L^{Y} − tπ_{∗}L^{Y}. Then we set

σ^{Y} := t^{α}σ ∈ π_{∗}L^{Y} and σ_{Y} := σ_{|_{Y}}^{Y} ∈ H^{0}(Y, L_{Y}).

For a different component Z of the special fibre X_{0} meeting Y at a point p, we define similarly L^{Z}, L_{Z}, σ^{Z} and σ_{Z}. If we write σ^{Z} = t^{β}σ^{Y} ∈ π_{∗}L^{Z} for a unique integer β, we have the following compatibility relation between σ_{Y} and σ_{Z} (cf. [EH1], Proposition 2.2):

(3)  deg(L_{Y}) − ord_{p}(σ_{Y}) ≤ β ≤ ord_{p}(σ_{Z}).

An immediate consequence of this is the inequality

ord_{p}(σ_{Y}) + ord_{p}(σ_{Z}) ≥ deg(L_{Y}) = deg(L_{Z}).

Assume from now on that we have two line bundles L_{η} and M_{η} on X_{η} and we choose an element ρ ∈ H^{0}(X_{η}, L_{η}) ⊗_{R_{η}} H^{0}(X_{η}, M_{η}). If Y and Z are components of X_{0} meeting at p as above, we define ρ^{Y} := t^{γ}ρ ∈ H^{0}(X, L^{Y}) ⊗_{R} H^{0}(X, M^{Y}), where γ is the minimal integer with this property. We have a similar definition for ρ^{Z} ∈ H^{0}(X, L^{Z}) ⊗_{R} H^{0}(X, M^{Z}). Between the sections ρ^{Y} and ρ^{Z} there is a relation ρ^{Z} = t^{α}ρ^{Y} for a uniquely determined integer α. To determine α we proceed as follows: we choose bases of sections {σ_{i} = σ_{i}^{Y} } for H^{0}(X, L^{Y}) and {τ_{j} = τ_{j}^{Y} } for H^{0}(X, M^{Y}) such that ord_{p}(σ_{i,Y}) = a_{i}^{L_{Y}} (p) and ord_{p}(τ_{j,Y}) = a_{j}^{M_{Y}} (p), for all relevant i and j (cf. e.g. [EH1], Lemma 2.3, for the fact that this can be done). Then there are integers α_{i} and β_{j} defined by σ_{i}^{Z} = t^{α_{i}}σ_{i} and τ_{j}^{Z} = t^{β_{j}}τ_{j}. To obtain a formula for the integer α we write ρ^{Y} = ∑_{i,j} f_{ij}σ_{i} ⊗ τ_{j}, where f_{ij} ∈ R. We have the identity

ρ^{Z} = ∑_{i,j}(t^{α−α_{i}−β_{j}}f_{ij})(t^{α_{i}}σ_{i}) ⊗ (t^{β_{j}}τ_{j}),

from which we easily deduce that α = max_{i,j}{α_{i} + β_{j} − ν(f_{ij})}, where ν denotes the valuation on R (see also [EH2], Lemma 3.2).

Lemma 4.1. With the above notations, if ρ_{Y} := ρ_{|_{Y}}^{Y} ∈ H^{0}(Y, L_{Y}) ⊗ H^{0}(Y, M_{Y}) and ρ_{Z} := ρ_{|_{Z}}^{Z} ∈ H^{0}(Z, L_{Z}) ⊗ H^{0}(Z, M_{Z}), then

ord_{p}(ρ_{Y}) + ord_{p}(ρ_{Z}) ≥ deg(L_{Y}) + deg(M_{Y}).

Proof. By definition, there exists a pair of indices (i_{1}, j_{1}) such that ν(f_{i_{1}j_{1}}) = 0 and

ord_{p}(ρ_{Y}) = ord_{p}(σ_{i_{1},Y}) + ord_{p}(σ_{j_{1},Y})

and clearly α ≥ α_{i_{1}} + β_{j_{1}}. To get an estimate on ord_{p}(ρ_{Z}) we only have to take into account the pairs of indices (i, j) for which α_{i} + β_{j} = α + ν(f_{ij}) ≥ α_{i_{1}} + β_{j_{1}}. For at least one such pair (i, j) we have that

ord_{p}(ρ_{Z}) = ord_{p}(t^{α_{i}}σ_{i,Z}) + ord_{p}(t^{β_{j}}τ_{j,Z}) ≥ α_{i} + β_{j}.

On the other hand, by applying (3) we can write

ord_{p}(ρ_{Y}) = ord_{p}(σ_{i_{1},Y}) + ord_{p}(τ_{j_{1},Y}) ≥ deg(L_{Y}) + deg(M_{Y}) − α_{i_{1}} − β_{j_{1}},

whence we finally have that ord_{p}(ρ_{Z}) + ord_{p}(ρ_{Y}) ≥ deg(L_{Y}) + deg(M_{Y}). □

We now fix integers g and k such that g ≥ 4 and (g + 2)/2 ≤ k ≤ g − 1 and consider the locus GP_{g,k}^{1} of curves [C] ∈ M_{g} for which the Gieseker-Petri Theorem fails for a base

point free pencil 𝔤_{k}^{1}. We denote by \overline{GP} _{g,k}^{1} the closure of GP_{g,k}^{1} in \overline{M}_{g} and we study GP_{g,k}^{1}

inductively by understanding the intersection \overline{GP} _{g,k}^{1} ∩ Δ_{1}.

Definition 4.2. For a smooth curve C of genus g, a Gieseker-Petri (gp)_{k}^{1}-relation consists of a linear series (L, V) ∈ G_{k}^{1}(C), V ⊂ H^{0}(L), together with an element

ρ ∈ PKer{μ_{0}(V) : V ⊗ H^{0}(K_{C} ⊗ L^{−1}) → H^{0}(K_{C})}.

