SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF \overline{M}_{g}

GAVRIL FARKAS

1. Introduction

The aim of this paper is to describe a systematic way of constructing effective divisors on \overline{M}_{g} having exceptionally small slope. In particular, these divisors provide a string of counterexamples to the Harris-Morrison Slope Conjecture (cf. [HMo]). In a previous paper [FP], we showed that the divisor \overline{K}_{10} on \overline{M}_{10} consisting of sections of K3 surfaces contradicts the Slope Conjecture on \overline{M}_{10}. Since the moduli spaces \overline{M}_{g} are known to behave erratically for small g and since the condition that a curve of genus g lie on a K3 surface is divisorial only for g = 10, the question remained whether \overline{K}_{10} is an isolated example or the first in a series of counterexamples. Here we prove that any effective divisor on \overline{M}_{g} consisting of curves satisfying a Green-Lazarsfeld syzygy type condition for a linear system residual to a pencil of minimal degree, violates the Slope Conjecture. A consequence of the existence of these effective divisors is that various moduli spaces \overline{M}_{g,n} with g ≤ 22, are proved to be of general type.

We recall that the slope s(D) of an effective divisor D on \overline{M}_{g} is defined as the smallest rational number a/b ≥ 0 such that the divisor class aλ − b(δ_{0} + ⋯ + δ_{[g/2]}) − [D] is an effective combination of boundary divisors. The Slope Conjecture predicts that s(D) ≥ 6 + 12/(g + 1) for all effective divisors D on \overline{M}_{g} (cf. [HMo]). More generally, the question of finding a good lower bound for the slope of \overline{M}_{g}

s_{g} := inf\{s(D) : D ∈ Eff(\overline{M}_{g})\}

is of great interest for a variety of reasons, for instance it would provide a new geometric solution to the Schottky problem. In a different direction, since s(K_{\overline{M}_{g}}) = 13/2 (cf. [HM]), to prove that \overline{M}_{g} is of general type it suffices to exhibit a single effective divisor D on \overline{M}_{g} of slope s(D) < 13/2.

The Slope Conjecture is true for all \overline{M}_{g} with g ≤ 9 but in [FP] we proved that on \overline{M}_{10}, we have the equality s(\overline{K}_{10}) = 7 < 6 + 12/11 (in fact \overline{K}_{10} is the only effective divisor on \overline{M}_{10} having slope < 6 + 12/11). In [FP] we also showed that K_{10} has four incarnations as a geometric subvariety of M_{10}. In particular, K_{10} can be thought of as either

(1) the locus of curves [C] ∈ M_{10} for which the rank 2 Mukai type Brill-Noether locus

SU_{2}(C, K_{C}, 6) := {E ∈ SU_{2}(C, K_{C}) : h^{0}(C, E) ≥ 7}

is not equal to ∅, or (2) the locus of curves [C] ∈ M_{10} carrying a pencil A ∈ W_{6}^{1}(C) such that the multiplication map μ_{A} : Sym^{2}H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, (K_{C} ⊗ A^{∨})^{⊗2}) is not surjective.

Research partially supported by the NSF Grant DMS-0450670 and by the Sloan Foundation.

The geometric conditions (1) and (2), unlike the original definition of K_{10}, can be extended to other genera. We fix g = 2k − 2 and denote by σ : 𝔊_{k}^{1} → M_{g} the Hurwitz scheme of k-sheeted coverings of P^{1} of genus g parametrizing pairs (C, A) with A ∈ W_{k}^{1}(C). For each i ≥ 0 we introduce the cycle U_{g,i} consisting of pairs (C, A) ∈ 𝔊_{k}^{1} such that K_{C} ⊗ A^{∨} fails the Green-Lazarsfeld property (N_{i}). By setting Z_{g,i} := σ(U_{g,i}) we obtain an induced geometric stratification of M_{g}

Z_{g,0} ⊂ Z_{g,1} ⊂ … ⊂ Z_{g,i} ⊂ … ⊂ M_{g}.

If for each (C, A) ∈ 𝔊_{k}^{1} we consider the map C ⟶^{|K_{C}⊗A^{∨}|} P^{k−2} induced by the residual linear system, we can define two vector bundles A and B on 𝔊_{k}^{1} such that

A(C, A) = H^{0}(Ω_{P^{k−2}}^{i}(i + 2)) and B(C, A) = H^{0}(Ω_{P^{k−2}}^{i}(i + 2) ⊗ O_{C}).

For g = 6i + 10 (hence k = 3i + 6), it turns out that rank(A) = rank(B) and U_{g,i} is the degeneracy locus of the natural vector bundle map ϕ : A → B, that is, Z_{g,i} is a virtual divisor on M_{g}, with virtual class σ_{∗}c_{1}(B − A). The main result of this article is the computation of the compactification inside \overline{M}_{g} of this degeneracy locus (see Theorem 4.1 for a more precise statement):

Theorem A. If σ : \overline{𝔊}_{3i+6}^{1} → \overline{M}_{6i+10} is the compactification of the Hurwitz stack by limit linear series, then there is a natural extension of the vector bundle map ϕ : A → B over \overline{𝔊}_{3i+6}^{1} such that \overline{Z}_{g,i} is the degeneracy locus of ϕ. Moreover the class of the pushforward to \overline{M}_{g} of the virtual degeneracy locus of ϕ is given by

σ_{∗}(c_{1}(B) − c_{1}(A)) ≡ a λ − b_{0} δ_{0} − b_{1} δ_{1} − ⋯ − b_{3i+5}δ_{3i+5},

where a, b_{0}, …, b_{3i+5} are explicitly determined coefficients such that b_{j} ≥ b_{0} for j ≥ 1 and

\frac{a}{b_{0}} = \frac{3(4i + 7)(6i^{2} + 19i + 12)}{(i + 2)(12i^{2} + 31i + 18)} < 6 + \frac{12}{g + 1}.

Corollary. If the degeneracy locus Z_{6i+10,i} is an actual divisor on M_{6i+10}, then we have that

s(\overline{Z}_{6i+10,i}) = \frac{3(4i + 7)(6i^{2} + 19i + 12)}{(i + 2)(12i^{2} + 31i + 18)} < 6 + \frac{12}{g + 1},

thus contradicting the Slope Conjecture on \overline{M}_{6i+10}.

The idea of the proof is to define a whole host of vector bundles G_{a,b} and H_{a,b} over 𝔊_{k}^{1} for a, b ≥ 0, having fibres

G_{a,b}(C, A) = H^{0}(Ω_{P^{k−2}}^{a}(a + b) ⊗ O_{C}) and H_{a,b}(C, A) = H^{0}(Ω_{P^{k−2}}^{a}(a + b)).

These bundles are related to one another by certain exact sequences (4) and (6) over 𝔊_{k}^{1} (see Section 3). After an analysis over each boundary divisor σ^{−1}(Δ_{j}), where 0 ≤ j ≤ 3i + 5, we find in Section 3 a unique way of extending G_{a,b} and H_{a,b} to vector bundles over \overline{𝔊}_{k}^{1} such that (4) and (6) continue to make sense and be exact and then use this to compute the Chern numbers of A = H_{i,2} and B = G_{i,2}.

We expect Z_{6i+10,i} to be always a divisor on M_{6i+10} but we are able to check this only for small i. To verify this in general one would have to prove that if [C] ∈ M_{6i+10} is a general curve, then one (or equivalently all) of the finitely many linear systems 𝔤_{9i+12}^{3i+4} = K_{C}(−𝔤_{3i+6}^{1}) satisfies property (N_{i}). This is a direct generalization of Green’s Conjecture to the case of curves and line bundles L with h^{1}(L) = 2 (see Section 2 for more on this analogy).

For g = 10 we recover of course the results from [FP]. In the next two cases, g = 16 and 22 we have complete results:

Theorem 1.1. The following subvariety of M_{16}

Z_{16,1} := {[C] ∈ M_{16} : ∃L ∈ W_{21}^{7}(C) such thatC ↪^{|L|} P^{7} is not cut out by quadrics },

is an effective divisor, and the class of its compactification is given by the formula

\overline{Z}_{16,1} ≡ 286(407λ − 61 δ_{0} − 325 δ_{1} − b_{2} δ_{2} − ⋯ − b_{8} δ_{8}),

where b_{j} ≥ b_{1} for all 2 ≤ j ≤ 8. In particular s(\overline{Z}_{16,1}) = 407/61 = 6.6721... < 6 + 12/17 = 6.705..., hence \overline{Z}_{16,1} provides a counterexample to the Slope Conjecture on \overline{M}_{16}.

In a similar manner we have the following example of a geometric divisor on \overline{M}_{22} of very small slope:

Theorem 1.2. The following subvariety of M_{22}

Z_{22,2} := {[C] ∈ M_{22} : ∃L ∈ W_{30}^{10}(C) such thatC ↪^{|L|} P^{10} fails property(N_{2})},

is an effective divisor on M_{22} and the class of its compactification is given by the formula

\overline{Z}_{22,2} ≡ 25194(1665 λ − 256 δ_{0} − 1407 δ_{1} − b_{2}δ_{2} − ⋯ − b_{11} δ_{11}),

where b_{j} ≥ b_{1} for all 2 ≤ j ≤ 11. In particular s(\overline{Z}_{22,2}) = 1665/256 = 6.50390... < 6 + 12/23 = 6.52173..., and \overline{Z}_{22,2} gives a counterexample to the Slope Conjecture on \overline{M}_{22}.

We note that while Theorem A gives a sequence of (virtual) examples of divisors contradicting the Slope Conjecture, for any given g one can construct other (actual) divisors on \overline{M}_{g} of slope < 6 + 12/(g + 1) also defined in terms of syzygies. For instance, the locus consisting of curves [C] ∈ \overline{M}_{20} for which there exists L ∈ W_{20}^{4}(C) such that

C ↪^{|L|} P^{4} is not cut out by quartics (or equivalently, the map I_{4}(L) ⊗ H^{0}(L) → I_{5}(L) is not bijective), is a divisor on \overline{M}_{20} violating the Slope Conjecture.

We record the following consequence of Theorems 1.1 and 1.2:

Theorem 1.3. (1) The moduli space of n-pointed curves \overline{M}_{22,n} is of general type for all n ≥ 2.

(2) The moduli space \overline{M}_{21,n} is of general type for all n ≥ 4.

(3) The moduli space \overline{M}_{16,n} is of general type for all n ≥ 9.

(4) The moduli space \overline{M}_{20,n} is of general type for n ≥ 6.

It is also possible to give examples of divisors of small slope on the moduli spaces \overline{M}_{g,n} of n-pointed stable curves. We have only pursued this for genera g ≤ 22 with the goal of proving that various moduli spaces \overline{M}_{g,n} are of general type (this is automatic for g ≥ 23). One of our examples is the following:

Theorem 1.4. The following subvariety of \overline{M}_{14,1}

Z_{14,0}^{1} := {[C, p] ∈ M_{14,1} : ∃L ∈ W_{18}^{6}(C) such thatC ⟶^{|L(−p)|} P^{5} fails property(N_{0})}

is an effective divisor on M_{14,1}. The class of its compactification \overline{Z}_{14,0}^{1} is given by the formula

\overline{Z}_{14,0}^{1} ≡ 33(237 λ + 14 ψ − 35 δ_{0} − 169δ_{1} − 183δ_{13} − b_{2}δ_{2} − ⋯ − b_{12}δ_{12}),

where b_{j} ≥ 15 + 27j for 3 ≤ j ≤ 12, j ≠ 4, b_{2} ≥ 325 and b_{4} ≥ 271. In particular [\overline{Z}_{14,0}^{1}] lies outside the cone of Pic(\overline{M}_{14,1}) spanned by pullbacks of effective divisors from \overline{M}_{14}, the boundary divisors δ_{0}, …, δ_{13} and the Weierstrass divisor \overline{W} on \overline{M}_{14,1}.

An equivalent formulation of the Slope Conjecture on \overline{M}_{g} for g such that g + 1 is composite (which we learned from S. Keel), is to say that the Brill-Noether divisors \overline{M}_{g,d}^{r} consisting of curves with a 𝔤_{d}^{r} when g − (r + 1)(g − d + r) = −1, lie on a face of the effective cone Eff(\overline{M}_{g}). It is well-known that s(\overline{M}_{g,d}^{r}) = 6 + 12/(g + 1) (cf. [EH3]), so the Slope Conjecture singles out these divisors as being of minimal slope. One can ask a similar question on \overline{M}_{g,1}. Are the pullbacks π^{∗}(\overline{M}_{g,d}^{r}) of the Brill-Noether divisors from \overline{M}_{g} and the Weierstrass divisor \overline{W} := \{[C, p] ∈ \overline{M}_{g,1} : p ∈ C is a Weierstrass point\} on a face of the effective cone Eff(\overline{M}_{g,1})? The question makes sense especially since in

[EH2] it is proved that the class of any generalized Brill-Noether divisor on \overline{M}_{g,1} (that is, any codimension 1 locus of curves (C, p) ∈ \overline{M}_{g,1} having a linear series with special ramification at p) lies inside the cone of Pic(\overline{M}_{g,1}) spanned by by [π^{∗}(\overline{M}_{g,d}^{r})] and [\overline{W}]. Theorem 1.4 shows that at least for g ≥ 14 the answer to the question raised above is emphatically negative.

Acknowledgments: I had useful conversations with many people on subjects related to this paper. I especially benefitted from discussions with J. Harris, S. Keel, D. Khosla and M. Popa.

2. Syzygies of algebraic curves

In this paragraph we review a few facts about the resolution of the ideal of a curve embedded in a projective space. As a general reference for syzygies and Koszul cohomology we recommend [L] or [GL].

Suppose that C is a smooth curve of genus g and L is a very ample line bundle on C giving an embedding C → P^{r} = P(V), where V = H^{0}(L). We denote by I_{C/P^{r}} the ideal of C in P^{r} and consider its minimal resolution of free S = Sym(V)-modules

0 → F_{r+1} → ⋯ → F_{2} → F_{1} → I_{C/P^{r}} → 0.

Then one can write F_{j} = ⊕_{l∈ℤ}S(−j − l)^{b_{j,l}(C)}, where b_{j,l}(C) = dim_{ℂ}Tor_{j}^{S}(S(C), ℂ)_{j+l} is the graded Betti number of C that comes on the l-th row and j-th column in the Betti diagram of C. Following Green and Lazarsfeld we say that the pair (C, L) satisfy the property (N_{i}) for some integer i ≥ 1, if F_{j} = ⊕S(−j − 1) for all j ≤ i (or equivalently in terms of graded Betti numbers, b_{i,l}(C) = 0 for all l ≥ 2). Using the computation of b_{j,l}(C) in terms of Koszul cohomology, there is a well-known cohomological interpretation of property (N_{i}): we denote by M = Ω_{P}^{r}(1) and M_{L} = M ⊗ O_{C}, hence we have an exact sequence

0 → M_{L} → H^{0}(L) ⊗ O_{C} → L → 0.

By taking exterior powers, for each i ≥ 0 we obtain the exact sequence:

(1)  0 → ∧^{i+1}M_{L} → ∧^{i+i}H^{0}(L) ⊗ O_{C} → ∧^{i}M_{L} ⊗ L → 0.

If L is a normally generated line bundle on C, then (C, L) satisfies property (N_{i}) if and only if for all j ≥ 1, the natural map

u_{i,j} : ∧^{i+1}H^{0}(L) ⊗ H^{0}(L^{⊗j}) → H^{0}(∧^{i}M_{L} ⊗ L^{⊗(j+1)})

obtained by tensoring the sequence (1) and taking global sections, is surjective (cf. e.g. [GL], Lemma 1.10).

We will be interested in the vector bundle M_{L} in the case when C is a curve of genus g = 2k − 2 and L = K_{C} ⊗ A^{∨} is residual to a base point free pencil A ∈ W_{k}^{1}(C). We start by establishing a more general technical result used throughout the paper:

Proposition 2.1. Let C be a curve of genus g and L a globally generated 𝔤_{d}^{r} on C. If p ≥ 1 is an integer such that g + min{r, p} ≤ d, then for all j ≥ 2 and 0 ≤ i ≤ p + 2 − j, we have the vanishing H^{1}(∧^{i}M_{L} ⊗ L^{⊗j}) = 0.

Proof. We use a filtration argument due to Lazarsfeld [L]. We choose general points x_{1}, …, x_{r−1} ∈ C such that h^{0}(C, L ⊗ O_{C}(−x_{1} − ⋯ − x_{r−1})) = 2 and then there is an exact sequence

0 ⟶ L^{∨}(x_{1} + ⋯ + x_{r−1}) ⟶ M_{L} ⟶ ⊕_{j=1}^{r−1}O_{C}(−x_{j}) ⟶ 0.

Taking exterior powers, we can write the exact sequence

0 → ∧^{i−1}(⊕_{j=1}^{r−1}O_{C}(−x_{j})) ⊗ L^{∨}(x_{1} + ⋯ + x_{r−1}) ⟶ ∧^{i}M_{L} ⟶ ∧^{i}(⊕_{j=1}^{r−1}O_{C}(−x_{j})) ⟶ 0.

