KOSZUL DIVISORS ON MODULI SPACES OF CURVES

GAVRIL FARKAS

1. Introduction

In this paper we describe a general method of constructing special effective divisors on various moduli spaces using the syzygies of the parametrized objects. The method can be applied to a wide range of moduli problems with the property that the coarse moduli space has canonical singularities hence pluricanonical forms extend over any desingularization of the moduli space. Here we treat the case of the moduli stacks \overline{M}_{g,n} and we develop the intersection theory machinery necessary to understand the compactification and compute the class of these Koszul divisors. Our main result (Theorem 1.1) provides the first infinite sequence of actual (as opposed to virtual) counterexamples to the Harris-Morrison Slope Conjecture and encodes in a single formula virtually all known divisor class calculations on \overline{M}_{g}.

The idea of using geometric divisors to study the geometry of a moduli space can be traced back to Harris and Mumford (cf. [HM]) who, in the course of their proof that \overline{M}_{g} is of general type for odd genus g = 2k − 1 ≥ 25, studied the Hurwitz divisor M_{g,k}^{1} := {[C] ∈ M_{g} : ∃ C →^{k:1} P^{1}} consisting of curves with a pencil 𝔤_{k}^{1}. By computing

the class of \overline{M}_{g,k}^{1} and comparing it in to K_{\overline{M}_{g}}, they showed that when g ≥ 25, the

canonical class is a combination with positive coefficients of [\overline{M}_{g,k}^{1}], the Hodge class λ (which is big and nef) and various boundary classes. Later, numerous other divisor class calculations were carried out. Eisenbud and Harris considered the Petri divisors on \overline{M}_{2k−2} consisting of curves C of genus 2k−2 having a pencil A ∈ W_{k}^{1}(C) which violates the Petri Theorem, which then they used to show that \overline{M}_{g} is of general type for even g ≥ 24 (cf. [EH3]). Logan introduced pointed Brill-Noether divisors on \overline{M}_{g,g} consisting of curves [C, x_{1}, …, x_{g}] ∈ \overline{M}_{g,g} with the property that h^{0}(C, O_{C}(x_{1} + ⋯ + x_{g})) ≥ 2 (cf. [Log]) and used them to determine the Kodaira type of \overline{M}_{g,n} for various g and n.

More recently, in [FP] in our work on the Harris-Morrison Slope Conjecture, we reinterpreted the condition that a curve [C] ∈ M_{10} lie on a K3 surface as saying that there exists a linear system L = K_{C}(−𝔤_{6}^{1}) ∈ W_{12}^{4}(C) such that the embedded curve

C ↪^{|L|} P^{4} is not projectively normal. Using this description we computed the class of the compactification of the divisor K_{10} of curves with this property and showed that s(\overline{K}_{10}) = 7, thus contradicting the Slope Conjecture. In [F1] we generalized this construction to cover all cases g = 6i + 10 and we obtained a (sometimes virtual) Hurwitz type divisor on \overline{M}_{6i+10} defined in terms of linear series 𝔤_{9i+12}^{3i+4} = K_{C}(−𝔤_{3p+6}^{1}) residual to a pencil of minimal degree. This locus, when a divisor, always has slope < 6+12/(g+1) thus violating the Slope Conjecture (see [HMo] and [FP] for background on the effective cone of \overline{M}_{g} and for the significance of the Harris-Morrison Conjecture). Around the same, Khosla provided a different type of example of a divisor on \overline{M}_{g} having exceptionally small slope (cf. [Kh]): on \overline{M}_{21} the closure of the locus of curves [C] ∈ \overline{M}_{21} possessing an embedding C ↪ P^{6} given by a 𝔤_{24}^{6} such that C lies on a quadric, is a divisor whose slope is less than the slope of the Harris-Mumford divisor \overline{M}_{21,11}^{1}.

Research partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment.

The aims of this paper are (1) to give a unified framework for doing divisor class calculation on \overline{M}_{g,n} and (2) to provide (empirical) evidence that syzygy divisors may be the answer to the riddle: Given a moduli space, what is the most intrinsic, most natural and from the point of view of birational geometry, most useful effective divisor on it? We prove that virtually all interesting known divisors on \overline{M}_{g} (the Harris-Mumford divisor, the Petri divisor and all known counterexamples to the Harris-Morrison Conjecture) can be treated in a unified way and are particular instances of a single syzygy type construction. In [F3] we shall further illustrate this ideology by studying moduli spaces of curves with various level structures from the point of view of syzygies.

We fix integers i ≥ 0 and s ≥ 1 and set r := 2s + si + i, g := rs + s and d := rs + r.