If C = C_{1}∪_{p}C_{2} is of compact type with C_{1} and C_{2} smooth of genus i and g−i respectively, a (gp)_{k}^{1}-relation on C is a collection (l, m, ρ_{1}, ρ_{2}), where l = {(L_{C_{1}}, V_{C_{1}}), (L_{C_{2}}, V_{C_{2}})} is a limit 𝔤_{k}^{1} on C, m = {(M_{C_{1}} = K_{C_{1}}(2(g − i)p) ⊗ L_{C_{1}}^{−1}, W_{1}), (M_{C_{2}} = K_{C_{2}}(2ip) ⊗ L_{C_{2}}^{−1}, W_{2})} is a limit 𝔤_{2g−2−k}^{g−k} on C, and elements

ρ_{1} ∈ PKer{V_{C_{1}} ⊗ W_{C_{1}} → H^{0}(K_{C_{1}}(2(g − i)p))}, ρ_{2} ∈ PKer{V_{C_{2}} ⊗ W_{C_{2}} → H^{0}(K_{C_{2}}(2ip))}

satisfying the relation ord_{p}(ρ_{1}) + ord_{p}(ρ_{2}) ≥ 2g − 2.

For a curve C of compact type, we denote by Q_{k}^{1}(C) the variety of (gp)_{k}^{1}-relations on C together with the scheme structure coming from its natural description as a determinantal variety. The discussion above shows that if [C] ∈ \overline{GP} _{g,k}^{1} then Q_{k}^{1}(C) ≠ ∅. Our strategy is to construct (gp)_{k}^{1}-relations on certain singular curves and prove that they can be deformed to nearby smooth curves filling up a divisor in \overline{M}_{g}. The most important technical result of this section is the construction of the moduli space of (gp)_{k}^{1}-relations over the versal deformation space of a curve of compact type inside the divisor Δ_{1}:

Theorem 4.3. Fix integers g ≥ 4 and k such that (g + 2)/2 ≤ k ≤ g − 1. Let C be a smooth curve of genus g − 1, p ∈ C and X_{0} := C ∪_{p} E, where E is an elliptic curve. We denote by π : X → B the versal deformation space of X_{0}, with X_{0} = π^{−1}(0) and 0 ∈ B. Then there exists a scheme Q_{k}^{1} → B, quasi-projective over B and compatible with base change, such that the fibre over b ∈ B parametrizes (gp)_{k}^{1}-relations over X_{b}. Furthermore each component of Q_{k}^{1} has dimension ≥ dim(B) − 1 = 3g − 4.

Proof. The scheme Q_{k}^{1} is going to be the disjoint union of subschemes where the vanishing sequences of the aspects of the two underlying limit linear series of a (gp)_{k}^{1}-relation are also specified. We will prove the existence for the component corresponding to vanishing sequences (1, 2) and (k − 2, k − 1) for the limit 𝔤_{k}^{1} and (1, 2, …, g − k + 1) and (g − 3, g − 2, …, 2g −3−k) for the limit 𝔤_{2g−2−k}^{g−k} respectively. The construction is entirely similar for the other compatible vanishing sequences. In our proof we will use Theorem 3.3 in [EH1] where a moduli space of limit linear series over the versal deformation space of a curve of compact type is constructed.

We start by setting some notations. We denote by Δ ⊂ B the “boundary” divisor corresponding to curves in which the node p is not smoothed. We denote by C_{p} and E_{p} the closures in X of the components of π^{−1}(Δ) containing C − {p} and E − {p} respectively. By shrinking B if necessary we can assume that O_{X}(C_{p} + E_{p}) = O_{X}. We denote by π_{P^{C}} : P^{C} → B the relative Picard variety corresponding to the family X → B such that for b ∈ Δ and π^{−1}(b) = X_{b} = C_{b} ∪ E_{b} with C_{b} ⊂ C_{p} and E_{b} ⊂ E_{p}, the fibre of P^{C} over b consists of line bundles L_{b} on X_{b} with deg_{C_{b}}(L_{b}) = k and deg_{E_{b}}(L_{b}) = 0. Interchanging the role of C and E we get another Picard variety P^{E} → B and tensoring with O_{X}(kC_{p}) gives an isomorphism P^{C} → P^{E}. We denote by P the inverse limit of P^{C} and P^{E} under this isomorphism. For b ∈ B and any line bundle L on X_{b}, we define two new line bundles L_{C} and L_{E} as follows: if b ∈ B − Δ then L_{C} = L_{E} = L. If b ∈ Δ and X_{b} = C_{b} ∪_{q} E_{b}, then L_{C} is the restriction to C of the unique line bundle on X_{b} obtained from L by tensoring with a divisor based at q and whose restriction to E_{b} is of degree 0 (and a similar definition for L_{E} with C and E reversed). Proceeding in a a way identical to [EH1], pp. 356-360, we construct a space of compatible frames ϕ : F → B factoring through π_{P} : P → B, and which parametrizes objects

x = {b, L, (σ_{i}^{C})_{i=0,1}, (σ_{i}^{E})_{i=0,1}, (τ_{j}^{C})_{j=0,...,g−k}, (τ_{j}^{E})_{j=0,...,g−k}},