In order to conclude that H^{0}(∧^{i}M_{L} ⊗ L^{⊗j}) = 0, it suffices to show that for all 1 ≤ i ≤ min{r − 1, p} we have the following:

(1) H^{0}(L^{⊗j} ⊗ O_{C}(−D_{i})) = 0 for each effective divisor D_{i} ∈ C_{i} with support in the set {x_{1}, …, x_{r−1}}, and

(2) H^{0}(L^{⊗(j−1)} ⊗ O_{C}(D_{r−i})) = 0, for any effective divisor D_{r−i} ∈ C_{r−i} with support contained in {x_{1}, …, x_{r−1}}.

Statement (2) follows because of degree reasons. The only way (1) could fail is when j = 2 and h^{0}(K_{C} ⊗ L^{∨}) ≥ r − i + 1, which by the Riemann-Roch theorem implies that g + i ≥ d + 1 and this contradicts our numerical assumption.

□

Remark 2.2. (1) For p ≥ r, the condition g +min{p, r} ≤ d is equivalent with H^{1}(L) = 0.

(2) Since we work in characteristic 0, the same filtration argument from the proof of Proposition 2.1 can be used to show that M_{L} is a stable vector bundle on C.

We fix g = 2k − 2 where k ≥ 3. We denote by M_{g}^{0} the open substack of M_{g} consisting of curves C of genus g which carry no 𝔤_{k−1}^{1}’s and which have no automorphisms (clearly the complement of M_{g}^{0} inside M_{g} has codimension ≥ 2). Over M_{g}^{0} we have a universal curve π : C → M_{g}^{0} and we also consider the Hurwitz stack σ : 𝔊_{k}^{1} → M_{g}^{0} parametrising pairs (C, A), where [C] ∈ M_{g}^{0} and A ∈ W_{k}^{1}(C) is a (necessarily base point free) 𝔤_{k}^{1} on C. It is a classical result that 𝔊_{k}^{1} is a smooth irreducible stack which is finite over M_{g}^{0}. The isomorphism 𝔊_{k}^{1} ∋ (C, A) ↦ (C, K_{C} ⊗ A^{∨}) ∈ 𝔊_{3k−6}^{k−2} will be used throughout the paper.

Let us consider now an element (C, A) ∈ 𝔊_{k}^{1} and set L := K_{C} ⊗A^{∨}. The genericity assumption on C implies that L is very ample and gives an embedding C ⊂ P^{k−2} of degree 3k − 6. We denote by I_{C} the ideal sheaf of C in this embedding. Proposition 2.3. Fix integers g = 2k − 2 with k ≥ 3 and i ≥ 0 with 2k ≥ 3i + 7. If (C, A) ∈ 𝔊_{k}^{1} and L = K_{C} ⊗ A^{∨}, then (C, L) fails to satisfy property (N_{i}) if and only if

h^{0}(P^{k−2}, ∧^{i}M_{P}_{k−2} ⊗ I_{C}(2)) ≥ \binom{k − 2}{i + 1}(k − 3i − 6) + 2\binom{k − 1}{i + 1} − \binom{k − 1}{i + 2} + 1.

Proof. We use the following diagram of exact sequences for the embedding C ↪^{|L|} P^{k−2}:

0  0  0
↓  ↓  ↓
∧^{i+1}M_{P}_{k−2} ⊗ I_{C}(1) ⟶ ∧^{i+1}H^{0}(O_{P}_{k−2}(1)) ⊗ I_{C}(1) ⟶ ∧^{i}M_{P}_{k−2} ⊗ I_{C}(2)
↓  ↓  ↓
∧^{i+1}M_{P}_{k−2}(1)  ⟶ ∧^{i+1}H^{0}(O_{P}_{k−2}(1)) ⊗ O_{P}_{k−2}(1) ⟶  ∧^{i}M_{P}_{k−2}(2)
↓^{a}  ↓_{b}  ↓^{c}
∧^{i+1}M_{L} ⊗ L  ⟶  ∧^{i+1}H^{0}(L) ⊗ L  ⟶  ∧^{i}M_{L} ⊗ L^{2}
↓  ↓  ↓
0  0  0

For simplicity we denote by the same symbol the map on global sections corresponding to a sheaf morphism and the morphism itself. Since H^{1}(P^{k−2}, ∧^{i+1}M_{P}_{k−2}(1)) = 0 (because of Bott vanishing, see [OSS], pg. 8), we can apply the snake lemma to the diagram obtained by taking global sections in the two lower rows. The map induced by b is an isomorphism, which gives an isomorphism

H^{0}(P^{k−2}, ∧^{i}M_{P}_{k−2} ⊗ I_{C}(2)) ≅ H^{1}(P^{k−2}, ∧^{i+1}M_{P}_{k−2} ⊗ I_{C}(1)).

From Proposition 2.1 we have that H^{1}(∧^{i}M_{L} ⊗ L^{2}) = 0, and thus C satisfies (N_{i}) if and only if the map

H^{1}(∧^{i+1}M_{L} ⊗ L) → ∧^{i+1}H^{0}(L) ⊗ H^{1}(L)

is an isomorphism, or equivalently h^{1}(∧^{i+1}M_{L} ⊗ L) ≤ h^{1}(∧^{i+1}H^{0}(L) ⊗ L) = 2\binom{k−1}{i+1}. From the Riemann-Roch theorem, this is the same thing as having

dim(Coker(a)) ≤ \binom{k − 2}{i + 1}(k − 3i − 6) + 2\binom{k − 1}{i + 1} − \binom{k − 1}{i + 2}.

But Coker(a) ≅ H^{0}(∧^{i}M_{P}_{k−2} ⊗ I_{C}(2)), so failure of (N_{i}) is equivalent to the map c having a kernel of dimension at least

\binom{k − 2}{i + 1}(k − 3i − 6) + 2\binom{k − 1}{i + 1} − \binom{k − 1}{i + 2} + 1 =

= \binom{k − 2}{i + 1}(k − 3i − 6)(1 − \frac{1}{(k − i − 2)(i + 2)}) + 1.

□

Remark 2.4. An immediate consequence of this proof is that when k − 3i − 6 < 0 (or equivalently g ≤ 6i + 8), the pair (C, A) always fails property (N_{i}). When k = 3i + 6 we have that (C, A) fails (N_{i}) if and only if h^{0}(P^{k−2}, ∧^{i}M_{P}_{k−2} ⊗ I_{C}(2)) ≥ 1, which we expect to be a divisorial condition on 𝔊_{k}^{1}.

This proposition allows us to define a determinantal substack of 𝔊_{k}^{1} consisting of pairs (C, A) for which L fails to satisfy property (N_{i}). On the fibre product C ×_{M_{g}^{0}} 𝔊_{k}^{1} there is a universal Poincaré bundle L whose existence is guaranteed by the universal property of 𝔊_{k}^{1}. If p_{1} : C ×_{M_{g}^{0}} 𝔊_{k}^{1} → C and p_{2} : C ×_{M_{g}^{0}} 𝔊_{k}^{1} → 𝔊_{k}^{1} are the natural projections, then E := p_{2∗}(p_{1}^{∗}ω_{π}⊗L^{∨}) is a vector bundle of rank k−1 with fibre E(C, A) = H^{0}(C, K_{C} ⊗ A^{∨}) over each point (C, A) ∈ 𝔊_{k}^{1}. We have a tautological embedding of the pullback of the universal curve C ×_{M_{g}^{0}} 𝔊_{k}^{1} into the projective bundle u : P(E) → 𝔊_{k}^{1} and we denote by J ⊂ O_{P(E)} the ideal sheaf of the image. Next, we define the vector bundle N on P(E) by the sequence

0 ⟶ N → u^{∗}(E) ⟶ O_{P(E)}(1) ⟶ 0,

and we further introduce two vector bundles A and B over 𝔊_{k}^{1} by setting

A := u_{∗}(∧^{i}N ⊗ O_{P(E)}(2)), and B := u_{∗}(∧^{i}N ⊗ O_{C×_{M^{0}_{g}} 𝔊_{k}^{1}}(2)).

If C ⊂ P^{k−2} is the embedding given by L = K_{C} ⊗ A^{∨}, then

A(C, A) = H^{0}(P^{k−2}, ∧^{i}M_{P}_{k−2}(2)) and B(C, A) = H^{0}(C, ∧^{i}M_{L} ⊗ L^{⊗2})

and there is a natural vector bundle morphism ϕ : A → B. From Grauert’s Theorem we see that both A and B are vector bundles over 𝔊_{k}^{1} (for B use that H^{1}(C, ∧^{i}M_{L} ⊗ L^{2}) = 0 because of Proposition 2.1). Moreover from Bott’s Theorem (see again [OSS]) and the Riemann-Roch Theorem respectively, we obtain that

rank(A) = (i + 1)\binom{k}{i + 2} and rank(B) = \binom{k − 2}{i}(4k − 9 − 3i).

Then Proposition 2.3 can be restated as follows:

Proposition 2.5. The cycle

U_{g,i} := {(C, A) ∈ 𝔊_{k}^{1} : (C, K_{C} ⊗ A^{∨}) fails property(N_{i})},

can be realized as the degeneracy locus of rank (i−1)\binom{k−1}{i+1}+(i+2)\binom{k−1}{i+2}−\binom{k−2}{i+1}(k−3i−6)−1 of the vector bundle map ϕ : A → B over 𝔊_{k}^{1}.

In this way we obtain a stratification of M_{g}^{0} with strata Z_{g,i} := σ(U_{g,i}) consisting of those curves C for which there exists a pencil A ∈ W_{k}^{1}(C) such that (C, K_{C} ⊗A^{∨}) fails property (N_{i}). Note that when g = 6i + 10 then rank(A) = rank(B) and U_{g,i} is simply the degeneracy locus of ϕ and we expect Z_{g,i} to be a divisor on M_{g}^{0}.

It is of course very natural to compare this newly defined stratification

Z_{g,0} ⊂ Z_{g,1} ⊂ … ⊂ Z_{g,i} ⊂ … ⊂ M_{g}^{0}

with the more classical stratification of M_{g} defined in terms of syzygies of K_{C}. If for each i ≥ 0 we set

N_{g,i} := {[C] ∈ M_{g} : (C, K_{C}) fails property (N_{i})},

then Green’s Conjecture for generic curves of fixed gonality (cf. [V3], [V2]) can be read as saying that for all 0 ≤ i ≤ (g − 2)/2, the locus N_{g,i} coincides with the (i + 2)-gonal locus M_{g,i+2}^{1} of curves with a 𝔤_{i+2}^{1}, while N_{g,i} = M_{g} for all i ≥ (g − 1)/2. One of the morals of this paper is that the stratification Z_{g,i} is very different in nature from the one given by gonality: whereas N_{g,0} is the locus of hyperelliptic curves of genus g, the smallest stratum Z_{g,0} is intimately related to the locus of curves lying on K3 surfaces which is well-known to be transversal to any Brill-Noether locus (a curve of genus g lying on a general K3 surface satisfies the Brill-Noether Theorem, cf. [L]).

We have already seen that Z_{g,i} = M_{g}^{0} when g ≤ 6i + 8 (cf. Remark 2.4). A result due to Mukai and Voisin (cf. [V1], Proposition 3.2) states that if C is a smooth curve of genus g = 2k − 2 sitting on a K3 surface S with Pic(S) = ℤ [C], then for all A ∈ W_{k}^{1}(C) the multiplication map Sym^{2}H^{0}(K_{C} ⊗ A^{∨}) → H^{0}((K_{C} ⊗ A^{∨})^{⊗2}) is not surjective, in other words, [C] ∈ Z_{g,0}. In a forthcoming paper we will show that the converse also holds, that is, the closure in M_{g} of the smallest stratum Z_{g,0} coincides with the locus

K_{g} := {[C] ∈ M_{g} : C lies on a K3 surface}.

The possibility of such an equality of cycles has already been raised in Voisin’s paper (cf. [V1], Remarques 4.13). Its main appeal lies in the fact that it gives an intrinsic characterization of a curve lying on a K3 surface which makes no reference to the K3 surface itself! We now make the following:

Conjecture 2.6. For an even genus g ≥ 6i + 10, the stratum Z_{g,i} is a proper subvariety of M_{g}^{0}. In particular, when g = 6i + 10, the stratum Z_{g,i} is a divisor on M_{g}^{0}.

For g ≤ 8 the conjecture is trivially true because K_{g} = M_{g}, that is, every curve of (even) genus g ≤ 8 sits on a K3 surface. The first interesting case is g = 10 when the conjecture holds and the identification K_{10} = Z_{10,0} is part of a more general picture (see [FP] for details). We have checked Conjecture 2.6 for all genera g ≤ 24 sometimes using the program Macaulay. We will describe this in detail in the most interesting cases, g = 16 and g = 22, when the strata Z_{16,1} and Z_{22,2} are divisors on M_{16} and M_{22} respectively.

Theorem 2.7. Let C be a general curve of genus 16. Then every linear series A ∈ W_{9}^{1}(C) gives

an embedding C ⟶^{|K_{C}⊗A^{∨}|} P^{7} of degree 21 such that the ideal of C is generated by quadrics (that is, it satisfies property (N_{1})). In particular Z_{16,1} is a divisor on M_{16}.

Proof. From the irreducibility of the Hurwitz scheme of coverings of P^{1} it follows that the variety 𝔊_{9}^{1} = 𝔊_{16,9}^{1} parametrizing pairs (C, A) with [C] ∈ M_{16} and A ∈ W_{9}^{1}(C) is irreducible. To prove that Z_{16,1} is a divisor it is enough to exhibit an element (C, A) ∈ 𝔊_{9}^{1}

with K_{C} ⊗ A^{∨} very ample, such that the embedded curve C ↪^{|K_{C}⊗A^{∨}|} P^{7} is cut out by quadrics.

We consider 13 general points in P^{2} denoted by p_{1}, p_{2}, q_{1}, …, q_{7} and r_{1}, …, r_{4} respectively, and define the linear system

H ≡ 8h − 3(E_{p_{1}} + E_{p_{2}}) − 2 ∑_{i=1}^{7} E_{q_{i}} − ∑_{j=1}^{4} E_{r_{j}}

on the blow-up S = Bl_{13}(P^{2}). Here h denotes the pullback of the line class from P^{2}.

Using the program Macaulay it is easy to check that S ↪^{|H|} P^{7} is an embedding and the graded Betti diagram of S is

\begin{matrix}1 & − & − & − \\ − & 7 & − & − \\ − & − & 35 & 56 \end{matrix}

Thus S is cut out by quadrics. To carry out this calculation we chose the 13 points in P^{2} randomly using the Hilbert-Burch theorem so that they satisfy the Minimal Resolution Conjecture (see [SchT] for details on how to pick random points in P^{2} using Macaulay). Next we consider a curve C ⊂ S in the linear system

(2)  C ≡ 14h − 5(E_{p_{1}} + E_{p_{2}}) − 4 ∑_{i=1}^{6} E_{q_{i}} − 3E_{q_{7}} − 2 ∑_{j=1}^{3} E_{r_{j}} − E_{r_{4}}.

By using Macaulay we pick C randomly in its linear system and then check that C is smooth, g(C) = 16 and deg(C) = 21. To show that C is cut out by quadrics one can compute directly the Betti diagram of C. Otherwise, since S is cut out by quadrics, to conclude the same thing about C, it suffices to show that the map

m : H^{0}(S, H) ⊗ H^{0}(S, 2H − C) → H^{0}(S, 3H − C)

is surjective (or equivalently injective). Since h^{0}(S, 2H − C) = 2, from the base point free pencil trick we get that Ker(m) = H^{0}(S, C − H) = 0, because C − H ≡ 6h − 2E_{p_{1}} − 2E_{p_{2}} − 2 ∑_{i=1}^{6} E_{q_{i}} − E_{q_{7}} − ∑_{j=1}^{3} E_{r_{j}} is clearly not effective for a general choice of the 13 points. □

Remark 2.8. A possible objection to this proof could be that while we defined the Hurwitz space 𝔊_{9}^{1} → M_{16}^{0} over the open part of M_{16} consisting of curves carrying no 𝔤_{8}^{1}’s, the curve C we constructed in the proof of Theorem 2.7 might lie outside M_{16}^{0}. However, this does not affect our conclusions because for every g and l < (g + 3)/2, the Hurwitz space of pairs (C, A) with [C] ∈ M_{g} an A ∈ W_{l}^{1}(C), is an irreducible variety, that is, there can be no component of the Hurwitz space mapping to a proper subvariety of the l-gonal locus M_{g,l}^{1} (This statement can be easily proved with the methods from [AC1]).

Remark 2.9. For later use we are going to record a slight generalization of Theorem 2.7. The conclusion of the theorem holds for the image in P^{7} of any plane curve C that is sufficiently general in the linear system given by (2). In particular one can choose C to have a cusp at a general extra point y ∈ P^{2}, in which case we obtain that a general one-cuspidal curve C ⊂ P^{7} of arithmetic genus 16 and degree 21 is cut out by quadrics (this again is quite easily checked with Macaulay). Since the Hilbert scheme of these curves is irreducible (use again the Hurwitz scheme), it follows that for a general (C, y) ∈ M_{15,1} and for an arbitrary linear system L ∈ W_{21}^{7}(C) having a cusp at y, the ideal of the

cuspidal image curve C →^{|L|} P^{7} is generated by quadrics.

In a somewhat similar manner we are going to show that for g = 22 the locus Z_{22,2} is a divisor.