We denote by 𝔊_{d}^{r} the stack parametrizing pairs [C, L] with [C] ∈ M_{g} and L ∈ W_{d}^{r}(C) and denote by σ : 𝔊_{d}^{r} → M_{g} the natural projection. Since ρ(g, r, d) = 0, by general Brill-Noether theory, the general curve of genus g will have finitely many 𝔤_{d}^{r}’s and there exists a unique irreducible component of 𝔊_{d}^{r} which maps onto M_{g}.

We denote by K_{i,j}(C, L) the (i, j)-th Koszul cohomology group of the pair [C, L] ∈ 𝔊_{d}^{r} and define a stratification of 𝔊_{d}^{r} with strata U_{g,i} := {(C, L) ∈ 𝔊_{d}^{r} : K_{i,2}(C, L) ≠ 0}. We then set Z_{g,i} := σ_{∗}(U_{g,i}).

Theorem 1.1. If σ : 𝔊̃_{d}^{r} → M̃_{g} is the compactification of 𝔊_{d}^{r} given by limit linear series on tree-like curves, then there exists a natural morphism between torsion free sheaves of the same rank ϕ : A → B over 𝔊̃_{d}^{r} such that \overline{Z}_{g,i} is the image of the degeneracy locus of ϕ. The class of the pushforward to M̃_{g} of the virtual degeneracy locus of ϕ is given by

σ_{∗}(c_{1}(B − A)) ≡ aλ − b_{0} δ_{0} − b_{1} δ_{1} − ⋯ − b_{[g/2]} δ_{[g/2]},

where a, b_{0}, …, b_{[g/2]} are explicitly given coefficients such that b_{1} = 12b_{0} − a and

s(σ_{∗}(c_{1}(B − A))) = \frac{a}{b_{0}} = 6\frac{f(s, i)}{(i + 2) s h(s, i)}, with

f(s, i) = (i^{4}+8i^{3}+24i^{2}+32i+16)s^{7}+(i^{4}+4i^{3}−16i−16)s^{6}−(i^{4}+7i^{3}+13i^{2}−12)s^{5}−(i^{4}+ 2i^{3}+i^{2}+14i+24)s^{4}+(2i^{3}+2i^{2}−6i−4)s^{3}+(i^{3}+17i^{2}+50i+41)s^{2}+(7i^{2}+18i+9)s+2i+2

and

h(s, i) = (i^{3} + 6i^{2} + 12i + 8)s^{6} + (i^{3} + 2i^{2} − 4i − 8)s^{5} − (i^{3} + 7i^{2} + 11i + 2)s^{4} − − (i^{3} − 5i)s^{3} + (4i^{2} + 5i + 1)s^{2} + (i^{2} + 7i + 11)s + 4i + 2.

Furthermore, we have that 6 < \frac{a}{b_{0}} < 6 + \frac{12}{g+1} whenever s ≥ 2. If the morphism ϕ is generically non-degenerate, then \overline{Z}_{g,i} is a divisor on \overline{M}_{g} which gives a counterexample to the Slope Conjecture for g = s(2s + si + i + 1).

For a precise definition of the partial compactification M̃_{g} ⊂ \overline{M}_{g} of M_{g} we re-

fer to Section 2. Since codim(\overline{M}_{g} − M̃_{g}, \overline{M}_{g}) ≥ 2, it makes no difference whether the computation of [\overline{Z}_{g,i}] is carried out over M̃_{g} or \overline{M}_{g}. Despite its complicated appearance, the slope computed in Theorem 1.1 encodes a surprising amount of information about \overline{M}_{g}. In particular, for suitable choices of s and i it specializes to the divisor class calculations carried out in [HM], [EH3], [Kh], [FP] and [F1] which were originally obtained using a variety of ad hoc techniques. The first interesting case is s = 1, g = 2i + 3 when 𝔤_{d}^{r} = 𝔤_{2g−2}^{g−1} = K_{C} (the canonical bundle is the only 𝔤_{2g−2}^{g−1} on a curve of genus g). We can relate the locus Z_{2i+3,i} to more classical loci in M_{2i+3} using Green’s Conjecture which predicts that for any smooth curve C one has the equivalence K_{l,2}(C, K_{C}) = 0 ⇔ l < Cliff(C). Although Green’s Conjecture for arbitrary curves is still open, Voisin proved it for generic curves of given gonality (cf. [V1], [V2]). In our case this gives a set-theoretic identification between Z_{2i+3,i} and the locus M_{2i+3,i+2}^{1} of (i + 2)-gonal curves. Thus Theorem 1.1 provides a new way of calculating the class of the compactification of the Brill-Noether divisor first computed by Harris and Mumford (cf. [HM]):

Corollary 1.2. The slope of the Harris-Mumford divisor \overline{M}_{2i+3,i+2}^{1} on \overline{M}_{2i+3} consisting of curves which cover P^{1} with degree ≤ i + 2 is given by the formula

s(\overline{M}_{2i+3,i+2}^{1}) = \frac{6(i + 3)}{i + 2} = 6 + \frac{12}{g + 1}.