where b ∈ B, L is a line bundle of degree k on X_{b}, (σ_{i}^{C}) (resp. (σ_{i}^{E})) is a projective frame inside H^{0}(L_{C}) (resp. H^{0}(L_{E})), while (τ_{j}^{C}) (resp. (τ_{j}^{E})) is a projective frame inside H^{0}((ω_{X_{b}} ⊗ L^{−1})_{C}) (resp. H^{0}((ω_{X_{b}} ⊗ L^{−1})_{E})), subject to the following identifications: if b ∈ B − Δ, so X_{b} is smooth and L_{E} = L_{C} = L, then we identify σ_{i}^{C} = σ_{1−i}^{E} for i = 0, 1 and τ_{j}^{C} = τ_{g−k−j}^{E} for j = 0, …, g − k (that is, there are only two frames, one inside H^{0}(L), the other inside H^{0}(K_{X_{b}} ⊗ L^{−1})). If b ∈ Δ and X_{b} = C_{b} ∪_{q} E_{b} then we require that ord_{q}(σ_{i}^{C}) ≥ i + 1, ord_{q}(σ_{i}^{E}) ≥ k − 2 + i for i = 0, 1, while ord_{q}(τ_{j}^{C}) ≥ j + 1 and ord_{q}(τ_{j}^{E}) ≥ g + j − 3. In this latter case l = {(L_{C}, < σ_{i}^{C} >_{i}), (L_{E}, < σ_{i}^{E} >_{i})} is a limit 𝔤_{k}^{1} and m = {((ω_{X} ⊗ L^{−1})_{C}, < τ_{j}^{C} >_{j}), ((ω_{X} ⊗ L^{−1})_{E}, < τ_{j}^{E} >_{j})} is a limit 𝔤_{2g−2−k}^{g−k} on X_{b}.

The scheme F is determinantal and each of its components has dimension ≥ dim(B)+ g+2+(g−k+1)(g−k−2), which is consistent with the naive dimension count for the fibre over b ∈ B − Δ. We also have tautological line bundles σ̃_{i}^{C}, σ̃_{i}^{E}, σ̃_{j}^{C} and σ̃_{j}^{E} over F, with fibres over each point being the 1-dimensional vector space corresponding to the frame denoted by the same symbol. For 2 ≤ i ≤ g − k + 2, we consider the rang g vector bundle bundle Ψ_{i} := π_{∗}(ω_{X/B} ⊗ O_{X}(iC_{p})); hence Ψ_{i}(b) = H^{0}(X_{b}, L_{b}) for b ∈ B − Δ, while for b ∈ Δ the fibre Ψ_{i}(b) consists of those sections in H^{0}(K_{C_{b}}(−(i−1)q))⊕H^{0}(O_{E_{b}}((i+1)q)) that are compatible at the node q.

For 1 ≤ i ≤ g − k we define a subscheme G_{i} of F by the equations

(4)  σ̃_{0}^{C} · τ̃_{i}^{C} = σ̃_{1}^{C} · τ̃_{i−1}^{C} and σ̃_{1}^{E} · τ̃_{g−k−i}^{E} = σ̃_{0}^{E} · τ̃_{g−k−i+1}^{E}.

Here by (σ̃_{α}^{C} · τ̃_{β}^{C})(x) we denote the element in PH^{0}((ω_{X_{b}})_{C}) obtained by multiplying representatives of σ̃_{α}^{C}(x) and of τ̃_{β}^{C}(x) for each x ∈ F, b = ϕ(x). To make more sense of (4), for each x ∈ F the element ((σ̃_{0}^{C} · τ̃_{i}^{C})(x), (σ̃_{1}^{E} · τ̃_{g−k−i}^{E})(x)) gives rise canonically to a point in P((ϕ^{∗}Ψ_{i+1})(x)) and abusing the notation we can consider (σ̃_{0}^{C} · τ̃_{i}^{C}, σ̃_{1}^{E} · τ̃_{g−k−i}^{E}) and (σ̃_{1}^{C} · τ̃_{i−1}^{C}, σ̃_{0}^{E} · τ̃_{g−k−i+1}^{E}) as sections of the P^{g−1} bundle P(ϕ^{∗}Ψ_{i+1})) → F. Then G_{i} is the locus in F where these sections coincide and therefore each component of G_{i} has dimension ≥ dim(F) − g + 1.

We define Q_{k}^{1} as the union of the scheme theoretic images of G_{i} for 1 ≤ i ≤ g − k under the map

G_{i} ∋ x ↦^{χ_{i}} (b, l, m, ρ_{1} = (σ_{0}^{C} ⊗ τ_{i}^{C} − σ_{1}^{C} ⊗ τ_{i−1}^{C}), ρ_{2} = (σ_{1}^{E} ⊗ τ_{g−k−i}^{E} − σ_{0}^{E} ⊗ τ_{g−k−i+1}^{E})),

where we recall that l and m denote the underlying limit 𝔤_{k}^{1} and 𝔤_{2g−2−k}^{g−k} respectively. From the base point free pencil trick applied on both C_{b} and E_{b}, it is easy to see that Q_{k}^{1} contains all (gp)_{k}^{1}-relations on the curves X_{b} = C_{b} ∪_{q} E_{b}, the points coming from G_{i} corresponding to those (b, l, m, ρ_{1}, ρ_{2}) for which ord_{q}(ρ_{1}) ≥ i + 2 and ord_{q}(ρ_{2}) ≥ 2g − i − 4.

We are left with estimating dim(Q_{k}^{1}): having fixed (b, l, m, ρ_{1}, ρ_{2}) inside χ_{i}(G_{i}), there are two cases to consider depending on whether X_{b} is smooth or not. In each case we obtain the same estimate for the fibre dimension of χ_{i} but here we only present the case b ∈ Δ, when X_{b} = C_{b} ∪_{q} E_{b}. We have a one dimensional family of choices for each of (σ_{0}^{C}, σ_{1}^{C}) and (σ_{0}^{E}, σ_{1}^{E}), and after choosing these, (τ_{i}^{C}, τ_{i−1}^{C}) and (τ_{g−k−i}^{E}, τ_{g−k−i+1}^{E}) are uniquely determined (again, use the base point free pencil trick). For choosing the remaining τ_{α}^{C}, α ≠ i, i − 1 we have a (g − k)(g − k + 1)/2 − (2g − 2k − 2i + 1)-dimensional family of possibilities, while for τ_{β}^{E}, β ≠ g−k−i, g−k−i+1 we get another (g−k)(g−k+1)/2−2i+1 dimensions. Adding these together we get that each component of Q_{k}^{1} has dimension ≥ 3g − 4. □