Theorem 2.10. Let C be a general curve of genus 22. Then for every linear series A ∈ W_{12}^{1}(C)

we have that |K_{C} ⊗ A^{∨}| is very ample and the resulting embedding A ↪^{|K_{C}⊗A^{∨}|} P^{10} satisfies property (N_{2}). In particular, the locus of curves C of genus 22 carrying a linear series 𝔤_{30}^{10} = K_{C}(−𝔤_{12}^{1}) which fails to satisfy property (N_{2}), is a divisor on M_{22}.

Proof. Ideally we would like to construct a rational surface S ⊂ P^{10} which satisfies property (N_{2}) and then consider a suitable curve C ⊂ S with g(C) = 22 and deg(C) =

30. For numerical reasons this will turn out not to be possible but we will be able to construct a smooth curve C′ ⊂ S with g(C′) = 20 and deg(C′) = 28 which satisfies (N_{2}). The desired curve C will be a smoothing in P^{10} of C ∪ L ∪ L′, where L and L′ are general chords of C′.

We start by choosing random points p, p_{1}, …, p_{7}, q_{1}, …, q_{5} ∈ P^{2}. Then the linear

system

H = 11h − 4E_{p} − 3 ∑_{i=1}^{7} E_{p_{i}} − 2 ∑_{j=1}^{5} E_{q_{j}}

yields an embedding S ⊂ P^{10} of Bl_{13}(P^{2}). Using Macaulay we get that the upper left corner of the Betti diagram of S is

\begin{matrix}1 & − & − & − & − \\ − & 29 & 98 & 72 & − \\ − & − & − & 264 & − \end{matrix}

that is, S satisfies property (N_{2}), hence H^{1}(S, ∧^{3}M_{S}(1)) = 0. If Γ ≡ H is a general hyperplane section of S, then g(Γ) = 13, deg(Γ) = 22 and an argument very similar to the one in Proposition 2.1 shows that the vector bundle M_{Γ} is stable. In particular H^{1}(Γ, ∧^{2}M_{Γ}(2)) = 0. Since M_{S|Γ} ≅ M_{Γ} ⊕ O_{Γ} one can write the exact sequence

0 ⟶ ∧^{i}M_{S}(1) ⟶ ∧^{i}M_{S}(2) ⟶ ∧^{i}M_{Γ}(2) ⊕ ∧^{i−1}M_{Γ}(2) ⟶ 0,

which leads to the vanishing H^{2}(S, ∧^{3}M_{S}(1)) = 0. Suppose now that C ⊂ S is a smooth non-degenerate curve. We have a commutative diagram:

H^{1}(C, ∧^{3}M_{C}(1)) ⟶ ∧^{3}H^{0}(O_{C}(1)) ⊗ H^{1}(O_{C}(1))
↓^{=} ↓^{=}
H^{1}(∧^{2}M_{S}(2H − C)) ↪ H^{2}(∧^{3}M_{S}(H − C)) → ∧^{3}H^{0}(O_{S}(1)) ⊗ H^{2}(H − C),

where the left vertical map is isomorphic because H^{1}(S, ∧^{3}M_{S}(1)) = H^{2}(S, ∧^{3}M_{S}(1)) = 0. To conclude that C satisfies (N_{2}), it suffices to show that H^{1}(S, ∧^{2}M_{S}(2H − C)) = 0, or equivalently, that the map f : ∧^{2}H^{0}(O_{S}(1)) ⊗ H^{0}(2H − C) → H^{0}(M_{S}(3H − C)) obtained from the Koszul complex, is surjective. If (deg(C), g(C)) = (30, 22), then one can check easily that h^{0}(S, 2H−C) = 3 and h^{0}(S, M_{S}(3H−C)) = 207, hence f cannot be surjective for dimensional reasons. The closest we can get to these numerical invariants is when (deg(C), g(C)) = (28, 20) and this is the type of curve on S we will be looking for. Take

C′ ≡ 17h − 6E_{p} − 5 ∑_{i=}^{7} E_{p_{i}} − 3 ∑_{j=1}^{5} E_{q_{j}},

and consider the embedding C′ ⊂ P^{10} given by |H|. Using Macaulay we check that C′ is a smooth curve of genus 20 and degree 28 with graded Betti diagram

\begin{matrix}1 & − & − & − \\ − & 27 & 80 & − \\ − & − & − & 432 \end{matrix}

Hence C′ satisfies (N_{2}). Now choose two general chords L and L′ of C and define C := C′ ∪ L ∪ L′. Clearly g(C) = 22, deg(C) = 30 and one last check with Macaulay shows that b_{2j}(C) = 0 for j ≥ 2, that is, C satisfies (N_{2}).

□

Remark 2.11. Just like in the case g = 16 we have a slight variation of the last Theorem. The same proof shows that for a general (C, y) ∈ M_{21,1} and for an arbitrary linear

series L ∈ W_{30}^{10}(C) that has a cusp at y, the one-cuspidal image curve C →^{|L|} P^{10} satisfies property (N_{2}).

3. Intersection theory calculations on \overline{M}_{g}

Recall that we have realized Z_{6i+10,i} as the image of the degeneracy locus U_{6i+10,i} of a vector bundle morphism ϕ : A → B over 𝔊_{k}^{1}. To compute the class of the compactification \overline{Z}_{6i+10,i} we are going to extend the determinantal structure of Z_{6i+10,i} over the boundary divisors in \overline{M}_{6i+10}.

We set g := 6i + 10, k := 3i + 6 and denote by M̃_{g} := M_{g}^{0} ∪ (∪_{j=0}^{3i+5}Δ_{j}^{0}) the locally closed substack of \overline{M}_{g} defined as the union of the locus M_{g}^{0} of smooth curves carrying no linear systems 𝔤_{k−1}^{1} to which we add the open subsets Δ_{j}^{0} ⊂ Δ_{j} for 1 ≤ j ≤ 3i + 5 consisting of 1-nodal genus g curves C ∪_{y} D, with [C] ∈ M_{g−j} and [D, y] ∈ M_{j,1} being Brill-Noether general curves, and the locus Δ_{0}^{0} ⊂ Δ_{0} containing 1-nodal irreducible genus g curves C′ = C/q ∼ y, where [C, q] ∈ M_{g−1} is a Brill-Noether general pointed curve and y ∈ C, together with their degenerations consisting of unions of a smooth genus g − 1 curve and a nodal rational curve. One can then extend the finite covering σ : 𝔊_{3k−6}^{k−2} → M_{g}^{0} to a proper, generically finite map

σ : 𝔊̃_{3k−6}^{k−2} → M̃_{g}

by letting 𝔊̃_{3k−6}^{k−2} be the variety of limit 𝔤_{3k−6}^{k−2}’s on the treelike curves from M̃_{g} (see [EH1], Theorem 3.4 for the construction of the space of limit linear series).

We will be interested in intersecting the divisors \overline{Z}_{6i+10,i} on \overline{M}_{g} with test curves

in the boundary of \overline{M}_{6i+10} which are defined as follows: we fix a Brill-Noether general curve C of genus 2k − 3 = 6i + 9, a general point q ∈ C and a general elliptic curve E. We define two 1-parameter families

(3) C^{0} := \{\frac{C}{y ∼ q} : y ∈ C\} ⊂ Δ_{0} ⊂ \overline{M}_{2k−2} and C^{1} := \{C∪_{y}E : y ∈ C\} ⊂ Δ_{1} ⊂ \overline{M}_{2k−2}.

It is well-known that these families intersect the generators of Pic(\overline{M}_{2k−2}) as follows:

C^{0} · λ = 0, C^{0} · δ_{0} = −(4k − 6), C^{0} · δ_{1} = 1 and C^{0} · δ_{a} = 0 for a ≥ 2, and
C^{1} · λ = 0, C^{1} · δ_{0} = 0, C^{1} · δ_{1} = −(4k − 8), C^{1} · δ_{a} = 0 for a ≥ 2.

Next, we fix an integer 2 ≤ j ≤ k − 1, a general curve C of genus 2k − 2 − j and a general curve pointed curve (D, y) of genus j. We define the 1-parameter family C^{j} := \{C ∪_{y} D : y ∈ C\} ⊂ Δ_{j} ⊂ \overline{M}_{2k−2}. We have that

C^{j} · λ = 0, C^{j} · δ_{a} = 0 for a ≠ j and C^{j} · δ_{j} = −(4k − 6 − 2j).

During our calculations we will often need the following enumerative result (cf. [HM], pg. 71):

Proposition 3.1. Let C be a general curve of genus j, y ∈ C a general fixed point and an integer 0 ≤ α ≤ j/2.

(1) The number of pencils A ∈ W_{j−α+1}^{1}(Y) satisfying h^{0}(Y, A(−(j + 1 − 2α)y)) ≥ 1 is equal to

a(j, α) = \frac{j + 1 − 2α}{j + 1 − α}\binom{j}{α}.

(2) The number of pencils A ∈ W_{j−α}^{1}(C) satisfying h^{0}(Y, A(−(j − 2α)q)) ≥ 1 for some unspecified point q ∈ Y is equal to

b(j, α) = (j − 2α − 1)(j − 2α)(j − 2α + 1)\binom{j}{α}.

Definition 3.2. (1) For a fixed general curve C of genus 2k − 3 we denote by δ(k, 1) the (finite) number of linear series L ∈ W_{3k−6}^{k−2}(C) for which there exists a point y ∈ C with h^{0}(L(−2y)) ≥ k − 2 and h^{0}(L(−ky)) ≥ 1.

(2) We fix 2 ≤ j ≤ k − 1, 0 ≤ α ≤ j/2 and a general curve C of genus 2k − 2 − j. We denote by δ(k, j, α) the number of linear series L_{C} ∈ W_{3k−j−4}^{k−2}(C) for which there exists a point y ∈ C such that h^{0}(L(−(α+1)y)) ≥ k−1−α, h^{0}(L(−(j−α+2)y)) ≥ k−1+α−j and h^{0}(L(−(k + 1)y)) ≥ 1.

The numbers δ(k, 1) and δ(k, j, α) can be computed by degeneration. Since we do not need their actual values we skip this calculation. Next we describe 𝔊̃_{3k−6}^{k−2} set theoretically. To set notation, if X is a treelike curve and l is a limit 𝔤_{d}^{r} on X, for a component Y of X we denote by l_{Y} = (L_{Y} , V_{Y} ⊂ H^{0}(L_{Y})) the Y -aspect of l. For a point y ∈ Y we denote by {a_{s}^{l_{Y}} (C)}_{s=0...r} the vanishing sequence of l at y and by ρ(l_{Y} , y) := ρ(g, r, d) − ∑_{i=0}^{r}(a_{i}^{l_{Y}} (y) − i) the adjusted Brill-Noether number with respect to y

Proposition 3.3. (1) Let C_{y}^{1} = C ∪_{y} E be an element of Δ_{1}^{0}. If (l_{C}, l_{E}) is a limit 𝔤_{3k−6}^{k−2} on C_{y}^{1}, then V_{C} = H^{0}(L_{C}) and L_{C} ∈ W_{3k−6}^{k−2}(C) has a cusp at y, that is, h^{0}(C, L(−2y)) = k − 2. If y ∈ C is a general point, then l_{E} = (O_{E}((3k − 6)y), (2k − 5)y + |(k − 1)y|), that is, l_{E} is uniquely determined. If y ∈ C is one of the finitely many points for which there exists L_{C} ∈ W_{3k−6}^{k−2}(C) such that ρ(L_{C}, y) = −1, then l_{E}(−(2k − 6)y) is a 𝔤_{k}^{k−2} with vanishing sequence at y being ≥ (0, 2, 3, …, k − 2, k). Moreover, at the level of 1-cycles we have the identification σ^{∗}(C^{1}) ≡ X + δ(k, 1) T, where

X := {(y, D) ∈ C × C_{k−2} : h^{0}(C, D + 2y) ≥ 2}

and T is the curve consisting of 𝔤_{k}^{k−2}’s on E with vanishing ≥ (0, 2, …, k − 2, k) at the fixed point y ∈ E.

(2) Let C_{y}^{0} = C/y ∼ q be an element of Δ_{0}^{0}. Then limit linear series of type 𝔤_{3k−6}^{k−2} on C_{y}^{0} are in 1:1 correspondence with complete linear series L on C of type 𝔤_{3k−6}^{k−2} satisfying the condition h^{0}(C, L⊗O_{C}(−y−q)) = h^{0}(C, L)−1. Thus there is an isomorphism between the cycle σ^{∗}(C^{0}) of 𝔤_{3k−6}^{k−2} on all curves C_{y}^{0} with y ∈ C and the smooth curve

Y := {(y, D) ∈ C × C_{k−2} : h^{0}(C, D + y + q) ≥ 2}.

Proof. We only prove (1) the remaining case being similar. Suppose l = (l_{C}, l_{E}) is a limit 𝔤_{3k−6}^{k−2} on C_{y}^{1}. Using the additivity of the Brill-Noether number (see e.g. [EH1], Lemma 3.6), we get that 0 = ρ(2k − 2, k − 2, 3k − 6) ≥ ρ(l_{C}, y) + ρ(l_{E}, y). Since the vanishing sequence at y of the E-aspect of l is

a^{l_{E}}(y) ≤ (2k − 5, 2k − 4, …, 3k − 8, 3k − 6),

it follows that we also have the inequality a^{l_{C}}(y) ≥ (0, 2, …, k − 1). If y ∈ C is general, then ρ(l_{C}, y) = 0 and a^{l_{C}}(y) = (0, 2, …, k − 1). Moreover, l_{C} corresponds to a complete linear series |L_{C}| on C such that h^{1}(C, L_{C}) = 1 and h^{1}(L_{C} ⊗ O_{C}(−2y)) = 2. If y ∈ C is one of the points such that ρ(l_{C}, y) = −1, then since C is Brill-Noether general we must have a_{0}^{l_{C}}(y) = 0, which implies that a^{l_{C}}(y) = (0, 2, …, k − 2, k), hence a^{l_{E}}(y) ≥ (2k − 6, 2k − 4, …, 3k − 8, 3k − 6), or equivalently, l_{E} = (2k − 6)y + |V_{E}|, where V_{E} ⊂ H^{0}(O_{E}(ky)) is a 𝔤_{k}^{k−2} with vanishing sequence ≥ (0, 2, 3, …, k − 2, k) at y. An easy argument shows that the variety Y of such V_{E}’s is isomorphic to P(O_{E}(ky)_{|2y}). □

Proposition 3.4. Let C_{y}^{j} := C ∪_{y} D be an element from Δ_{j}^{0}, where 2 ≤ j ≤ k − 1, g(C) = 2k − 2 − j and g(D) = j. Suppose (l_{C}, l_{D}) is a limit 𝔤_{3k−6}^{k−2} on C_{y}^{j}. If y ∈ C is a general point, then l_{C} has the divisor (j − 2)y as base locus and there exists an integer 0 ≤ α ≤ [j/2] such that l_{C}(−(j − 2)y) = |L_{C}|, with L_{C} ∈ W_{3k−j−4}^{k−2}(C) satisfying the inequality

a^{L_{C}}(y) ≥ (0, 1, …, α − 1, α + 1, α + 2, …, j − α, j − α + 2, j − α + 3, …, k − 1, k).

Moreover l_{D} is one of the a(j, α) linear systems on D of type (2k − j − 4)y + |L_{D}|, with L_{D} ∈ W_{k−2+j}^{k−2}(D) such that h^{0}(L_{D}(−(k+1−α)y)) = α andh^{0}(L_{D}(−(k+α−j)y)) = j−α.

If for some integer 0 ≤ α ≤ [j/2], y ∈ C is one of the b(2k − 2 − j, k − 1 + α − j) points for which there exists L_{C} ∈ Pic^{3k−j−3}(C) such that h^{0}(L_{C}(−(α + 2)y)) = k − 1 − α and h^{0}(L_{C}((j − α + 2)y)) = k − j + α, then there are two possibilities:

(1) The aspect l_{C} is of the form (j − 2)y + |L_{C}(−y)|, whereas l_{D} = (2k − j − 4)y + |L_{D}|, where L_{D} ∈ W_{k−2+j}^{k−2}(D) satisfies h^{0}(L_{D}(−(k + 1 − α)y)) ≥ α andh^{0}(L_{D}(−(k + 1 + α − j)y)) ≥ j − α − 1.

(2) The aspect l_{C} is of the form (j − 3)y + (L_{C}, V_{C}), where V_{C} ⊂ H^{0}(L_{C}) is such that H^{0}(L_{C}(−2y)) ⊂ V_{C}, and l_{D} = (2k − j − 4)y + |L_{D}| with L_{D} ∈ W_{k−2+j}^{k−2}(D) and

a^{L_{D}}(y) = (0, 1, …, k − 1 − j + α, k + 1 − j + α, …, k − 1 − α, k + 1 − α, …, k − 1, k + 1).

Finally, if for some 0 ≤ α ≤ [j/2], y ∈ C is one of the δ(k, j, α) points for which there exists L_{C} ∈ Pic^{3k−4−j}(C) such that

a^{L_{C}}(y) = (0, 1, …, α − 1, α + 1, …, j − α, j − α + 2, …, k − 2, k − 1, k + 1),

then l_{C} = (j −2)y +|L_{C}| and l_{D} = (2k −j −5)y +lD^{′}, where lD^{′} is a (non-complete) 𝔤_{k−1+j}^{k−2} on

D satisfying a^{l′_{D}}(y) = (0, 2, …, k −1+α−j, k +1+α−j, …, k −α, k +2−α, …, k, k +1).