We can now prove Theorem 1.4. More precisely we have the following inductive result:

Theorem 4.4. Fix integers g, k such that g ≥ 4 and (g + 2)/2 ≤ k ≤ g − 1. Suppose GP_{g−1,k−1}^{1} has a divisorial component Z for which a general [C] ∈ Z is such that there exists a 0-dimensional component of Q_{k−1}^{1}(C) whose general point corresponds to a base point free 𝔤_{k−1}^{1}. Then GP_{g,k}^{1} has a divisorial component Z′, for which a general curve [C′] ∈ Z′ is such that Q_{k}^{1}(C′) has a 0-dimensional component corresponding to a base point free 𝔤_{k}^{1}. Moreover, if ϵ : \overline{M}_{g−1,1} → \overline{M}_{g−1} is the forgetful morphism, then using the identification Δ_{1} = \overline{M}_{g−1,1} × \overline{M}_{1,1}, we have that \overline{Z}′ ∩ Δ_{1} ⊃ ϵ^{∗}(Z) × \overline{M}_{1,1}.

Proof. We choose a general curve [C] ∈ Z ⊂ GP_{g−1,k−1}^{1}, a general point p ∈ C and we set X_{0} := C ∪_{p} E, where E is an elliptic curve. By assumption, there exists a base point free (A, V) ∈ G_{k−1}^{1}(C) and ρ ∈ PKer(μ_{0}(V)) such that dim_{(A,V,ρ)}Q_{k−1}^{1}(C) = 0. In particular Ker(μ_{0}(V)) is 1-dimensional and h^{0}(A) = 2. Let π : X → B be the versal deformation space of X_{0}, Δ ⊂ B the boundary divisor corresponding to singular curves, and we consider the scheme ν : Q_{k}^{1} → B parametrizing (gp)_{k}^{1}-relations, which was constructed in Theorem 4.3. We construct a (gp)_{k}^{1}-relation z = (l, m, ρ_{1}, ρ_{2}) on X_{0} as follows: the C-aspect of the limit 𝔤_{k}^{1} denoted by l is obtained by adding p as a base point to (A, V), while the E-aspect of l is constructed by adding (k − 2)p as a base locus to |O_{E}(p + q)|, where q ∈ E − {p} satisfies 2(p − q) ≡ 0. Thus the vanishing sequences a^{l_{C}}(p) and a^{l_{E}}(p) are (1, 2) and (k − 2, k − 1) respectively. The C-aspect of the limit 𝔤_{2g−2−k}^{g−k} we denote by m, is the complete linear series |M_{C}| = |K_{C}(p) ⊗ A^{−1}| which by Riemann-Roch has vanishing sequence (1, 2, …, g − k + 1) at p. Finally the E-aspect of m is the subseries of |O_{E}((2g − 1 − k)p − q)| with vanishing (g − 3, g − 2, …, 2g − k − 3) at p. From the base point free pencil trick it follows that we can choose uniquely the relations ρ_{C} on C and ρ_{E} on E such that ord_{p}(ρ_{C}) = 3 and ord_{p}(ρ_{E}) = 2g − 5 (we use that h^{0}(C, K_{C} ⊗ A^{−2}) = dim(Ker(μ_{0}(V)) = 1 by assumption, hence ρ_{C} is essentially ρ up to subtracting the base locus).

From Theorem 4.3, every component of Q_{k}^{1} passing through z has dimension ≥ 3g − 4. On the other hand we claim that every component of ν^{−1}(Δ) passing through z has dimension ≤ 3g − 5 and that z is an isolated point in ν^{−1}([X_{0}]). Assuming this for a moment, we obtain that z is a smoothable (gp)_{k}^{1}-relation in the sense that there is a component of Q_{k}^{1} through z which meets ν^{−1}(B − Δ). From this it follows that [X_{0}] ∈ \overline{GP} _{g,k}^{1} ∩ Δ_{1}. Since by construction the curves [X_{0}] fill up a divisor inside Δ_{1}, we conclude that GP_{g,k}^{1} has a divisorial component Z′ such that \overline{Z}′ ∩ Δ_{1} ⊃ ϵ^{∗}(Z) × \overline{M}_{1,1}.

Furthermore, because the vanishing sequences of the C and E-aspects of l add up precisely to k, every 𝔤_{k}^{1} on a smooth curve “near”X_{0} which specializes to l, is base point free (cf. [EH1], Proposition 2.5). We obtain that a point z′ ∈ ν^{−1}(B − Δ) near z will satisfy dim_{z′}(Q_{k}^{1}) = 3g − 4 and will correspond to a smooth curve [C′] ∈ GP_{g,k}^{1}, satisfying all the required conditions.

We return now to the estimate for dim_{z}(ν^{−1}(Δ)): we consider a curve X_{b} = C_{b} ∪_{q} E_{b} with b ∈ Δ, and let (l, m, ρ_{C_{b}}, ρ_{E_{b}}) ∈ ν^{−1}(b). Hence the underlying limit linear series l and m have vanishing sequences a^{l_{C_{b}}} (q) = (1, 2), a^{l_{E_{b}}} (q) = (k − 2, k − 1) and a^{m_{C_{b}}} (q) = (1, 2, …, g − k + 1), a^{m_{E_{b}}} (q) = (g − 3, g − 2, …, 2g − 3 + k) respectively.