Proof. Since by definition [D, y] ∈ M_{j,1} is Brill-Noether general, we have that ρ(l_{D}, y) ≥

0. The locus of those (C, y) ∈ M_{2k−2−j,1} carrying a 𝔤_{3k−6}^{k−2} with ρ(𝔤_{3k−6}^{k−2}, y) ≤ −2 is of codimension ≥ 2, hence we must also have that ρ(l_{C}, y) ≥ −1. Using Proposition 1.2 from [EH3] we have that a_{k−2}^{l_{C}}(y) ≤ k − 2 + j and a_{0}^{l_{C}}(y) ≥ j − 3. There are two cases to consider:

(a) If ρ(l_{C}, y) = 0, then ρ(l_{D}, y) = 0. Assume first that a_{0}^{l_{C}}(y) = j − 2. Since we have that 0 ≤ a_{s}^{l_{C}}(y)−s−(j −2) ≤ 2 for all 0 ≤ s ≤ k−2, there must exist an integer 0 ≤ α ≤ [j/2], such that if L_{C} denotes the 𝔤_{3k−4−j}^{k−2} obtained from l_{C} by removing (j − 2)y, then the vanishing at y is a^{L_{C}}(y) = (0, 1, …, α − 1, α + 1, α + 2, …, j − α, j − α + 2, …, k). By compatibility, l_{D} has (2k−j −4)y in its base locus and we write l_{D} = (2k−j −4)y+|L_{D}|, where a^{L_{D}}(y) = (0, 1, …, k − 2 + j − α, k + j − α, …, k − 1 − α, k + 1 − α, …, k). By Riemann-Roch, the number of such L_{D}’s equals the number of 𝔤_{j−α+1}^{1}’s on D with a (j − 2α + 1)-fold point at y which is computed by a(j, α). If a_{0}^{l_{C}}(y) = j − 3, we denote by L_{C} ∈ Pic^{3k−3−j}(C) the line bundle obtained from l_{C} by subtracting (j − 3)y, and again we see that there must exist 0 ≤ α ≤ [j/2] such that h^{0}(L_{C}(−(j − α + 2)y)) = k − j + α and h^{0}(L_{C}(−(α + 2)y)) = k − 1 − α. By duality, K_{C} ⊗ L_{C}^{∨} ⊗ O_{C}((j − α + 2)y) is one of the finitely many 𝔤_{k−1−α}^{1}’s on C with a (j − 2α)-fold point at y. This number equals b(2k − 2 − j, k − 1 + α − j) and this explains case (2) of our statement.

(b) If ρ(l_{C}, y) = −1, then ρ(l_{D}, y) = 1. An argument similar to that used above gives two possibilities for a^{l_{C}}(y) and the rest follows because l_{C} and l_{D} are compatible at y. □

Retaining the notation from Proposition 3.4, for each integer 0 ≤ α ≤ [j/2] we define the following 1-cycles:

X_{j,α} := {(y, L_{C}) ∈ C × Pic^{3k−j−4}(C) : a_{α}^{L_{C}}(y) ≥ α + 1, a_{j−α}^{L_{C}}(y) ≥ j − α + 2},

Yj,α^{′} := {(y, l) ∈ C × G_{3k−j−3}^{k−2}(C) : a_{1}^{l}(y) ≥ 2, a_{α}^{l}(y) ≥ α + 2, a_{j−α−1}^{l}(y) ≥ j − α + 2},

Yj,α^{′′} := {L_{D} ∈ Pic^{k−2+j}(D) : a_{k+α−j}^{L_{D}}(y) ≥ k + α − j + 1, a_{k−α−1}^{L_{D}}(y) ≥ k − α + 1}

and

Yj,α^{′′′} := {l ∈ G_{k−1+j}^{k−2}(D) : a_{1}^{l}(y) ≥ 2, a_{k−1+α−j}^{l}(y) ≥ k + α − j + 1, a_{k−1−α}^{l}(y) ≥ k − α + 2}.

Via Proposition 3.4 we view X_{j,α}, Yj,α^{′}, Yj,α^{′′} and Yj,α^{′′′} as 1-cycles on 𝔊̃_{3k−6}^{k−2}. Note that X_{j,α} projects onto C while Yj,α^{′} is isomorphic to the union of b(2k − 2 − j, k − 1 + α − j) copies of P^{1}. We can then rephrase Proposition 3.4 as follows:

Proposition 3.5. If σ : 𝔊̃_{3k−6}^{k−2} → M̃_{g} is the natural projection and [C] ∈ M_{2k−2−j} where 2 ≤ j ≤ k − 1, then we have the following numerical equivalence relation between 1-cycles:

σ^{∗}(C^{j}) ≡ ∑_{α=0}^{[j/2]}(a(j, α) X_{j,α} + δ(k, 2k − 2 − j, k + α − j) Yj,α^{′}+

+b(2k − 2 − j, k − 1 + α − j) Yj,α^{′′} + δ(j, α) Yj,α^{′′′}).

To compute the class of the cycles introduced in Propositions 3.3 and 3.5, we introduce some notation. Let C be a Brill-Noether general curve of genus g. We denote by C_{d} the d-th symmetric product of C, by U ⊂ C × C_{d} the universal divisor, and by L the Poincare bundle on C ×Pic^{d}(C). We also introduce the projections π_{1} : C ×C_{d} → C and π_{2} : C × C_{d} → C_{d} (and we use the same notation for the projections from C × Pic^{d}(C) onto the factors). We define the cohomology classes

η = π_{1}^{∗}([point]) ∈ H^{2}(C × C_{d}) and x = π_{2}^{∗}(D_{x_{0}}) ∈ H^{2}(C × C_{d}),

where x_{0} ∈ C is an arbitrary point and D_{x_{0}} := {D ∈ C_{d} : x_{0} ∈ D} (the definition of x is obviously independent of the point x_{0}). Finally, if δ_{1}, …, δ_{2g} ∈ H^{1}(C, ℤ) ≅ H^{1}(C_{d}, ℤ) is a symplectic basis, then we define the class

γ := − ∑_{α=1}^{g}(π_{1}^{∗}(δ_{α})π_{2}^{∗}(δ_{g+α}) − π_{1}^{∗}(δ_{g+α})π_{2}^{∗}(δ_{α})).

With these notations we have the formula (cf. [ACGH], p. 338) [U] ≡ dη + γ + x, corresponding to the Hodge decomposition of [U]. We also record the formulas γ^{3} = γη = 0, η^{2} = 0 and γ^{2} = −2ηπ_{2}^{∗}(θ), where θ ∈ H^{2}(C_{d}, ℤ) is the pullback of the class of the theta divisor on Jac(C). Similarly, c_{1}(L) = dη + γ, where η, γ ∈ H^{2}(C × Pic^{d}(C)) pull back to η, γ ∈ H^{2}(C × C_{d}). When d = k − 2 and g = 2k − 3, we set M := π_{1}^{∗}(K_{C}) ⊗ O_{C×C_{k−2}}(−U), hence c_{1}(M) = (3k − 6)η − γ − x.

We now compute the class of the curves X and Y defined in Proposition 3.3:

Proposition 3.6. Let C be a Brill-Noether general curve of genus g = 2k − 3 and q ∈ C a general point.

(1) The class of the curve X = {(y, D) ∈ C × C_{k−2} : h^{0}(C, D + 2y) ≥ 2} is given by

[X] ≡ \frac{θ^{k−2}}{(k − 2)!} − \frac{θ^{k−3}x}{(k − 3)!} − \frac{2θ^{k−3}γ}{(k − 3)!}+

+\frac{2xθ^{k−4}γ}{(k − 4)!} − \frac{4(k + 1)}{(k − 4)!}xθ^{k−4}η + \frac{4k − 2}{(k − 3)!}θ^{k−3}η.

(2) The class of the curve Y = {(y, D) ∈ C × C_{k−2} : h^{0}(C, D + y + q) ≥ 2} is given by

[Y] ≡ \frac{θ^{k−2}}{(k − 2)!} − \frac{θ^{k−3}x}{(k − 3)!} − \frac{θ^{k−3}γ}{(k − 3)!}+

+\frac{xθ^{k−4}γ}{(k − 4)!} − \frac{k + 1}{(k − 4)!}xθ^{k−4}η + \frac{k − 1}{(k − 3)!}θ^{k−3}η.

Proof. We will realize both X and Y as pullbacks of degeneracy loci and compute their classes using the Thom-Porteous formula. We consider the map ϵ : C × C_{k−2} → C_{k} given by ϵ(y, D) := 2y + D. It is easy to check that

ϵ^{∗}(x) = 2η + x and ϵ^{∗}(θ) = 4gη + θ − 2γ.

Then X = ϵ^{∗}(C_{k}^{1}) and to compute the class of C_{k}^{1} = {E ∈ C_{k} : h^{0}(E) ≥ 2} we introduce the rank k vector bundle P := (π_{2})_{∗}(π_{1}^{∗}K_{C} ⊗ O_{U}) on C_{k} having total Chern class c_{t}(P) = (1 − x)^{−k+4}e^{\frac{θ}{1−x}} (cf. [ACGH], pg. 240). There is a natural bundle map ϕ^{∨} : ϵ^{∗}(P^{∨}) → H^{0}(K_{C})^{∨} ⊗ O_{C×C_{k−2}}, and X = Z_{k−1}(ϕ^{∨}). Then by Thom-Porteous, we can write

[X] = [\frac{1}{c_{t}(ϵ^{∗}(P^{∨}))}]_{k−2} = ϵ^{∗}(\frac{θ^{k−2}}{(k − 2)!} − \frac{xθ^{k−3}}{(k − 3)!}) =

= \frac{(4gη + θ − 2γ)^{k−2}}{(k − 2)!} − \frac{(4gη + θ − 2γ)^{k−3}(2η + x)}{(k − 3)!},

which quickly leads to the desired expression for [X].

To compute the class of Y we proceed in a similar manner: we consider the map χ : C × C_{k−2} → C_{k} given by χ(y, D) := y + q + D. Then χ^{∗}(x) = x + η and χ^{∗}(θ) = gη + θ − γ. For each (y, D) ∈ C × C_{k−2} we have a natural map

H^{0}(K_{C|y+q+D})^{∨} → H^{0}(K_{C})^{∨}

which globalizes to a vector bundle map ψ^{∨} : χ^{∗}(P^{∨}) → H^{0}(K_{C})^{∨} ⊗ O_{C×C_{k−2}} and then it is clear that Y = Z_{k−1}(ψ^{∨}). Applying Thom-Porteous again, we obtain

[Y] = [\frac{1}{c_{t}(χ^{∗}(P^{∨}))}]_{k−2} = χ^{∗}(\frac{θ^{k−2}}{(k − 2)!} − \frac{xθ^{k−3}}{(k − 3)!}) =

= \frac{(gη + θ − γ)^{k−2}}{(k − 2)!} − \frac{(gη + θ − γ)^{k−3}(η + x)}{(k − 3)!},

which after a straightforward calculation gives the class of Y. □

We also need the following intersection theoretic result:

Lemma 3.7. For j ≥ 1 we denote by F_{j} := (π_{2})_{∗}(M^{⊗j}) the vector bundle on C_{k−2} with fibre F_{j}(D) = H^{0}(C, jK_{C} − jD) over each D ∈ C_{k−2}. Then c_{1}(F_{j}) = −j^{2}θ − j(3j − 2)(k − 2)x.

Proof. We apply the Grothendieck-Riemann-Roch Theorem for the map π_{2} : C×C_{k−2} → C_{k−2}. We obtain that

c_{1}(F_{j}) = [(π_{2})_{∗}((1 + j((3k − 6)η − γ − x) + \frac{j^{2}}{2} ((3k − 6)η − γ − x)^{2}) · (1 − (2k − 4)η))]_{1} =

= −j^{2}θ − j(3j − 2)(k − 2)x.

□

Next we compute the class of X_{j,α} when 2 ≤ j ≤ k − 1. It is convenient to state our result as follows:

Proposition 3.8. Let C be a general curve of genus 2k − 2 − j such that 2 ≤ j ≤ k − 1. We have the following relation in H^{4k−4−2j}(C × Pic^{3k−j−4}(C)):

∑_{α=0}^{[j/2]} a(j, α) X_{j,α} ≡ \frac{2(2k − 3)!}{k!(k − 2)!(2k − 3 − j)!}(\frac{θ^{2k−2−j}}{2k − 2 − j} +\frac{(3kj + 12k − 9j − 18)θ^{2k−3−j}γ}{2k − 3} +

\frac{θ^{2k−3−j}η}{(2k − 5)(2k − 3)}(60−64k+16k^{2} +j(60− \frac{67}{2}k−10k^{2} +6k^{3})+j^{2}(15+2k−5k^{2})+ \frac{3}{2}kj^{3})).

Proof. We fix an integer 0 ≤ α ≤ [j/2] and a divisor D ∈ C_{e} of degree e ≥ k−j−1. We set d := 3k −4−j, Γ := D ×Pic^{d}(C) and denote by u, v : C ×C ×Pic^{d}(C) → C ×Pic^{d}(C) the two projections and by J_{α}(L) the α-th jet bundle of the Poincare bundle on C × Pic^{d}(C). Then P(D) := (π_{2})_{∗}(L ⊗ O_{C×Pic^{d}(C)}(Γ)) is a vector bundle of rank e + k − 1. For

each α ≥ 0 we define the vector bundle J_{α}(L, D) := u_{∗}(v^{∗}(L) ⊗ O_{(α+1)Δ+v^{∗}(Γ)}), where Δ ⊂ C × C × Pic^{d}(C) is the diagonal, and consider the vector bundle map

ϕ_{α} : J_{α}(L, D)^{∨} → (π_{2})^{∗}(P(D))^{∨},

which on fibres is the dual of the map H^{0}(L_{C} ⊗ O(D)) → H^{0}(L_{C} ⊗ O_{(α+1)y+D}(D)) for each pair (y, L_{C}) ∈ C × Pic^{d}(C).

The cycle X_{j,α} consists of pairs (y, L_{C}) such that dim Ker(ϕ_{α}(y, L_{C})) ≥ 1 and dim Ker(ϕ_{j−α+1}(y, L_{C})) ≥ 2 and its class is given by the formula (cf. [Fu],Theorem 14.3):

X_{j,α} ≡ c_{j+1−α}(α) c_{k−1+α−j}(j − α + 1) − c_{k−α}(α) c_{k−2+α−j}(j − α + 1),

where c_{l}(β) := c_{l}((π_{2})^{∗}(P(D))^{∨} − J_{β}(L, D)^{∨}). Since c_{tot}((π_{2})^{∗}(P(D)) = e^{θ} and c_{1}(L) = dη + γ (independent of D!), after a short calculation we obtain that

c_{l}(β) = \frac{θ^{l}}{l!} + \frac{θ^{l−1}}{(l − 1)!}((β + 1)γ + (β + 1)(d + β(2k − 3 − j))η) − \frac{θ^{l−1}}{(l − 2)!}(β + 1)(β + 2)η.

A straightforward computation now leads to the stated formula. □

For each integers 0 ≤ a ≤ k − 2 and b ≥ 2 we shall define vector bundles G_{a,b} over 𝔊̃_{3k−6}^{k−2} with fibre

G_{a,b}(C, L) = H^{0}(∧^{a}M_{L} ⊗ L^{⊗b})

over every point (C, L) ∈ 𝔊_{3k−6}^{k−2}. Note that G_{i,2|𝔊_{3k−6}^{k−2}} = B, where B is the vector bundle we introduced in Proposition 2.5. Of course, the question is how to extend this description over the divisors Δ_{j}^{0}. First we will extend G_{a,b} over σ^{−1}(M_{g}^{0} ∪Δ_{0}^{0} ∪Δ_{1}^{0}) and we start by constructing the vector bundles G_{0,b}:

Proposition 3.9. For each b ≥ 2 there exists a vector bundle G_{0,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{1}^{0} ∪ Δ_{0}^{0}) ⊂ 𝔊̃_{3k−6}^{k−2} of rank k(3b − 2) − 6b + 3 whose fibres admit the following description:

• For (C, L) ∈ 𝔊_{3k−6}^{k−2}, we have that G_{0,b}(C, L) = H^{0}(C, L^{⊗b}).

• For t = (C ∪_{y} E, L) ∈ σ^{−1}(Δ_{1}^{0}), where L is the linear series 𝔤_{3k−6}^{k−2} on C determining

a limit 𝔤_{3k−6}^{k−2} on C ∪_{y} E, we have that
G_{0,b}(t) = H^{0}(C, L^{⊗b}(−2y)) + ℂ · u^{b} ⊂ H^{0}(C, L^{⊗b}),
where u ∈ H^{0}(C, L) is any section such that ord_{y}(u) = 0.

• For t = (C/y ∼ q, L) ∈ σ^{−1}(Δ_{0}^{0}), where q, y ∈ C and L is a 𝔤_{3k−6}^{k−2} on C, we have that

G_{0,b}(t) = H^{0}(C, L^{⊗b}(−y − q)) ⊕ ℂ · u^{b} ⊂ H^{0}(C, L^{⊗b}),
where u ∈ H^{0}(C, L) is a section such that ord_{y}(u) = ord_{q}(u) = 0.