Clearly ord_{q}(ρ_{C_{b}}) ≥ 3(= 1 + 2 = 2 + 1). We set (A_{b} = L_{C_{b}}(−q), V_{C_{b}} := V_{C_{b}}(−q)) ∈ G_{k−1}^{1}(C_{b}) and (B_{b} = L_{E_{b}}(−(k − 2)q), V_{E_{b}} := V_{E_{b}}(−(k − 2)q) ∈ G_{2}^{1}(E_{b}). We claim that in fact ord_{q}(ρ_{C_{b}}) = 3 and therefore ord_{q}(ρ_{E_{b}}) = 2g − 5(= k − 2 + (2g − 3 − k) = k − 1 + (2g − 4 − k)). Indeed assuming that ord_{q}(ρ_{C_{b}}) ≥ 4, from the base point free pencil trick we have that h^{0}(C_{b}, K_{C_{b}} ⊗ A_{b}^{−2}(−q)) ≥ 1. But h^{0}(C, K_{C} ⊗ A^{−2}(−p)) = 0 (use the assumption on C and the fact that p ∈ C is a general point), which implies that we can assume that h^{0}(C_{b}, K_{C_{b}} ⊗ A_{b}^{−2}(−q)) = 0 for any point in a component of ν^{−1}(Δ) passing through z.

After subtracting base points ρ_{C_{b}} can be viewed as an element in the projectivization of the kernel of the map μ_{0}(V_{C_{b}}) : V_{C_{b}}⊗H^{0}(K_{C_{b}}⊗A_{b}^{−1}) → H^{0}(K_{C_{b}}), while ρ_{E_{b}} is in the projectivized kernel of the map μ_{0}(V_{E_{b}}) : H^{0}(E_{b}, B_{b}) ⊗ H^{0}(E_{b}, B_{b}^{−1}(4q)) → H^{0}(E_{b}, O_{E_{b}}(4q)). In other words [C_{b}] ∈ GP_{g−1,k−1}^{1} and from the base point free pencil trick we get that H^{0}(E_{b}, O_{E_{b}}(4q) ⊗ B_{b}^{−2}) ≠ 0, which leaves only finitely many choices for B_{b} and ρ_{E_{b}}. It follows that dim_{z}ν^{−1}(Δ) ≤ dim_{[C]}(GP_{g−1,k−1}^{1}) + 1 + 1 = 3g − 5. □

Proof of Theorem 1.4. We apply Theorem 4.4 starting with the base case k ≥ 3, g = 2k−2. In this situation the locus GP_{2k−2,k}^{1} is a divisor in M_{g} which can also be viewed as the

branch locus of the map to M_{g} from the Hurwitz scheme of coverings C ⟶^{k:1} P^{1} having a genus g source curve (cf. [EH3], Section 5). The locus of [C] ∈ M_{g} having infinitely many base point free 𝔤_{k}^{1}’s is of codimension ≥ 2, hence by default the general point of GP_{2k−2,k}^{1} corresponds to a curve with finitely many (A, V) ∈ G_{k}^{1}(C). The fact that for each of these pencils, dim Ker(μ_{0}(V)) ≤ 1, also follows from [EH3]. Applying now Theorem 4.4 repeatedly we construct divisorial components of GP_{2k−2+a,k+a}^{1} for all k ≥ 3 and a ≥ 0. It is easy to check that in this way we fill all the cases claimed in the statement. □

One could also define the loci GP_{g,k}^{1} for k ≤ (g + 1)/2. In this case GP_{g,k}^{1} coincides with the locus of k-gonal curves, which is irreducible of dimension 2g + 2k − 5. When g is odd, GP_{g,(g+1)/2}^{1} is the well-known Brill-Noether divisor on M_{g} introduced by Harris and Mumford (see [EH3]). The Gieseker-Petri divisors GP_{g,k}^{1} with k ≥ (g + 2)/2 that we introduced, share certain properties with the Brill-Noether divisor. For instance the following holds (compare with [EH3], Proposition 4.1):

Proposition 4.5. We denote by j : \overline{M}_{2,1} → \overline{M}_{g} the map obtained by attaching a fixed general pointed curve (C_{0}, p) of genus g − 2. Then for (g + 1)/2 ≤ k ≤ g − 1 we have the relation j^{∗}(\overline{GP} _{g,k}^{1}) = q\overline{W}, where q ≥ 0 and W is the divisor of Weierstrass points on M_{2,1}.

Sketch of proof. We can degenerate (C_{0}, p) to a string of elliptic curves (E_{1} ∪…∪E_{g−2}, p), where p lies on the last component E_{g−2}. We assume that for all 2 ≤ i ≤ g − 2, the points of attachment between E_{i−1} and E_{i} are general. Fix now [B, p] ∈ M_{2,1} and assume that [X_{0} := C_{0} ∪_{p} B] ∈ \overline{GP} _{g,k}^{1}. We denote by (l_{B}, m_{B}, ρ_{B}) the B-aspect of a (gp)_{k}^{1}-relation on X_{0}. Then using the setup described at the beginning of Section 4 we obtain that ord_{p}(ρ_{B}) ≥ 2g − 4. Since l_{B} is a 𝔤_{k}^{1} and m_{B} is a 𝔤_{2g−2−k}^{g−k}, the only way this could happen is if a^{l_{B}}(p) = (k − 2, k) and a^{m_{B}}(p) = (…, 2g − 4 − k, 2g − 2 − k), which implies that h^{0}(O_{B}(2p)) ≥ 2, that is, [B, p] ∈ W. □

Remark 4.6. Using methods developed in this section we can also prove the following result useful for the computation of the class [\overline{GP} _{g,k}^{1}] ∈ Pic(\overline{M}_{g}): if ϵ : \overline{M}_{g−1,1} → \overline{M}_{g−1} is the forgetful morphism and ϕ : \overline{M}_{g−1,1} → \overline{M}_{g} denotes the map attaching an elliptic tail at the marked point, then ϕ^{∗}(\overline{GP} _{g,k}^{1}) is set-theoretically the union of two divisors: ϵ^{∗}(\overline{GP} _{g−1,k−1}^{1}), and the closure D in \overline{M}_{g−1,1} of the locus of curves [C, p] ∈ M_{g−1,1} for which there exists a base point free A ∈ W_{k}^{1}(C) such that h^{0}(C, A(−2p)) ≥ 1 and the multiplication map H^{0}(C, A) ⊗ H^{0}(K_{C} ⊗ A^{∨}(2p)) → H^{0}(K_{C}(2p)) is not injective. It is natural to view D as a “pointed” Gieseker-Petri divisor on \overline{M}_{g,1}.

We consider now the moduli space S_{g,n} of n-pointed spin curves of genus g and its subvariety S_{g,n}^{r} consisting of elements (C, p_{1}, …, p_{n}, L), where [C, p_{1}, …, p_{n}] ∈ M_{g,n} and L ∈ Pic^{k}(C) is a line bundle such that L^{2} ⊗ O_{C}(p_{1} + ⋯ + p_{n}) = K_{C} and h^{0}(L) ≥ r + 1. Of course we assume that 2k + n = 2g − 2. The base point free pencil trick relates these loci to the loci GP_{g,k}^{i} we introduced before. Precisely, if f : S_{g,n} → M_{g} is given by [C, p_{1}, …, p_{n}, L] ↦ [C], then f(S_{g,n}^{1}) = GP_{g,k}^{1}.

We now look at the divisor Z ⊂ GP_{g,k}^{1} constructed in Theorem 4.4. The condition that for a general [C] ∈ Z, the scheme Q_{k}^{1}(C) has a 0-dimensional component with general point corresponding to a base point free 𝔤_{k}^{1}, can be translated into saying that f^{−1}[C] has a zero-dimensional component. We obtain in this way that there exists a component Y of S_{g,n}^{1} of dimension 3g − 4 such that f(Y) = Z. This proves Corollary 1.5.

5. Injectivity of Gaussian maps

We are going to prove Theorem 1.3 by degeneration. Our proof is inspired by the work of Eisenbud and Harris on the Gieseker-Petri Theorem (cf. [EH2]). Suppose we have a family of genus g curves π : X → B over a base B = Spec(R) with R being a complete DVR with local parameter t and let 0 and η respectively, denote the special and the generic point of R. Assume furthermore that X_{η} is smooth and that X_{0} is a curve of compact type consisting of a string of components of which g of them E_{1}, … E_{g}, are elliptic curves, while the rest are rational curves, glued in such a way that the stable model of X_{0} is the curve E_{1} ∪_{p_{1}} E_{2} ∪_{p_{2}} E_{3} ∪ … ∪ E_{g−1} ∪_{p_{g−1}} E_{g}. Slightly abusing the notation, for 2 ≤ i ≤ g − 1 we will consider p_{i−1} and p_{i} ∈ E_{i} to be the points of attachment of E_{i} to \overline{X_{0} − E_{i}} and we will choose X_{0} in such a way that p_{i} − p_{i−1} is not a torsion class in Pic^{0}(E_{i}).

We proceed by contradiction and assume that there exists a line bundle L_{η} on X_{η} of degree d, together with a non-zero element

ρ_{η} ∈ Ker{ψ_{Lη} : ∧^{2}H^{0}(X_{η}, L_{η}) → H^{0}(X_{η}, Ω_{X_{η}}^{1} ⊗ L_{η}^{2})}.

(Note that we because the shape of X_{0} does not change if we blow-up the surface X, we can assume that we have a bundle L_{η} on X_{η} rather than on the geometric generic fibre X\overline{η}.) As in Section 4, for each component Y of X_{0} we have the line bundle L^{Y} on X extending L_{η} and having degree 0 restriction to all components Z ≠ Y of X_{0} and we set L_{Y} := L_{|Y}^{Y}. Starting with ρ_{η} ∈ ∧^{2}π_{∗}(L_{η}) we obtain elements ρ^{Y} = t^{α}ρ_{η} ∈ ∧^{2}π_{∗}(L^{Y}) − t ∧^{2} π_{∗}(L^{Y}) for uniquely determined integers α, and we define ρ_{Y} := ρ_{|Y}^{Y} ∈ ∧^{2}H^{0}(Y, L_{Y}).

Lemma 5.1. For each component Y of X_{0} we have that

ρ_{Y} ∈ Ker{ψ_{L_{Y}} : ∧^{2}H^{0}(Y, L_{Y}) → H^{0}(Y, Ω_{Y}^{1} ⊗ L_{Y}^{2} )}.

Proof. We use the commutative diagram

∧^{2}H^{0}(X_{0}, L_{|X_{0}}^{Y})  ⟶^{res} ∧^{2}H^{0}(Y, L_{Y})
↓^{ψ_{L{Y \atop |X_{0}}}}  ↓^{ψ_{L_{Y}}}
H^{0}(X_{0}, Ω_{X_{0}}^{1} ⊗ L_{|X_{0}}^{Y^{⊗^{2}}} ) ⟶^{res} H^{0}(Y, Ω_{Y}^{1} ⊗ L_{Y}^{2} )

and keep in mind that the upper restriction map is injective. □

We will use the following observation (similar to the one for ordinary multiplication maps): let C be a smooth curve, p ∈ C and M a line bundle on C. If ρ ∈ Ker(ψ_{M}) and {σ_{i}} is a basis of H^{0}(M) such that ord_{p}(σ_{i}) = a_{i}^{M}(p) = a_{i}, then there are distinct pairs of integers (i_{1}, j_{1}) ≠ (i_{2}, j_{2}) with i_{1} ≠ j_{1} and i_{2} ≠ j_{2}, such that ord_{p}(ρ) = ord_{p}(σ_{i_{1}}) + ord_{p}(σ_{j_{1}}) = ord_{p}(σ_{i_{2}}) + ord_{p}(σ_{j_{2}}). This follows from a local calculation: if t is a local parameter for C at p, then

ψ_{M}(σ_{i} ∧ σ_{j}) = ((a_{i} − a_{j})t^{a_{i}+a_{j}−1} + h.o.t.)dt,

and since ψ_{M}(ρ) = 0, the number ord_{p}(ρ) must be attained for at least two pairs (i, j).