Proof. Recalling that σ : 𝔊̃_{d}^{r} → M̃_{g} denotes the variety of limit 𝔤_{d}^{r}’s over all treelike curves of genus g, the morphism 𝔊_{3k−6}^{k−2} → 𝔊_{b(3k−6)}^{k(3b−2)−6b+2} given by (C, L) ↦ (C, L^{⊗b}) can be extended to a morphism ν_{b} : 𝔊̃_{3k−6}^{k−2} → 𝔊̃_{b(3k−6)}^{k(3b−2)−6b+2} over M_{g}∪Δ_{0}^{0}∪Δ_{1}^{0} as follows. We fix a point t = (C ∪_{y} E, l = (l_{C}, l_{E})) a limit 𝔤_{3k−6}^{k−2}, where l_{C} = (L_{C}, H^{0}(L_{C})) ∈ G_{3k−6}^{k−2}(C) is such that H^{0}(L_{C}) = H^{0}(L_{C}(−2y)) ⊕ ℂ · u, for a certain u ∈ H^{0}(L_{C}) with ord_{y}(u) = 0, while l_{E} = (O_{E}((3k − 6)y), (2k − 6)p + |V_{E}|) (any limit 𝔤_{3k−6}^{k−2} on C ∪_{y} E is of this form). Then we set ν_{b}(t) := (l_{C}^{b}, l_{E}^{b}), where l_{C}^{b} = (L_{C}^{⊗b}, H^{0}(L_{C}^{⊗b}(−2y)) ⊕ ℂ · u^{b} ⊂ H^{0}(L_{C}^{⊗b})) and l_{E}^{b} = (O_{E}(b(3k − 6)p), (2k − 3)p + |(k(3b − 2) − 6b + 3)p|). In other words, (l_{C}^{b}, l_{E}^{b}) is the b-th power of the original limit linear series 𝔤_{3k−6}^{k−2}.

For a point z = (C/y ∼ q, L_{C}) ∈ Δ_{0}^{0} with L_{C} being a 𝔤_{3k−6}^{k−2} such that H^{0}(L_{C}) = H^{0}(L_{C}(−y − q)) ⊕ ℂ · u, we define ν_{b}(z) := (L_{C}^{⊗b}, H^{0}(L_{C}^{⊗b}(−y − q)) ⊕ ℂ · u^{b} ⊂ H^{0}(L_{C}^{⊗b})). The fact that ν_{b} can be constructed algebraically follows easily from the equations of the scheme 𝔊̃_{3k−6}^{k−2} described in [EH1], pg. 358. The variety 𝔊̃_{b(3k−6)}^{k(3b−2)−6b+2} carries a tautological vector bundle T = T_{C} with fibre over each point corresponding to a curve from M_{g}^{0} ∪ Δ_{0}^{0} being the space of global sections of the linear series, while the fibre over a point corresponding to a curve from Δ_{1}^{0} is the space of sections of the aspect of the limit linear series corresponding to the curve of genus g − 1. We define G_{0,b} := ν_{b}^{∗}(T ). The description of the fibres of the vector bundle G_{0,b} is now immediate. □

Having defined the vector bundles G_{0,b} we now define inductively all vector bundles G_{a,b}: first we define G_{1,b} as the kernel of the multiplication map G_{0,1} ⊗ G_{0,b} → G_{0,b+1}, that is, by the exact sequence

0 ⟶ G_{1,b} ⟶ G_{0,1} ⊗ G_{0,b} ⟶ G_{0,b+1} ⟶ 0.

Having defined G_{l,b} for all l ≤ a − 1, we define the vector bundle G_{a,b} by the exact sequence

(4)  0 ⟶ G_{a,b} ⟶ ∧^{a}G_{0,1} ⊗ G_{0,b} ⟶^{d_{a,b}} G_{a−1,b+1} ⟶ 0.

We now prove the right-exactness of the sequence (4) which will ensure the correctness of the definition of G_{a,b}:

Proposition 3.10. For integers b ≥ 2 and 1 ≤ a ≤ i, the Koszul multiplication map d_{a,b} : ∧^{a}G_{0,1} ⊗ G_{0,b} → G_{a−1,b+1} is well-defined and surjective. In particular one can define the vector bundle G_{a,b} := Ker(d_{a,b}) over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}) and the sequence (4) makes sense.

Proof. We do induction on a and start with the case a = 1. We only check the surjectivity of the map G_{0,1}⊗G_{0,b} → G_{0,b+1} over the locus σ^{−1}(Δ_{1}^{0}), the other cases, namely σ^{−1}(M_{g}^{0}) and σ^{−1}(Δ_{0}^{0}) being quite similar. It is enough to show that if C is a sufficiently general curve of genus 2k − 3, y ∈ C is a point and L ∈ W_{3k−6}^{k−2}(C) is a linear system with a cusp at y, then the map H^{0}(L) ⊗ H^{0}(L^{⊗b}(−2y)) → H^{0}(L^{⊗(b+1)}(−2y)) is onto. This is equivalent to H^{1}(M_{L} ⊗ L^{⊗b}(−2y)) = 0 which follows because of Proposition 2.1.

We now treat the general case 1 ≤ a ≤ i and we want to show that the Koszul map d_{a,b} : ∧^{a}H^{0}(L) ⊗ G_{0,b}(L) → G_{a−1,b+1}(L) is surjective for each b ≥ 2 and for each (C, L) as above. For simplicity we have denoted G_{a,b}(L) = G_{a,b}(t), where t ∈ σ^{−1}(Δ_{1}^{0}) is the point (C ∪_{y} E, l), E is an arbitrary elliptic tail and l is a limit linear series having L as its C-aspect. We can then write H^{0}(L)/H^{0}(L(−2y)) = ℂ · u, where u ∈ H^{0}(L) is uniquely determined up to translation by an element from H^{0}(L(−2y)). We first claim that the restriction of the Koszul differential

d_{a,b}^{0} : ∧^{a}H^{0}(L) ⊗ H^{0}(L^{⊗b}(−2y)) → H^{0}(∧^{a−1}M_{L} ⊗ L^{⊗(b+1)}(−2y))

is surjective. This is so because Coker(d_{a,b}^{0}) = H^{1}(∧^{a}M_{L} ⊗ L^{⊗b}(−2y)) = 0 (use again that M_{L} is a stable bundle). But then since

H^{0}(∧^{a−1}M_{L} ⊗ L^{⊗(b+1)}(−2y)) ⊂ G_{a−1,b+1}(L)

is a linear subspace of codimension \binom{k−2}{a−1}, to prove that d_{a,b} itself is surjective it suffices to notice that the image of the injective map ∧^{a−1}H^{0}(L(−2y)) → ∧^{a−1}H^{0}(L)⊗G_{0,b+1}(L) given by

f_{1} ∧ … ∧ f_{a−1} ↦ f_{1} ∧ … ∧ f_{a−1} ⊗ u^{b+1} − ∑_{l=1}^{a−1}(−1)^{l−1}u ∧ f_{1} ∧ … ∧ f̌_{l} ∧ … ∧ f_{a−1} ⊗ f_{l}u^{b}

is entirely contained in Im(d_{a,b}) and is clearly disjoint from H^{0}(∧^{a−1}M_{L}⊗L^{⊗(b+1)}(−2y)). □

To compute the intersection numbers of the divisors \overline{Z}_{g,i} with the test curves C^{0} and C^{1} we need to understand the restriction of the vector bundle G_{i,2} to X and Y.

Proposition 3.11. Let C be a general curve of genus 2k − 3 and k = 3i + 6 with i ≥ 0. If C^{1} is the test curve in Δ_{1} obtained by attaching to C a fixed elliptic tail at a varying point of C and X = σ^{−1}(C^{1}), then we have the following formula:

c_{1}(G_{2,i|X}) = \binom{3i + 4}{i}(c_{x} x + c_{η} η + c_{γ} γ + c_{θ} θ),

where

c_{η} = −27i + 40, c_{x} = −\frac{27i^{4} + 153i^{3} + 331i^{2} + 323i + 120}{(3i + 4)(i + 1)}

c_{γ} = \frac{5i + 8}{3i + 4}, c_{θ} = −\frac{27i^{3} + 101i^{2} + 124i + 48}{(3i + 3)(3i + 4)}.

Proof. For j ≥ 2 we define the jet bundle J_{1}(M^{⊗j}) := (u_{∗})(v^{∗}(O_{C×C_{k−2}}(M^{⊗j})) ⊗ O_{2Δ}), where u, v : C ×C ×C_{k−2} → C ×C_{k−2} are the projections. We have two exact sequences of vector bundles on X:

0 ⟶ u_{∗}(v^{∗}(M^{⊗j}) ⊗ O(−2Δ))_{|X} ⟶ G_{0,j|X} ⟶ M_{|X}^{⊗j} ⟶ 0

and

0 ⟶ u_{∗}(v^{∗}(M^{⊗j}) ⊗ O(−2Δ))_{|X} ⟶ π_{2}^{∗}(F_{j})_{|X} ⟶ J_{1}(M^{⊗j})_{|X} ⟶ 0,

from which we can write that

c_{1}(G_{0,j |X}) = c_{1}(π_{2}^{∗}(F_{j})_{|X}) − c_{1}((π_{1}^{∗}K_{C} ⊗ M^{⊗j})_{|X}) =
= −j^{2}θ − (3j + 4)(k − 2) η + jγ − j((3j − 2)(k − 2) − 1) x.

From the exact sequences defining G_{i,2} we then obtain that

c_{1}(G_{i,2|X}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1|X} ⊗ G_{0,l+2|X}).

Note that Riemann-Roch gives that rk(G_{0,l+2}) = (3k−6)l+4k−9, while c_{1}(∧^{i−l}G_{0,1|X}) = \binom{k−2}{i−l−1}(−θ − (k − 2)x). To obtain a closed formula for c_{1}(G_{i,2|X}) we now specialize to the case k = 3i + 6 and we write that

c_{1}(G_{i,2|X}) = ∑_{l=0}^{i}(−1)^{l}[((9i + 12)l + (12i + 15))\binom{3i + 4}{i − l − 1}(−θ − (3i + 4)x)+

+\binom{3i + 5}{i − l}(−(l + 2)^{2}θ + (l + 2)γ − (3i + 4)(3l + 10)η − (l + 2)((3l + 4)(3i + 4) − 1)x)].

A long but elementary calculation leads to the stated formula.

□

A similar calculation yields the first Chern class of G_{i,2|Y}:

Proposition 3.12. Let (C, q) be a general pointed curve of genus 2k − 3 and C^{0} the test curve in Δ_{0} obtained by identifying the fixed point q with a varying point y on C. If Y = σ^{−1}(C^{0}), then we have the formula

c_{1}(G_{i,2|Y}) = \binom{3i + 4}{i}(d_{x} x + η + d_{θ} θ),

where

d_{x} = −\frac{27i^{4} + 153i^{3} + 331i^{2} + 323i + 120}{(3i + 4)(i + 1)}, d_{θ} = −\frac{27i^{3} + 101i^{2} + 124i + 48}{(3i + 3)(3i + 4)}.

Proof. We define Γ_{q} := C × {q} × C_{k−2} and denote by ι : C × C_{k−2} ↪ Γ_{q} the inclusion map. Then we have the exact sequences on Y

0 ⟶ (u_{∗})(v^{∗}(M^{⊗j}) ⊗ O(−Δ − Γ_{q}))_{|Y} ⟶ G_{0,j|Y} ⟶ M_{|Y}^{⊗j} ⟶ 0,

and

0 ⟶ ι^{∗}(v^{∗}(M^{⊗j}) ⊗ O(−Δ))_{|Y} ⟶ (u_{∗})(v^{∗}(M^{⊗j}) ⊗ O_{Δ+Γ_{q}})_{|Y} ⟶ π_{2}^{∗}(F_{j})_{|Y} ⟶ 0.

Using that c_{1}(M) = (3k − 6)η − γ − x, it is now straightforward to check that

c_{1}(ι^{∗}(v^{∗}(M^{⊗j}) ⊗ O(−Δ))) = −jx − η,

hence c_{1}(G_{0,j|Y}) = c_{1}(π_{2}^{∗}(F_{j}))_{|Y} + jx + η = −j^{2}θ + η − j((3j − 2)(k − 2) − 1)x. But then as in the previous proposition we have that

c_{1}(G_{i,2|Y}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1|Y} ⊗ G_{0,l+2|Y}),

which after some calculations yields the stated formula in the special case k = 3i+6. □

In what follows we extend the vector bundles G_{a,b} over the whole 𝔊̃_{3k−6}^{k−2}. Roughly speaking, the fibre of G_{a,b} over a point corresponding to a singular curve C ∪_{y} D will be the space H^{0}(C ∪ D, ∧^{a}M_{L} ⊗ L^{⊗b}), where L will be a line bundle on C ∪ D obtained by twisting appropriately a limit 𝔤_{3k−6}^{k−2} on C ∪ D. The twisting is chosen such that, the vector bundles G_{a,b} sit in the exact sequences (4). In the next statement we use the notation introduced in Proposition 3.5:

Theorem 3.13. For integers 0 ≤ a ≤ i, b ≥ 2, or (a, b) = (0, 1), there exists a vector bundle G_{a,b} over 𝔊̃_{3k−6}^{k−2} having the following properties:

(1) If (C, L) ∈ 𝔊_{3k−6}^{k−2}, then G_{a,b}(C, L) = H^{0}(∧^{a}M_{L} ⊗ L^{⊗b}), that is, G_{a,b |𝔊_{3k−6}^{k−2}} = G_{a,b}.

If t = (C ∪_{y} D, L_{C}, L_{D}) ∈ σ^{−1}(Δ_{j}^{0}) ⊂ 𝔊̃_{3k−6}^{k−2}, with 2 ≤ g(D) = j ≤ k − 1, g(C) = 2k − 2 − j, L_{C} ∈ W_{3k−4−j}^{k−2}(C) and L_{D} ∈ W_{k−2+j}^{k−2}(D), then there are two situations:

(2) If 2 ≤ j ≤ 2i + 2, then G_{a,b}(t) = H^{0}(C ∪ D, ∧^{a}M_{L_{C∪D}} ⊗ L_{C∪D}^{⊗b}), where L_{C∪D} = (L_{C}(−(j + 2)y), L_{D}(−(k − 2 − j)y)) ∈ Pic^{3k−6−2j}(C) × Pic^{2j}(D) is a globally generated line bundle on C ∪ D.

(3) If 2i + 3 ≤ j ≤ 3i + 5 and c := [(j + 2)/2], then G_{a,b}(t) = H^{0}(C ∪ D, ∧^{a}M_{L_{C∪D}} ⊗ L_{C∪D}^{⊗b}), where L_{C∪D} = (L_{C}(−cy)), L_{D}(−(k − c)y)) ∈ Pic^{3k−4−j−c}(C) × Pic^{j+c−2}(D) is a globally generated line bundle on C ∪ D.

(4) There are exact sequences over 𝔊̃_{3k−6}^{k−2}: 0 ⟶ G_{a,b} ⟶ ∧^{a}G^{0,1} ⊗ G_{0,b} ⟶ G_{a−1,b+1} ⟶ 0.

Proof. For j ≥ 2 the divisors Δ_{j}^{0} are mutually disjoint and we can carry out the construction of G_{a,b} over each open set σ^{−1}(M_{g}^{0} ∪ Δ_{j}^{0}) ⊂ 𝔊̃_{3k−6}^{k−2} and glue the resulting bundles together. We denote by π : C_{j} → M_{g}^{0} ∪ Δ_{j}^{0} the restriction of the universal curve and by p : Y = C_{j} ×_{M_{g}^{0}∪Δ_{j}^{0}} 𝔊̃_{3k−6}^{k−2} → 𝔊̃_{3k−6}^{k−2}. Then (σp)^{−1}(Δ_{j}^{0}) = D_{j} + D_{2k−2−j}, where D_{j} (resp. D_{2k−2−j}) denotes the divisor in Y corresponding to the marked point being on the genus j (resp. 2k − 2 − j) component. If L = L_{0,3k−6} is the Poincare bundle on Y chosen such that it parametrizes bundles having bidegree (0, 3k − 6) on curves [D ∪_{y} C] ∈ Δ_{j}^{0}, then for b ≥ 1 we set

G_{0,b} := p_{∗}(L^{⊗b} ⊗ O_{Y}(−2bj D_{j})), when 2 ≤ j ≤ 2i + 2,

and G_{0,b} := p_{∗}(L^{⊗b} ⊗ O_{Y}(−b(j − 2 + c)D_{j})) when 2i + 3 ≤ j ≤ 3i + 5. For each point t = (R = C ∪_{y} D, L_{C}, L_{D}) ∈ σ^{−1}(Δ_{j}^{0}), we have that G_{0,b}(t) = H^{0}(R, L_{R}^{⊗b}), where in the case j ≤ 2i + 2 we have L_{R} = (L_{C}(−(j + 2)y), L_{D}(−(k − 2 − j)y)) ∈ Pic^{3k−6}(R) and since H^{1}(R, L_{R}^{⊗b}) = 0 for b ≥ 2, it follows that G_{0,b} is a vector bundle for all b ≥ 1. When 1 ≤ a ≤ i, the sheaves G_{a,b} are defined inductively using the sequences (4). For this to make sense and in order to conclude that G_{a,b} are vector bundles, one has to show that

(5)  H^{1}(R, ∧^{a}M_{L_{R}} ⊗ L_{R}^{⊗b}) = 0,

for all b ≥ 2 and 0 ≤ a ≤ i. To achieve this we use the Mayer-Vietoris sequence on R:

0 → ∧^{a}M_{L_{R}}⊗L_{R}^{⊗b} ⟶ (∧^{a}M_{L_{R}}⊗L_{R}^{⊗b})_{|C}⊕(∧^{a}M_{L_{R}}⊗L_{R}^{⊗b})_{|D} ⟶ ∧^{a}M_{L_{R}}⊗L_{R}^{⊗b}⊗ℂ(y) → 0,

together with the exact sequences

0 ⟶ H^{0}(L_{R|C}(−y)) ⊗ O_{D} ⟶ M_{L_{R}} ⊗ O_{D} ⟶ M_{L_{R|D}} ⟶ 0,

and

0 ⟶ H^{0}(L_{R|D}(−y)) ⊗ O_{C} ⟶ M_{L_{R}} ⊗ O_{C} ⟶ M_{L_{R|C}} ⟶ 0.