Proposition 5.2. Suppose Y and Z are two components of X_{0} meeting at a point q and let p be a general point on Y. We have the following inequalities:

(1) ord_{q}(ρ_{Z}) ≥ ord_{p}(ρ_{Y}).

(2) If Y is one of the elliptic components of X_{0}, then ord_{q}(ρ_{Z}) ≥ ord_{p}(ρ_{Y}) + 2.

Proof. Although (1) is essentially Proposition 3.1 from [EH2] we will briefly go through the proof and in doing so we will also prove (2). We pick a basis {σ_{i} = σ_{i}^{Y} } of π_{∗}(L^{Y}) such that ord_{p}(σ_{i|Y}) = a_{i}^{L}(p) and for which there are integers α_{i} with the property that {σ_{i}^{Z} = t^{α_{i}}σ_{i}} form a basis for H^{0}(X, L^{Z}) (see [EH1], Lemma 2.3 for the fact that such a basis can be chosen). We then write ρ^{Y} = ∑_{i≠j} f_{ij}σ_{i} ∧ σ_{j}, with f_{ij} ∈ R, and we can express ρ^{Z} = t^{γ}ρ^{Y}, where γ = max_{i≠j}{α_{i} + α_{j} − ν(f_{ij})}. Here ν denotes the valuation on the ring R. From the definition of γ it follows that there exists a pair (i, j), i ≠ j, with γ = α_{i} + α_{j} − ν(f_{ij}), such that we have a string of inequalities

(5)  ord_{q}(ρ_{Z}) = ord_{q}(σ_{i|Z}^{Z}) + ord_{q}(σ_{j|Z}^{Z}) ≥ α_{i} + α_{j} ≥ γ,

(see also Section 4). On the other hand there exists a pair (i′, j′), i′ ≠ j′ such that ν(f_{i′j′}) = 0, for which we can write the inequalities (6) ord_{p}(ρ_{Y}) = ord_{p}(σ_{i′|Y})+ord_{p}(σ_{j′|Y}) ≤ (d−ord_{q}(σ_{i′|Y}))+(d−ord_{q}(σ_{j′|Y})) ≤ α_{i′} +α_{j′} ≤ γ.

Combining (5) and (6) we get the first part of the Proposition. When moreover the curve Y is elliptic, since ψ_{L_{Y}} (ρ_{Y}) = 0, there must at least two pairs (i_{1}, j_{1}) and (i_{2}, j_{2}) for which (6) holds. On the other hand p − q ∈ Pic^{0}(Y) can be assumed not to be a torsion class, and we obtain that ord_{p}(σ_{i|Y}) + ord_{q}(σ_{i|Y}) ≤ d − 1 for all indices i except at most one. This and the fact that the vanishing orders ord_{p}(σ_{i|Y}) are all distinct, quickly lead to the inequality ord_{q}(ρ_{Z}) ≥ γ ≥ ord_{p}(ρ_{Y}) + 2. □

A repeated application of Proposition 5.2 gives the following result:

Proposition 5.3. Let X_{0} be the curve described in the degeneration above and which has the stable model ∪_{i=1}^{g}E_{i}, where E_{i} are elliptic curves. We denote by p_{i−1} and p_{i} the points of attachment of E_{i} to the rest of X_{0}. If ψ_{L_{η}}(ρ_{η}) = 0, then ord_{p_{g−1}}(ρ_{E_{g}}) ≥ ord_{p_{1}}(ρ_{E_{2}}) + 2g − 4.

We are now in a position to prove Theorem 1.3. In fact we have a more general result:

Theorem 5.4. For a general genus g curve C and for any line bundle L on C of degree d ≤ a + g + 2, where a ≥ 0, we have that dim Ker(ψ_{L}) ≤ a(a + 1). In particular, if d ≤ g + 2 then ψ_{L} is injective.

Proof. We apply Proposition 5.3 and degenerate C to X_{0} = E_{1} ∪…∪E_{g}. We assume that Ker(ψ_{C,L}) is at least 1 + a(a + 1)-dimensional. Then dim Ker(ψ_{X_{0},L_{|X_{0}}^{E_{2}}} ) ≥ 1 + a(a + 1)

and since the restriction map ∧^{2}H^{0}(X_{0}, L_{|X_{0}}^{E_{2}}) → ∧^{2}H^{0}(E_{2}, L_{E_{2}}) is injective we obtain that Ker(ψ_{E_{2},L_{E_{2}}} ) is at least 1 + a(a + 1)-dimensional as well. For simplicity let us denote E_{2} = E, L_{E_{2}} = L and p_{1} = p ∈ E_{2} (recall that p_{1} ∈ E_{2} ∩ E_{1}).

If we choose a basis {σ_{i}} of H^{0}(L) adapted to the point p, then as we noticed before for each ρ ∈ Ker(ψ_{L}) there will be at least two distinct pairs of integers (i_{1}, j_{1}) ≠ (i_{2}, j_{2}) where i_{1} ≠ j_{1}, i_{2} ≠ j_{2} such that

ord_{p}(ρ) = ord_{p}(σ_{i_{1}}) + ord_{p}(σ_{j_{1}}) = ord_{p}(σ_{i_{2}}) + ord_{p}(σ_{j_{2}}).

The vanishing sequence a^{L_{E_{1}}} (p) is ≤ (…, d − 3, d − 2, d), hence the vanishing sequence of L = L_{E_{2}} at p is ≥ (0, 2, 3, 4, 5, …), which yields that ord_{p}(ρ) ≥ 5(= 0 + 5 = 2 + 3) for every ρ ∈ Ker(ψ_{L}). Since dim Ker(ψ_{L}) ≥ 1 + a(a + 1), there is a subspace W_{1} ⊂ Ker(ψ_{L}) of dimension ≥ a(a + 1) such that ord_{p}(ρ) ≥ 6(= 0 + 6 = 2 + 4) for each ρ ∈ W_{1}.