We obtain that (5) holds if we can show that

H^{1}(C, ∧^{a}M_{L_{R|C}} ⊗ L_{R|C}^{⊗b}) = H^{1}(C, ∧^{a}M_{L_{R|C}} ⊗ L_{R|C}^{⊗b}(−y)) = 0

and

H^{1}(D, ∧^{a}M_{L_{R|D}} ⊗ L_{R|D}^{⊗b}) = H^{1}(D, ∧^{a}M_{L_{R|D}} ⊗ L_{R|D}^{⊗b}(−y)) = 0.

We only check this when j ≤ 2i + 2 the remaining case being similar. Since h^{0}(L_{R|C}) = k −1−j and h^{0}(L_{R|D}) = j +1, the inequalities g(C)+min{h^{0}(L_{R|C})−1, i} ≤ deg(L_{R|C}) and g(D) + min{h^{0}(L_{R|D}) − 1, i} ≤ deg(L_{R|D}) are satisfied precisely because j ≤ 2i + 2 and we can invoke Lemma 2.1. □

Next, for 0 ≤ a ≤ i and b ≥ 1 we define vector bundles H_{a,b} over 𝔊̃_{3k−6}^{k−2} having fibre H_{i,j}(C, L) = H^{0}(∧^{i}M_{P}_{k−2}(j)) over each point (C, L) ∈ 𝔊_{3k−6}^{k−2} corresponding to an

embedding C ↪^{|L|} P^{k−2}. First we set H_{0,1} := G_{0,1} and H_{0,b} := Sym^{b}H_{0,1} for b ≥ 1. Having already defined H_{a−1,b} for all b ≥ 1, we define H_{a,b} via the exact sequence

(6)  0 ⟶ H_{a,b} ⟶ ∧^{a}H_{0,1} ⊗ Sym^{b}H_{0,1} ⟶ H_{a−1,b+1} ⟶ 0.

The bundles H_{a,b} being defined entirely in terms of H_{0,1}, the right exactness of the sequence (6) is obvious. There is a natural vector bundle morphism ϕ_{a,b} : H_{a,b} → G_{a,b} over 𝔊̃_{3k−6}^{k−2}; when k = 3i + 6, a = i and b = 2, then rank(H_{i,b}) = rank(G_{i,b}) and the degeneracy locus of ϕ_{i,2} is the codimension 1 compactification over 𝔊̃_{3k−6}^{k−2} of the locus U_{g,i} defined in Proposition 2.5.

Next we compute the Chern classes of H_{a,b} along the curves X and Y:

Proposition 3.14. When k = 3i + 6, we have the following formulas for the first Chern class of H_{i,2}:

c_{1}(H_{i,2|X}) = c_{1}(H_{i,2|Y}) = −(3i + 6)\binom{3i + 4}{i}((3i + 4)x + θ).

Proof. Using repeatedly the sequence (6) we obtain the formula

c_{1}(H_{i,2|X}) = c_{1}(H_{i,2|Y}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1|X} ⊗ Sym^{l+2}G_{0,1|X}).

Since c_{1}(G_{0,1}) = −θ − (3i + 4)x and c_{1}(Sym^{j}(G_{0,1})) = \frac{j(3i+4+j)}{(3i+5)(3i+4)} \binom{3i+j+3}{j}c_{1}(G_{0,1}), the stated formula is obtained after a straightforward calculation. □

In the remainder of this section we will compute c_{1}(G_{i,2} − H_{i,2}) · σ^{∗}(C^{j}), where

C^{j} ⊂ Δ_{j} is the test curve associated to [C] ∈ M_{2k−2−j}. We retain the notation introduced in Propositions 3.4 and 3.5. There are two definitions of the bundles G_{a,b|σ^{∗}(Δ_{j}^{0})} depending whether j ≤ 2i+2 or j ≥ 2i+3 and we will explain in detail the calculations only in the first case, the second being similar. We start by describing the Chern number of the restriction of G_{0,1} to the 1-cycle ∑_{α=0}^{[j/2]} a(j, α) X_{j,α} on C × Pic^{3k−j−4}(C):

Proposition 3.15. Suppose C is a general curve of genus 2k − 2 − j where 2 ≤ j ≤ 2i + 2 and let C^{j} ⊂ Δ_{j} ⊂ \overline{M}_{2k−2} be the test curve associated to C. Then

c_{1}(G_{0,1| ∑_{α=0}^{[j/2]} a(j,α)X_{j,α}}) = −θ − j((3k − j − 4)η + γ) − j(j + 1)(2k − 3 − j)η + ℵ,

where ℵ = 6(j + 1)\begin{matrix}\binom{6i+10−j}{3i+6} & \binom{6i+8}{j} \\ ( & {3i+4 \atop j} & ) \end{matrix}.

Proof. For each 0 ≤ α ≤ j/2, we fix once and for all a linear series L_{D} ∈ W_{k−2+j}^{k−2}(D) such that h^{0}(L_{D}(−(k + 1 − α)y)) = α and h^{0}(L_{D}(−(k + α − j)y)) = j − α (there are a(j, α) such 𝔤_{k−2+j}^{k−2}’s and for each of them X_{j,α} × {L_{D}} is a component of σ^{∗}(C^{j})). Just like in Proposition 3.8, we introduce the Poincare bundle L on X_{j,α} and we define the sheaf on X_{j,α} given by P := π_{2}^{∗}(π_{2∗}L). Furthermore, for each a ≥ 0 we consider the Taylor map ν_{a} : P → J_{a}(L) (the target being the a-th jet bundle of L), and we set F_{a} := Im(ν_{a}) ⊂ J_{a}(L) and P_{a+1} := Ker(ν_{a}).

For each pair (y, L_{C}) ∈ X_{j,α} we have an exact sequence

0 ⟶ H^{0}(D, L_{R|D}(−y)) ⟶ H^{0}(C ∪_{y} D, L_{R}) ⟶ H^{0}(C, L_{R|C}) ⟶ 0,

where L_{R} = (L_{C}(−(j+2)y), L_{D}(−(k−2−j)y)) is the line bundle on C ∪D whose global sections give the fibre of G_{0,1} at each point in 𝔊̃_{3k−6}^{k−2}. This exact sequence globalizes to an exact sequence of vector bundles on X_{j,α}

0 ⟶ H^{0}(L_{D}(−(k − 1 − j)y)) ⊗ O_{X_{j,α}} ⟶ G_{0,1|X_{j,α}} ⟶ P_{j+2} ⟶ 0,

thus c_{1}(G_{0,1|X_{j,α}}) = c_{1}(P_{j+2}). We now study how the bundles F_{a} relate to one another via the exact sequences 0 ⟶ π_{1}^{∗}(K_{C}^{⊗a}) ⊗ L ⟶ J_{a}(L) ⟶ J_{a−1}(L) ⟶ 0, linking successive jet bundles. Recalling that for a generic point (y, L_{C}) ∈ X_{j,α} we have that a^{L_{C}}(y) = (0, …, α − 1, α + 1, …, j − α, j − α + 2, …, k) (cf. Proposition 3.4), it follows that F_{a} = J_{a}(L) for a ≤ α − 2, while F_{α−1} is obtained from J_{α−1}(L) via an elementary transformation along the divisor D_{1} ⊂ X_{j,α} consisting of the b(2k − 2 − j, k − 2 + α − j) points (y, L_{C}) ∈ X_{j,α} where a_{α−1}^{L_{C}}(y) = α. More precisely, one has the exact sequence

0 ⟶ π_{1}^{∗}(K_{C}^{⊗(α−1)}) ⊗ L ⊗ O_{X_{j,α}}(−D_{1}) ⟶ F_{α−1} ⟶ J_{α−2}(L) ⟶ 0.

Furthermore, we have exact sequences

0 ⟶ π_{1}^{∗}(K_{C}^{⊗(α+1)}) ⊗ L ⟶ F_{α+1} ⟶ F_{α−1} ⟶ 0 and
0 ⟶ π_{1}^{∗}(K_{C}^{⊗a}) ⟶ F_{a} ⟶ F_{a−1} ⟶ 0, for all α + 2 ≤ a ≤ j − α − 2.

If D_{2} ⊂ X_{j,α} is the divisor consisting of the b(2k − 2 − j, k − 1 + α − j) points (y, L_{C}) satisfying the condition a_{j−α−1}^{L_{C}}(y) = j − α + 1, then we also have the exact sequences

0 ⟶ π_{1}^{∗}(K_{C}^{⊗(j−α)}) ⊗ L ⊗ O_{X_{j,α}}(−D_{2}) ⟶ F_{j−α} ⟶ F_{j−α−1} ⟶ 0,
0 ⟶ π_{1}^{∗}(K_{C}^{⊗(j−α+2)}) ⊗ L ⟶ F_{j−α+2} ⟶ F_{j−α} ⟶ 0,

and finally

0 ⟶ π_{1}^{∗}(K_{C}^{⊗a}) ⊗ L ⟶ F_{a} ⟶ F_{a−1} ⟶ 0, for j − α + 2 ≤ a ≤ k − 1.

We gather all the information contained in these sequences to obtain that

c_{1}(F_{j+1}) = jc_{1}(L) + j(j + 1)(g(C) − 1)η − deg(D_{1} + D_{2}), and then

c_{1}(G_{0,1| ∑_{α} a(j,α)X_{j,α}}) = c_{1}(P)−c_{1}(F_{j+1}) = −θ−j((3k−j−4)η+γ)−j(j+1)(2k−3−j)η+ℵ,

where ℵ := ∑_{α=0}^{[j/2]} a(j, α)(b(2k − 2 − j, k − 2 + α − j) + b(2k − 2 − j, k − 1 + α − j)) =

= j!(6i + 10 − j)! ∑_{α=0}^{j+1} \frac{(j − 2α − 1)(j − 2α)(j − 2α + 1)^{2}}{(3i + 5 − α)! (3i + 5 + α − j)! (j − α + 1)! α!}.

This sum can be computed with Maple which finishes the proof. □

Next we compute the Chern class of G_{0,b} with b ≥ 2. The proof being similar to that in Proposition 3.15 (in fact simpler), we decided to omit it.

Proposition 3.16. Let b ≥ 2 and 2 ≤ j ≤ 2i + 2. Then for each 0 ≤ α ≤ j/2 we have that

c_{1}(G_{0,b|X_{j,α}}) = −b^{2}θ − b^{2}(j + 2)(γ + (3k − 4 − j)η) − \binom{bj + 2b}{2}(2k − 3 − j)η.

We turn our attention to the remaining components of σ^{∗}(C^{j}). First we show that Yj,α^{′′′} does not appear in the computation of c_{1}(G_{i,2} − H_{i,2}) · σ^{∗}(C^{j}):

Proposition 3.17. For 0 ≤ α ≤ j/2, a ≥ 0 and b ≥ 1, we have that c_{1}(G_{a,b}) · Yj,α^{′′′} =

c_{1}(H_{a,b}) · Yj,α^{′′′} = 0.

Proof. Clearly, it suffices to show that c_{1}(G_{0,b}) · Yj,α^{′′′} = 0 for all b ≥ 1 and then use the exact sequences (4) and (6). We carry this out only for b = 1 the case b ≥ 2 being analogous. Fix l = (L_{D}, V_{D} ⊂ H^{0}(L_{D})) ∈ Yj,α^{′′′}. By duality K_{D} ⊗ L_{D}^{∨}((k + 2 − α)y) is one of the a(j, α) (that is, finitely many) linear systems 𝔤_{j−α+1}^{1} on D with a (j − 2α + 1)-fold point at y. The space of sections V_{D} ⊂ H^{0}(L_{D}) is chosen such that H^{0}(L_{D}(−2y)) ⊂ V_{D}, hence Yj,α^{′′′} can be identified with the disjoint union of a(j, α) copies of the projective line P(H^{0}(L_{D})/H^{0}(L_{D}(−2y))), one for each choice of L_{D}. Then H^{0}(L_{C∪D|D}) = H^{0}(L_{D}(−(k − 1 − j)y)), for each l ∈ Yj,α^{′′′} (that is, G_{0,1}(l) is independent of V_{D}). Globalizing this we get that G_{0,1|Y′′′_{j,α}} is trivial. □

In a similar fashion we have the following:

Proposition 3.18. For 0 ≤ α ≤ j/2, a ≥ 0 and b ≥ 1, we have the equalities c_{1}(G_{a,b}) · Yj,α^{′} =

c_{1}(H_{a,b}) · Yj,α^{′} = 0.

Proposition 3.19. For 0 ≤ α ≤ j/2 and b ≥ 1, we have the equality

c_{1}(G_{0,b}) · Yj,α^{′′} = −\frac{b^{2}(j − 2α)j!}{α!(j − α)!}.

Proof. We start by recalling that Yj,α^{′′} is the locus of line bundles L_{D} ∈ Pic^{k−2+j}(D) such that h^{0}(L_{D}(−(k + 1 + α − j)y)) ≥ j − α − 1 and h^{0}(L_{D}(−(k − α + 1)y)) ≥ α. Using [Fu], Theorem 14.3, one finds that Yj,α^{′′} ≡ \frac{j−2α}{α!(j−α)!}θ^{j−1}. Having fixed once and for all one of

the finitely many linear series L_{C} ∈ W_{3k−j−4}^{k−2}(C) such that {L_{C}} × Yj,α^{′} ⊂ 𝔊̃_{3k−6}^{k−2}, there is an identification

G_{0,1}(L_{D}) = H^{0}(L_{D}(−(k − 2 − j))y),

for each L_{D} ∈ Yj,α^{′′}. Since the vector bundle Pic^{k−2+j}(D) ∋ L_{D} ↦ H^{0}(L_{D} ⊗ O_{β}y), is algebraically trivial for each β ≥ 0 (remember that y ∈ D is a fixed point), we obtain that c_{1}(G_{0,1|Y′′_{j,α}}) = −θ. Similarly, for b ≥ 2 we get that c_{1}(G_{0,b|Y′′_{j,α}}) = −b^{2}θ from which now the conclusion follows. □

Propositions 3.15, 3.16, 3.17, 3.18, 3.19 will enable us to compute the intersection numbers c_{1}(G_{i,2}−H_{i,2})·σ^{∗}(C^{j}) needed to determine the coefficient of δ_{j} in the expansion of [\overline{Z}_{6i+10,i}]. Repeatedly using (6) , one can write that

(7) c_{1}(H_{i,2}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1} ⊗ Sym^{l+2}G_{0,1}) = (3i + 6)\binom{3i + 4}{i}c_{1}(G_{0,1})

(note the similarity with Proposition 3.14). Using (4) we also have that

(8)  c_{1}(G_{i,2}) = ∑_{l=0}^{i}(−1)^{l}c_{1}(∧^{i−l}G_{0,1} ⊗ G_{0,l+2}),

and a simple calculation yields that

(9) c_{1}(G_{i,2} − H_{i,2}) = −\frac{13i^{2} + 35i + 24}{(3i + 4)(i + 1)}\binom{3i + 4}{i}c_{1}(G_{0,1}) + ∑_{l=0}^{i}(−1)^{l}\binom{3i + 5}{i − l}c_{1}(G_{0,l+2}).
4. The slope of \overline{Z}_{6i+10,i}

In this section we finish the calculation of the class of the virtual divisor \overline{Z}_{g,i}. As before, we fix i ≥ 0, k = 3i + 6 and g = 6i + 10, hence ϕ : H_{i,2} → G_{i,2} is a morphism between two vector bundles of the same rank defined over 𝔊̃_{9i+12}^{3i+4}. Theorem 4.1. We fix i ≥ 0 and denote by σ : 𝔊̃_{9i+12}^{3i+4} → M̃_{6i+10} the natural projection map. Then we have the following formula for the class of the pushforward to M̃_{6i+10} of the virtual degeneracy locus of the morphism ϕ : H_{i,2} → G_{i,2}:

σ_{∗}(c_{1}(G_{i,2} − H_{i,,2})) = \frac{1}{3i + 5}\binom{6i + 7}{3i + 4, 2i + 1, i + 2}(aλ − b_{0} δ_{0} − ⋯ − b_{3i+5} δ_{3i+5}),

where

a = \frac{(4i + 7)(6i^{2} + 19i + 12)}{(2i + 3)(i + 2)}, b_{0} = \frac{12i^{2} + 31i + 18}{3(2i + 3)}, b_{1} = \frac{12i^{2} + 33i + 20}{i + 2} and

b_{j} = \frac{jf(i, j)}{6(i + 1)(i + 2)(2i + 3)(6i + 9 − j)}, for2 ≤ j ≤ 2i + 2,

with

f(i, j) = 864 i^{5} − (240j − 5256)i^{4} + (16j^{2} − 998j + 11830)i^{3}+
+(31j^{2} − 1254j + 11585)i^{2} + (−5j^{2} − 286j + 3981)i − 24j^{2} + 240j − 216,

and

b_{j} = \frac{2(13i^{2} + 35i + 24)(3i + 5)}{(i + 1)(i + 2)\binom{6i+10}{j}} \binom{3i + 4}{(j − 1)/2}^{2} + \frac{g(i, j)}{12(i + 1)(i + 2)(2i + 3)(6i + 9 − j)}

for an odd j such that 2i + 3 ≤ j ≤ 3i + 5, where

g(i, j) = 1728ji^{5} − (576j^{2} − 10512j + 540)i^{4} + (48j^{3} − 2592j^{2} + 24120j − 3234)i^{3}+ (140j^{3}−3926j^{2}+25176j−7278)i^{2}+(109j^{3}−2107j^{2}+10875j−7263)i+12j^{3}−156j^{2}+972j−2700 and

b_{j} = \frac{(13i^{2} + 35i + 24)((3i + 5)(2j + 1) − j^{2})}{2(i + 1)(i + 2)(3i + 5)\binom{6i+10}{j}} \binom{3i + 5}{j/2}^{2}+\frac{h(i, j)}{12(i + 1)(i + 2)(2i + 3)(6i + 9 − j)},

for an even j such that 2i + 3 ≤ j ≤ 3i + 5, where h(i, j) = 1728ji^{5} − (576j^{2} − 10512j + 1020)i^{4} + (48j^{3} − 2592j^{2} + 24280j − 6034)i^{3}+ (140j^{3}−3942j^{2}+25830j−13400)i^{2}+(109j^{3}−2145j^{2}+11774j−13230)i+12j^{3}−180j^{2}+1392j−4896.