Repeating this reasoning for W_{1} instead of Ker(ψ_{L}) we obtain a subspace W_{2} ⊂ W_{1} with dim(W_{2}) ≥ dim(W_{1}) − 1 such that ord_{p}(ρ) ≥ 7(= 0 + 7 = 2 + 5 = 3 + 4) for every ρ ∈ W_{2}, and then a subspace W_{3} ⊂ W_{2} with dim(W_{3}) ≥ dim(W_{2}) − 2 with the property that ord_{p}(ρ) ≥ 8(= 0 + 8 = 2 + 6 = 3 + 5) for all ρ ∈ W_{3}. At the end of this argument we find at least one element ρ = ρ_{E_{2}} ∈ Ker(ψ_{L}) such that ord_{p}(ρ) ≥ 2a + 5. Since this reasoning works if we replace Ker(ψ_{L}) with any of its subspaces having dimension ≥ 1 + a(a + 1), we can assume that ρ_{E_{2}} is the restriction to E_{2} of an element ρ_{η} in the kernel of the corresponding Gaussian map on the general curve X_{η}, which according to the procedure described before Lemma 5.1 will produce elements ρ_{E_{i}} ∈ Ker(ψ_{L_{E_{i}}}) for 1 ≤ i ≤ g. Applying Proposition 5.3 we have that ord_{p_{g−1}}(ρ_{E_{g}}) ≥ ord_{p}(ρ) + 2g − 4 = 2(a + g) + 1. The vanishing sequence of L_{E_{g}} at p_{g} is ≤ (…, d − 3, d − 2, d) from which we obtain that on the other hand

ord_{p_{g−1}}(ρ_{E_{g}}) ≤ 2d − 5(= d + (d − 5) = (d − 2) + (d − 3)),

which combined with the previous inequality yields d ≥ a + g + 3 which is a contradiction. □

Note that Theorem 5.4 is valid for an arbitrary line bundle on a general genus g curve. It is clear that Proposition 5.3 would give better sufficient conditions for the injectivity of ψ_{L} if we restricted ourselves to line bundles on C having a prescribed ramification sequence at a given point p ∈ C. In this case we degenerate (C, p) to (X_{0} = E_{1} ∪ … ∪ E_{g}, p), where X_{0} is as in Theorem 5.4 and p ∈ E_{1} is such that p−p_{1} ∈ Pic^{0}(E_{1}) is not a torsion class. We leave it to the interested reader to work out the numerical details. We can also improve on Theorem 5.4 if we look only at a suitably general line bundle L on C:

Proposition 5.5. Fix integers g, d and r ≥ 2 such that d ≤ g+r, ρ = g−(r+1)(g−d+r) ≥ 0 and moreover d < g +3+ \frac{ρ}{2(r−1)}. Then if C is a general curve of genus g and L ∈ W_{d}^{r}(C) is general, the Gaussian map ψ_{L} is injective.

Proof. We degenerate C to X_{0}, fix a general point p ∈ E_{1} and set a := [ρ/(r − 1)] + 2. Our numerical assumptions imply that ρ − (a − 2)(r − 1) ≥ 0. From the general theory of limit linear series in [EH1] reducing the Brill-Noether theory of X_{0} to Schubert calculus, we know that there exists a smoothable limit linear series of type 𝔤_{d}^{r} on X_{0}, say l = {L_{E_{i}} ∈ W_{d}^{r}(E_{i})}_{i=0,...,g} having vanishing sequence ≥ (0, 1, a, a+1, a+2, …, a+r −2) at the point p.

Assume by contradiction that there are elements ρ_{E_{i}} ∈ Ker(ψ_{L_{E_{i}}}) coming from an element ρ ≠ 0 in the kernel of the corresponding Gaussian on the general curve. Then ord_{p}(ρ_{E_{1}}) ≥ a + 1(= 1 + a = 0 + (a + 1)) and from Proposition 5.2 we get that ord_{p_{g−1}}(ρ_{E_{g}}) ≥ ord_{p}(ρ_{E_{1}}) + 2g − 2 = 2g + a − 1. On the other hand, as we noticed before ord_{p_{g−1}}(ρ_{E_{g}}) ≤ 2d − 5 which gives a contradiction. □

Remark 5.6. The techniques from this section also allow us to study the kernel S_{2}(L) of the multiplication map μ_{L} : Sym^{2}H^{0}(L) → H^{0}(L^{2}). In a way similar to the proof of Theorem 5.4 we can show that if L is an arbitrary line bundle of degree d ≤ g + a + 1 on a general curve C of genus g then dim S_{2}(L) ≤ a(a + 1). The a = 0 case of this result has been established by Teixidor (cf. [T2]). We also note that this result as well as Theorem 5.4, are meaningful when the bundle L is special. On the other hand the case when L is nonspecial (when, under suitable assumptions, we expect surjectivity for both ψ_{L} and μ_{L}), has been extensively covered in the literature (see e.g. [Pa]).

Theorem 1.3 answers Question 5.8.1 from Wahl’s survey [W1], where the problem is raised in terms of self-correspondences on a curve. Suppose that C is a smooth curve and we consider the diagonal Δ ⊂ C × C and the projections p_{i} : C × C → C for i = 1, 2. For a line bundle L on C we denote L_{i} := p_{i}^{∗}L for i = 1, 2. We can rephrase Theorem 1.3 as follows:

Proposition 5.7. If L is a line bundle of degree d ≤ g + 1 on a general curve C of genus g, then H^{0}(C × C, L_{1} + L_{2} − 2Δ) = 0.

Proof. We use that H^{0}(C × C, L_{1} + L_{2} − 2Δ) = Ker(Φ_{L}) = S_{2}(L) ⊕ Ker(ψ_{L}). We have proved that Ker(ψ_{L}) = 0 while S_{2}(L) = 0 follows from [T2]. □

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Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544

      E-mail address: gfarkas@math.princeton.edu