In particular, b_{j} ≥ b_{0} for all 1 ≤ j ≤ 3i + 5 and

s(σ_{∗}(c_{1}(G_{i,2} − H_{i,2}))) = \frac{a}{b_{0}} = \frac{3(4i + 7)(6i^{2} + 19i + 12)}{(12i^{2} + 31i + 18)(i + 2)}.

Proof. Since codim(\overline{M}_{g} − M̃_{g}, \overline{M}_{g}) ≥ 2, it makes no difference whether we compute the class σ_{∗}(G_{i,2} − H_{i,2}) on M̃_{g} or on \overline{M}_{g} and we can write

σ_{∗}(G_{i,2} − H_{i,2}) = Aλ − B_{0} δ_{0} − B_{1} δ_{1} − ⋯ − B_{3i+5} δ_{3i+5} ∈ Pic(\overline{M}_{g}),

where λ, δ_{0}, …, δ_{3i+5} are the generators of Pic(\overline{M}_{g}). We start with the following:

Claim: One has the relation A − 12B_{0} + B_{1} = 0.

We pick a general curve [C, q] ∈ M_{6i+9,1} and at the fixed point q we attach to C a Lefschetz pencil of plane cubics. If we denote by R ⊂ \overline{M}_{g} the resulting curve, then R · λ = 1, R · δ_{0} = 12, R · δ_{1} = −1 and R · δ_{j} = 0 for j ≥ 2. The relation A − 12B_{0} + B_{1} = 0 follows once we show that σ^{∗}(R) · c_{1}(G_{i,2} − H_{i,2}) = 0. To achieve this we check that G_{0,b|σ^{∗}(R)} is trivial and then use (4) and (6). We take [C ∪_{q} E] to be an arbitrary curve from R, where E is an elliptic curve. Using that limit 𝔤_{9i+12}^{3i+4} on C∪_{q}E are in 1 : 1 correspondence with linear series L ∈ W_{9i+12}^{3i+4}(C) having a cusp at q (this being a statement that holds independent of E) and that G_{0,b|σ^{∗}(Δ_{1}^{0})} consists on each fibre of sections of the genus g − 1 aspect of the limit 𝔤_{9i+12}^{3i+4}, the claim now follows.

Now we determine A, B_{0} and B_{1} explicitly. We fix a general pointed curve (C, q) ∈ M_{6i+9,1} and construct the test curves C^{1} ⊂ Δ_{1} and C^{0} ⊂ Δ_{0}. Using the notation from Proposition 3.3, we get that σ^{∗}(C^{0}) · c_{1}(G_{i,2} − H_{i,2}) = c_{1}(G_{i,2|Y}) − c_{1}(H_{i,2|Y}) and σ^{∗}(C^{1}) · c_{1}(G_{i,2} − H_{i,2}) = c_{1}(G_{i,2|X}) − c_{1}(H_{i,2|X}) (the other component T of σ^{∗}(C^{1}) does not appear because G_{0,b|T} is trivial for all b ≥ 1). On the other hand

C^{0} · σ_{∗}(c_{1}(G_{i,2} − H_{i,2})) = (12i + 18)B_{0} − B_{1} and C^{1} · σ_{∗}(c_{1}(G_{i,2} − H_{i,2})) = (12i + 16)B_{1},

while we already know that A − 12B_{0} + B_{1} = 0. Using Propositions 3.11, 3.12, 3.14, the expressions for [X] and [Y] as well as the well-known formula θ^{j}x^{k−2−j} = g!/(g − j)! on C_{k−2}, we get a linear system of 3 equations in A, B_{0} and B_{1} which leads to the stated formulas for the first three coefficients.

To compute B_{j} for 2 ≤ j ≤ 3i + 5, we fix general curves [C] ∈ M_{6i+10−j} and [D, y] ∈ M_{j,1} and consider the test curve C^{j} ⊂ Δ_{j}. Then on one hand we have that 2(6i + 9 − j)B_{j} = σ^{∗}(C^{j}) · c_{1}(G_{i,2} − H_{i,2}), on the other hand

σ^{∗}(C^{j})·c_{1}(G_{i,2}−H_{i,2}) = c_{1}(G_{i,2}−H_{i,2})·∑_{α=0}^{[j/2]}((a(j, α)X_{j,α}+b(6i+10−j, 3i+5+α−j)Yj,α^{′′}).

Using (9) together with Proposition 3.19, we get that

c_{1}(G_{i,2} − H_{i,2}) · ∑_{α=0}^{[j/2]} b(6i + 10 − j, 3i + 5 + α − j)Yj,α^{′′} =

= \frac{(8i^{2} + 19i + 12)j!(6i + 10 − j)!(3i + 2)!}{i!(2i + 4)!}(∑_{α=0}^{j} \frac{(j − 2α − 1)(j − 2α)^{2}(j − 2α + 1)}{α! (j − α)! (3i + 5 − α)! (3i + 5 + α − j)!}),

which can be computed by Maple, whereas c_{1}(G_{i,2} − H_{i,2}) · (∑_{α=0}^{[j/2]} a(j, α)X_{j,α}) can be computed using (9) and Propositions 3.15 and 3.16 (note in particular that ℵ has to be multiplied by the coeffcient of c_{1}(G_{0,1}) in (9)). The rest now follows and checking that b_{j} ≥ b_{0} for j ≥ 1 is elementary. □

Remark 4.2. If we retain the notation of Theorem 4.1, we see that if we fix j ≥ 1 and let i vary, we have that lim_{i→∞} \frac{b_{j}}{b_{0}} = 6j. This is the same asymptotical estimate as in the

case of the classical Brill-Noether divisors where \frac{b_{j}}{b_{0}} = \frac{6j(g−j)}{g+3} for all j ≥ 1 (cf. [EH3]).

Theorem 4.3. Assume that vector bundle morphism ϕ : H_{i,2} → G_{i,2} is non-degenerate at a general point from σ^{−1}(Δ_{1}^{0}). Then \overline{Z}_{6i+10,i} is a divisor on \overline{M}_{6i+10} and

6 < s(\overline{Z}_{6i+10,i}) = \frac{3(4i + 7)(6i^{2} + 19i + 12)}{(12i^{2} + 31i + 18)(i + 2)} < 6 + \frac{12}{g + 1}.

Remark 4.4. Since σ^{−1}(Δ_{1}^{0}) is irreducible, to verify the assumption made in Theorem 4.3 it suffices to show that if C ⊂ P^{3i+4} is a general 1-cuspidal curve with p_{a}(C) = 6i+10 and deg(C) = 9i + 12, then C satisfies property(N_{i}). We checked this for g = 16, 22 (cf. Remarks 2.9 and 2.11), while the case g = 10 was treated in [FP]. The assumption in Theorem 4.3 implies that ϕ is non-degenerate over a general point from σ^{−1}(Δ_{0}^{0}) and also over a general point from σ^{−1}(M_{g}^{0}), which is equivalent to Z_{g,i} being a divisor on M_{g}^{0}. Therefore the assumption is slightly stronger than Conjecture 2.6.

Proof of Theorem 4.3. We denote by Z(ϕ) the degeneracy locus of ϕ, thus σ_{∗}(Z(ϕ)) is a divisor on \overline{M}_{6i+10,i} such that σ_{∗}(Z(ϕ)) = \overline{Z}_{g,i} + ∑_{j=0}^{3i+5} d_{j}Δ_{j}, for certain coefficients d_{j} ≥ 0. Using the Remark above we obtain that d_{0} = d_{1} = 0, hence s(\overline{Z}_{6i+10,i}) = s(σ_{∗}(c_{1}(G_{i,2} − H_{i,2}))) and the rest follows from Theorem 4.1. □

Remark 4.5. In the simplest case i = 0, we can compare the formula for [\overline{Z}_{6i+10,i}] with our findings in [FP]. Theorem 4.1 gives the formula

σ_{∗}(c_{1}(G_{0,2} − H_{0,2})) ≡ 42(7λ − δ_{0} − 5δ_{1} − \frac{4}{3}δ_{2} − 7δ_{3} − \frac{44}{7}δ_{4} − \frac{685}{42}δ_{5}),

whereas Theorem 1.6 from [FP] says that if K_{10} is the divisor on M_{10} of curves lying on a K3 surface then

\overline{K}_{10} = 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − b_{5}δ_{5},

where b_{5} ≥ 6. Since we have also established the set-theoretic equality Z_{10,0} = K_{10} (cf. [FP], Theorem 1.7), it follows that there is a scheme-theoretic equality Z_{10,0} = 42K_{10}. Here 42 is the number of pencils 𝔤_{6}^{1} on a general curve of genus 10, and its appearance in the formula of [\overline{Z}_{10,0}] has a clear geometric meaning: if a Brill-Noether general curve

[C] ∈ M_{10} fails (N_{0}) for one linear system 𝔤_{12}^{4} = K_{C}(−𝔤_{6}^{1}), then it fails (N_{0}) for all 42 linear systems 𝔤_{12}^{4} it possesses, that is, the map σ : U_{10,0} → K_{10,0} is 42 : 1. Moreover, the vector bundle morphism H_{0,2} → G_{0,2} is degenerate along each of the boundary divisors Δ_{2}, …, Δ_{5}. This situation presumably extends to higher genera as well hence the main thrust of Theorem 4.1 is that it computes s(\overline{Z}_{6i+10,i}).

To finish the proof of Theorems 1.1 and 1.2 we specialize to the cases g = 16 and 22 and note that the assumption made in Theorem 4.3 is satisfied in these situations (cf. Theorems 2.7 and 2.10).

5. An effective divisor on \overline{M}_{14,1}

In this section we describe how to construct effective divisors on \overline{M}_{g,n} using syzygy type conditions for pointed curves. We treat only one example. We denote by \overline{Z}_{14,0}^{1} the closure in \overline{M}_{14,1} of the locus Z_{14,0}^{1} of smooth pointed curves [C, p] ∈ M_{14,1} for which there exists a linear series L ∈ W_{18}^{6}(C) such that the map

μ_{L,p} : Sym^{2}H^{0}(C, L(−p)) → H^{0}(C, L^{⊗2}(−2p))

is not an isomorphism. Just like the in the case of the loci Z_{g,i} on M_{g}, the locus Z_{14,0}^{1} can be naturally viewed as the pushforward of the degeneracy locus of a morphism between two vector bundles of the same rank 21 over 𝔊_{18}^{6} ×_{M_{14}} M_{14,1}.

We are going to show that \overline{Z}_{14,0}^{1} is a divisor and in order to compute its class we need some preparations. Recall that for g ≥ 3, the group Pic(\overline{M}_{g,1}) is freely generated by λ, the tautological class ψ and the boundary classes δ_{i} = [Δ_{i}] with 0 ≤ i ≤ g − 1, where for i ≥ 1, the generic point of Δ_{i} is a union of two smooth curves of genus i and g − i meeting at a point, the marked point lying on the genus i component. We denote by π : \overline{M}_{g,1} → \overline{M}_{g} the natural forgetful map. We can then write the class of \overline{Z}_{14,0}^{1} on \overline{M}_{14,1} as

\overline{Z}_{14,0}^{1} ≡ a λ + c ψ − b_{0} δ_{0} − b_{1} δ_{1} − ⋯ − b_{13} δ_{13}.

As before, we will determine the relevant coefficients in the expression of [\overline{Z}_{14,0}^{1}] by intersecting our divisor with various test curves.

Proposition 5.1. Let C be a general curve of genus 14 and C̃ = π^{−1}([C]) ⊂ \overline{M}_{14,1}, the test curve obtained by letting the marked point vary along C. Then C̃ · \overline{Z}_{14,0}^{1} = 12012, hence c = C̃ · \overline{Z}_{14,0}^{1}/(2g − 2) = 462.

Proof. Let us fix a linear series L ∈ W_{18}^{6}(C). We count the number of points p ∈ C, for which the multiplication map μ_{L,p} not injective. If p_{1} : C × C → C and p_{2} : C × C → C are the two projections, we define the vector bundles

E := (p_{2})_{∗}(p_{1}^{∗}(L) ⊗ O_{C×C}(−Δ)) and F := (p_{2})_{∗}(p_{1}^{∗}(L^{⊗2}) ⊗ O_{C×C}(−2Δ)).

There is a natural multiplication map μ_{L} : Sym^{2}(E) → F, and the cardinality of its degeneracy locus is just c_{1}(F) − c_{1}(Sym^{2}(E)). A simple calculation shows that c_{1}(E) = −4x − θ = −18, hence c_{1}(Sym^{2}(E)) = −126, while c_{1}(F) = −98. We obtain that μ_{L,p} is not an isomorphism for precisely 28 points p ∈ C. Since C has 429 linear series 𝔤_{18}^{6} (cf. [ACGH]), we obtain c = (C̃ · \overline{Z}_{14,0}^{1})/26 = 429 · 28/26 = 462. □

For more relations among the coefficients of [\overline{Z}_{14,0}^{1}] we define the map ν : \overline{M}_{1,2} → \overline{M}_{g,1} obtained by attaching to each 2-pointed elliptic curve [E, q, p] a fixed general 1-pointed curve [C, q] ∈ M_{g−1} (the point of attachment being q). One has the pullback formulas

ν^{∗}(λ) = λ, ν^{∗}(ψ) = ψ_{p}, ν^{∗}(δ_{0}) = δ_{0}, ν^{∗}(δ_{1}) = −ψ_{q} and ν^{∗}(δ_{g−1}) = δ_{qp},

where ψ_{p} and ψ_{q} are the tautological classes corresponding to the marked points p and q, while δ_{qp} is the boundary component of curves having a rational tail containing both q and p. On \overline{M}_{1,2} these classes are not independent and we have the relations ψ_{q} = ψ_{p}, λ = ψ_{p} − δ_{qp} and δ_{0} = 12(ψ_{p} − δ_{qp}) (see e.g. [AC2], Proposition 1.9).

Proposition 5.2. If ν : \overline{M}_{1,2} → \overline{M}_{14,1} is as above, then ν^{∗}(\overline{Z}_{14,0}^{1}) = ∅. It follows that we have the relations

a − 12b_{0} + b_{13} = 0 andc + b_{1} = b_{13}.

Proof. We assume that [X = C ∪_{q} E, p ∈ E] ∈ \overline{Z}_{14,0}^{1}. Then there exists a limit linear series 𝔤_{18}^{6} on X determined by its aspects L_{C} ∈ W_{18}^{6}(C) and L_{E} ∈ W_{18}^{6}(E), together with compatible elements

ρ_{C} ∈ Ker{μ_{L_{C}} : Sym^{2}H^{0}(L_{C}) → H^{0}(L_{C}^{⊗2})}, ρ_{E} ∈ Ker{μ_{L_{E}} : Sym^{2}H^{0}(L_{E}) → H^{0}(L_{E}^{⊗2})},

satisfying the inequality

ord_{q}(ρ_{C}) + ord_{q}(ρ_{E}) ≥ deg(L_{C}) + deg(L_{E}) = 36

and such that ρ_{E} ∈ Sym^{2}H^{0}(L_{E}(−p)) (see [FP], Section 4, for how to study multiplication maps in the context of limit linear series). Because (C, q) ∈ \overline{M}_{13,1} is general, the vanishing sequence of L_{C} at q equals (0, 2, 3, 4, 5, 6, 7), the vanishing sequence of L_{E} at q is (11, 12, 13, 14, 15, 16, 18) and finally, the vanishing sequence of L_{E} at p is either (0, 1, 2, 3, 4, 5, 6) or (0, 1, 2, 3, 4, 5, 7), depending on whether p−q ∈ Pic^{0}(E) is a 7-torsion class or not.

We claim that ord_{q}(ρ_{E}) ≤ 29(= 13 + 16 = 14 + 15). Indeed, otherwise ord_{q}(ρ_{E}) ≥ 30(= 14+16 = 15+15), and since ρ_{E} ∈ Sym^{2}H^{0}(L_{E}(−p)), after subtracting base points ρ_{E} becomes a ≠ 0 element in the kernel of the map Sym^{2}H^{0}(N) → H^{0}(N^{⊗2}), where N = L_{E}(−p−14q) ∈ Pic^{3}(E). This is obviously impossible. Therefore ord_{q}(ρ_{E}) ≤ 29, so by compatibility, ord_{q}(ρ_{C}) ≥ 7(= 2+5 = 3+4). We now show that when (C, p) ∈ M_{13,1} is chosen generically, there can be no such element ρ_{C} which leads to a contradiction.

Claim: Suppose σ_{0}, σ_{2}, σ_{3}, σ_{4}, σ_{5}, σ_{6}, σ_{7} is a basis of H^{0}(L_{C}) adapted to the point q in the sense that ord_{q}(σ_{i}) = i. If W (q, L_{C}) ⊂ Sym^{2}H^{0}(L_{C}) denotes the 17-dimensional subspace spanned by the elements σ_{i} · σ_{j} with 4 ≤ i ≤ j ≤ 7, σ_{3} · σ_{j} for j ≥ 4 and σ_{2} · σ_{j} for j ≥ 5, then the restriction of the multiplication map W (q, L_{C}) → H^{0}(L_{C}^{2}(−7q)) is an isomorphism. (Note that W (q, L_{C}) does not depend on the chosen basis {σ_{i}}).

The proof of this claim is similar to the proof of Theorem 5.1 in [FP]. It is enough to construct a single 1-cuspidal curve X ⊂ P^{6} with p_{a}(X) = 14 and deg(X) = 18, such that if ν : C → X is the normalization of X and q ∈ C is the inverse image of the cusp, then X does not lie on any quadric contained in W (q, ν^{∗}(O_{X}(1))). We

construct the following cuspidal curve: define Γ ⊂ P^{6} to be the image of the map t ↦^{f} [1, t^{2}, t^{3}, t^{4}, t^{5}, t^{6}, t^{7}], then choose a general hyperplane H ⊂ P^{6} which intersects Γ in distinct points p_{1}, …, p_{7}. Take D ⊂ H to be a general smooth curve of genus g(D) = 7 and deg(D) = 11 which passes through p_{1}, …, p_{7}. Then X := Γ ∪ D is a curve of arithmetic genus 14 and degree 21 having a cusp at the point q = f(0) ∈ Γ. The quadrics in W (q, ν^{∗}(O_{X}(1))) can of course be written down explicitly and to show that D can be chosen such that it is not contained in any quadric from W (q, ν^{∗}(O_{X}(1))) amounts to a simple counting argument. □

Remark 5.3. The claim we have just proved also shows that \overline{Z}_{14,0}^{1} is a divisor on \overline{M}_{14,1}. Alternatively, this can be proved in the same way as Theorem 2.7.

We now determine the coefficient b_{1}:

Proposition 5.4. Consider general curves [E, p, q] ∈ M_{1,2} and [C] ∈ M_{13}, and denote by C^{2} ⊂ \overline{M}_{14,1} the test curve consisting of points \{[C ∪_{q} E, p ∈ E]\}_{\{q∈C\}}. Then C^{2} · \overline{Z}_{14,0}^{1} = 133848, therefore b_{1} = C^{2} · \overline{Z}_{14,0}^{1}/24 = 5577.

Proof. Throughout the proof we will use the notations introduced in Proposition 3.11. From the proof of Proposition 5.2 it is clear that the intersection number C^{2}·\overline{Z}_{14,0}^{1} equals the number of pairs (q, D) ∈ X ⊂ C × C_{6} such that the map

W (q, K_{C}(−D)) → H^{0}(K_{C}^{⊗2}(−2D − 7q))

is not an isomorphism. To compute this number we construct the rank 17 vector bundle W over X having fibre W (q, D) = W (q, K_{C}(−D)) over each point (q, D) ∈ X. The curve C being general, the vanishing sequence a^{K_{C}(−D)}(q) will be generically equal to (0, 2, 3, 4, 5, 6, 7), while at a finite number of points (q, D) ∈ X we will have that a^{K_{C}(−D)}(q) = (0, 2, 3, 4, 5, 6, 8). In order to compute c_{1}(W) we note that W has a subbundle W_{1} ⊂ W which fits into two exact sequences:

0 ⟶ Sym^{2}u_{∗}(v^{∗}(M) ⊗ O(−4Δ))_{|X} ⟶ W_{1} → u_{∗}(v^{∗}(M) ⊗ O(−4Δ))_{|X}⊗P ⟶ 0,

where P = u_{∗}(v^{∗}(M) ⊗ I_{Δ}^{3}/I_{Δ}^{4})_{|X}, and

0 ⟶ W_{1} ⟶ W ⟶ u_{∗}(v^{∗}(M) ⊗ O(−5Δ))_{|X}⊗u_{∗}(v^{∗}(M) ⊗ I_{Δ}^{2}/I_{Δ}^{3})_{|X} ⟶ 0.

Next, we consider the multiplication map W → u_{∗}(v^{∗}(M^{⊗2}) ⊗ O(−7Δ))_{|X} whose de-

generation locus we want to compute. The intersection number C^{2} · \overline{Z}_{14,0}^{1} is equal to

(c_{1}(u_{∗}(v^{∗}(M^{⊗2}) ⊗ O(−7Δ))) − c_{1}(W)) · [X] = (3θ − 7x − γ + 18η) · [X] = 133848,

as it turns out after a short calculation. Here we have used that c_{1}(W) can be computed from the two exact sequences involving W and W_{1}, while

c_{1}(u_{∗}(v^{∗}(M^{⊗2}) ⊗ O(−7Δ))) = c_{1}(F_{2}) − c_{1}(J_{6}(M^{⊗2})) = −4θ − 34x + 14γ − 756η.

□

Since b_{1} = 5577 we now obtain that b_{13} = b_{1}+c = 6039. To compute the coefficient b_{0} (and thus the λ-coefficient a) we use our last test curve:

Proposition 5.5. Let (C, q, p) ∈ M_{13,2} be a general 2-pointed curve. We denote by C^{3} ⊂ \overline{M}_{14,1} the family consisting of curves \{[C/y ∼ q, p]\}_{\{y∈C\}}. Then C^{3} ·\overline{Z}_{14,0}^{1} = c+26b_{0} −b_{13} = 24453. It follows that b_{0} = 1155 and a = 7821.

Proof. We retain the notations introduced in Proposition 3.12. We construct a vector bundle map Sym^{2}(E) → F over the curve Y ⊂ C ×C_{6}, where E = u_{∗}(v^{∗}(M)⊗O(−Γ_{p})) is the bundle with fibre E(y, D) = H^{0}(K_{C}(−D − p)), while F is the bundle with fibre

F(y, D) = H^{0}(K_{C}^{⊗2}(−2D − 2p − y − q)) ⊕ ℂ · t^{2} ⊂ H^{0}(K_{C}^{⊗^{2}}(−2D − 2p)),

where H^{0}(K_{C}(−D − p))/H^{0}(K_{C}(−D − p − y − q)) = ℂ · t, for every (y, D) ∈ Y.

It is easy to show that c_{1}(E) = −5x − θ, hence c_{1}(Sym^{2}(E)) = −35x − 7θ. The class c_{1}(F) can be computed from the exact sequence

0 ⟶ u_{∗}(v^{∗}(M^{⊗2}) ⊗ O(−2Γ_{p} − Γ_{q} − Δ))_{|Y} ⟶ F ⟶ (M^{⊗2} ⊗ O(−2Γ_{p}))_{|Y} (−A) ⟶ 0,

where A = Y ∩ π_{1}^{−1}(p) is the effective divisor on the curve Y consisting of all points (p, D) such that h^{0}(p + q + D) ≥ 2. Using the formula for [Y] (cf. Proposition 3.12), we get that |A| = 429. We also compute that c_{1}(u_{∗}(v^{∗}(M^{⊗2}) ⊗ O(−2Γ_{p} − Γ_{q} − Δ))) = −4θ − 40x − 33η + 2γ and of course c_{1}(M^{⊗2} ⊗ O(−2Γ_{p})) = 2(17η − γ − x). Therefore we can write that

C^{3} · \overline{Z}_{14,0}^{1} = (c_{1}(F) − c_{1}(Sym^{2}(E))) · [Y] − |A| = (−7x + 3θ + η) · [Y] − |A| = 24453.

□

We have thus far determined the coefficients a, c, b_{0}, b_{1}, b_{13} in the expansion of [\overline{Z}_{14,0}^{1}]. This is already enough to conclude that [\overline{Z}_{14,0}^{1}] lies outside the cone of Pic(\overline{M}_{14,1}) spanned by pullbacks of effective divisors from \overline{M}_{14}, Brill-Noether divisors on \overline{M}_{14,1} and boundary divisors (see the discussion after Theorem 1.4 for the relevance of this result). To get a bound on the remaining coefficients b_{j} for 2 ≤ j ≤ 12, we use a variant of Theorem 1.1 from [FP]. The boundary divisor Δ_{j} ⊂ \overline{M}_{14,1} with 3 ≤ j ≤ 12, j ≠ 4 is filled-up by pencils R_{j} obtained by attaching to a fixed 2-pointed curve [B, p, q] ∈ M_{j,2} a variable 1-pointed curve [C, q] ∈ \overline{M}_{14−j,1} moving in a Lefschetz pencils of curves of genus 14 − j sitting on a fixed K3 surface. Deformations of R_{j} cover Δ_{j} for j ≠ 2, 4, hence we have that R_{j} · \overline{Z}_{14,0}^{1} ≥ 0. Since one also has the relations (see [FP], Lemma 2.4)

R_{j} · λ = 15 − j, R_{j} · δ_{j} = −1, R_{j} · δ_{0} = 6(17 − j), R_{j} · ψ = 0 and R_{j} · δ_{i} = 0 for i ≠ 0, j,

we immediately get the estimate b_{j} ≥ 15 + 27j for all 3 ≤ j ≤ 12, j ≠ 4. To obtain the bounds b_{2} ≥ 325 and b_{4} ≥ 271 we use similar pencils filling up Δ_{2} and Δ_{4} respectively. We skip these details. This completes the proof of Theorem 1.4.

6. The Kodaira dimension of \overline{M}_{g,n}

In this last section we use the effective divisors \overline{Z}_{16,1} and \overline{Z}_{22,2} to improve Logan’s results about which moduli spaces \overline{M}_{g,n} are of general type. For a general reference about Pic(\overline{M}_{g,n}) we refer to [AC2] and [Log]. For each 1 ≤ i ≤ n we denote by π_{i} : \overline{M}_{g,n} → \overline{M}_{g,1} the morphism forgetting all marked points except the one labelled by i and we also consider the map π : \overline{M}_{g,n} → \overline{M}_{g} which forgets all marked points. We recall that the canonical class of \overline{M}_{g,n} is given by the formula

K_{\overline{M}_{g,n}} = 13λ − 2δ_{0} + ∑_{i=1}^{n} ψ_{i} − 2 ∑_{i≥0, S} δ_{i:S} − ∑_{S} δ_{1:S}.

Here ψ_{i} is the tautological class corresponding to the i-th marked point, while δ_{i:S} with i ≥ 0 and ∅ ≠ S ⊂ {1, …, n} denotes the class of the boundary divisor with generic point being a union of two curves of genus i and g − i such that the marked points on the genus i component are precisely those labelled by S.

Proof of Theorem 1.3. We start with the case g = 22 and we show that \overline{M}_{22,2} is of general type. On \overline{M}_{22,2} we consider the averaged pullback of the Weierstrass divisor

W_{12} := π_{1}^{∗}(\overline{W}) + π_{2}^{∗}(\overline{W}) ≡ −2 λ + \frac{g(g + 1)}{2}(ψ_{1} + ψ_{2}) − ∑_{i≥0,S≠∅} c_{i:S} δ_{i:S},

where c_{i:S} ≥ 0, and significantly, the coefficient of δ_{0} is 0. By Theorem 1.2, we have another effective class, namely [π^{∗}(\overline{Z}_{22,2})] = c(\frac{1665}{256}λ−δ_{0}−⋯), where c > 0. One can easily check that K_{\overline{M}_{22,2}} can be written as a positive combination of [W_{12}], [π^{∗}(\overline{Z}_{22,2})], ψ_{1} + ψ_{2} and some other boundary classes. Since ψ_{1} + ψ_{2} is big and nef, it follows that \overline{M}_{22,2} is of general type.

When g = 21 we use the maps χ_{i,j} : \overline{M}_{21,5} → \overline{M}_{22} for 1 ≤ i < j ≤ 5, where

χ_{i,j} identifies the marked points labelled by i and j and forgets those labelled by {i, j}^{c}. The ℚ-divisor class

∑_{i<j} χ_{i,j}^{∗}(\overline{Z}_{22,2}) ≡ c(\frac{1665}{256}λ − δ_{0} + \frac{2}{5} ∑_{i=1}^{5} ψ_{i} − ⋯), where c > 0,

is obviously effective on \overline{M}_{21,5}. Since the ℚ-class −λ + 11 ∑_{i=1}^{5} ψ_{i} − 0 · δ_{0} − ⋯, is also effective (cf. [Log], Theorem 5.4 - we have retained only the coefficients that play a role in our argument), once again we see that K_{\overline{M}_{21,5}} can be written as the sum of an effective divisor and a positive multiple of ∑_{i=1}^{5} ψ_{i}.

Finally we settle the case g = 16: we adapt Theorem 5.4 from [Log] to conclude that the ℚ-class −λ + \frac{23}{9}(∑_{i=1}^{9} ψ_{i}) − 0 · δ_{0} − ⋯, is effective on \overline{M}_{16,9} (precisely, this is the class of the S_{9}-orbit of the closure in \overline{M}_{16,9} of the effective divisor D on M_{16,9} consisting of points [C, p_{1}, …, p_{9}] such that h^{0}(C, 2p_{1} + ⋯ + 2p_{7} + p_{8} + p_{9}) ≥ 2 and we have explicitly indicated that the coefficient of δ_{0} is 0 and retained only the coefficients that are significant for this calculation). The class [π^{∗}(\overline{Z}_{16,1})] = c(\frac{407}{61} λ−δ_{0} −⋯), where c > 0, is also effective and one writes K_{\overline{M}_{16,9}} as a positive sum of these two effective classes, boundary classes and the big and nef class ∑_{i=1}^{9} ψ_{i}. □

Remark 6.1. Theorem 6.3 from [Log] claims that \overline{M}_{22,4} is of general type but the numerical argument used in the proof seems to be incorrect.

References

[AC1] E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341-362.

[AC2] E. Arbarello and M. Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Etudes Sci. Publ. Math. 88 (1998), 97-127.

[ACGH] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Grundlehren der mathematischen Wissenschaften 267, Springer.

[EH1] D. Eisenbud and J. Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), 337-371.

[EH2] D. Eisenbud and J. Harris, Irreducibility of some families of linear series with Brill-Noether number −1, Ann. Scient. Ec. Norm. Sup.(4) 22 (1989), 33–53.

[EH3] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), 359–387.

[F] G. Farkas, The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), 43-65.

[FP] G. Farkas and M. Popa, Effective divisors on \overline{M}_{g}, curves on K3 surfaces and the Slope Conjecture, J. Algebraic Geom. 14 (2005), 241-267.

[Fu] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 2, Springer 1998.

[GL] M. Green and R. Lazarsfeld, Some results on syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), 301-314.

[HM] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23-88.

[HMo] J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), 321–355.

[L] R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, in: Lectures on Riemann Surfaces, (M. Cornalba, X. Gomez-Mont, A. Verjovsky eds.), World Scientific 1989, 500-559.

[Log] A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. of Math. 125 (2003), 105-138.

[OSS] C. Okonek, M. Schneider and K. Spindler, Vector bundles on complex projective spaces, Progress in Math 3, Birkhauser 1980.

[SchT] F.-O. Schreyer and F. Tonoli, Needles in a Haystack: Special Varieties via Small Fields, in: Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computations in Math. 8 (2002) 251-279.

[V1] C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249-272.

[V2] C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface, J. Eur. Math. Soc. 4 (2002), 363-404.

[V3] C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163-1190.

Department of Mathematics, University of Texas, Austin, TX 78712

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