For s = 2 and g = 6i + 10 (that is, in the case h^{1}(L) = 2 when 𝔊_{d}^{r} is isomorphic to a Hurwitz stack parameterizing covers of P^{1}), we recover the main result from [F1]:

Corollary 1.3. The slope of the divisor \overline{Z}_{6i+10,i} on \overline{M}_{6i+10} consisting of curves possessing a pencil 𝔤_{3i+6}^{1} such that if L = K_{C}(−𝔤_{3i+6}^{1}) ∈ W_{9i+12}^{3i+4}(C) denotes the residual linear system,

then C ↪^{|L|} P^{3i+4} fails to satisfy the Green-Lazarsfeld property (N_{i}), is given by the formula:

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

In the case i = 0 we have complete results in the sense that (1) we show that \overline{Z}_{g,0} is an actual divisor on \overline{M}_{g} and (2) we can compute the entire class [\overline{Z}_{g,0}] rather than the λ, δ_{0} and δ_{1} coefficients. In particular we show that b_{j} ≥ b_{0} for j ≥ 1, hence the slope of \overline{Z}_{g,0} is always computed by the λ and δ_{0} coefficients.

Theorem 1.4. For g = s(2s + 1), r = 2s, d = 2s(s + 1) the slope of the virtual class of the locus of those curves [C] ∈ \overline{M}_{g} for which there exists L ∈ W_{d}^{r}(C) such that the embedded curve

C ↪^{|L|} P^{r} sits on a quadric hypersurface, is

s(\overline{Z}_{s(2s+1),0}) = \frac{a}{b_{0}} = \frac{3(16s^{7} − 16s^{6} + 12s^{5} − 24s^{4} − 4s^{3} + 41s^{2} + 9s + 2)}{s(8s^{6} − 8s^{5} − 2s^{4} + s^{2} + 11s + 2)}.

Note that this locus has been first considered by D. Khosla who, using a different approach, was able to compute the coefficients a and b_{0} (cf. [Kh]). Showing that the degeneration loci Z_{g,i} are actual divisors on M_{g} can be very difficult in practice (for instance, the statement that Z_{2i+3,i} is a divisor on M_{2i+3} is essentially Green’s Conjecture for a generic curve of odd genus). Apart from the case s = 1 (settled by Voisin in [V2]), the only cases where it was previously known that Z_{g,i} is an actual divisor were s = 2, i = 0 (cf. [FP], this being the K3 divisor on M_{10}), s = 2, i = 1, 2 (cf. [F1]) and when s = 3, i = 0 (cf. [Kh]) - these last three cases having been settled using Macaulay. Here we show that for i = 0 the degeneracy loci Z_{g,0} are honest divisors on \overline{M}_{g}, that is, the map ϕ : A → B described in Theorem 1.1 is generically non-degenerate. This provides the first infinite sequence of actual (as opposed to virtual) counterexamples to the Harris-Morrison Slope Conjecture [HMo]:

Theorem 1.5. For an integer s ≥ 2 we set r := 2s, d := 2s(s + 1) and g := s(2s + 1). Then ϕ : A → B is a generically non-degenerate map between vector bundles over 𝔊̃_{d}^{r} having the same rank and its degeneracy locus

Z_{g,0} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such thatC ↪^{|L|} P^{r} is not projectively normal }

is a divisor on \overline{M}_{g} of slope

s(\overline{Z}_{g,0}) = \frac{3(16s^{7} − 16s^{6} + 12s^{5} − 24s^{4} − 4s^{3} + 41s^{2} + 9s + 2)}{s(8s^{6} − 8s^{5} − 2s^{4} + s^{2} + 11s + 2)}

contradicting the Slope Conjecture.

As an application of the techniques developed for proving Theorem 1.1 we compute the class of the Gieseker-Petri divisors on \overline{M}_{g}. Recall that Petri’s Theorem asserts that for a general curve [C] ∈ M_{g} and for an arbitrary line bundle L on C, the multiplication map

μ_{0}(L) : H^{0}(L) ⊗ H^{0}(K_{C} ⊗ L^{∨}) → H^{0}(K_{C})

is injective (see [EH4] and [Laz] for two very different, relatively short proofs). The map μ_{0}(L) governs the deformation theory of sections of the line bundle L. It is well-known that G_{d}^{r}(C) is smooth of expected dimension ρ(g, r, d) at a point [L] ∈ W_{d}^{r}(C) if and only if μ_{0}(L) is injective. The locus in M_{g} where the Petri Theorem fails breaks up into numerous components and its geometry is still quite mysterious (see [F2], [EH3]). For integers r, s ≥ 1 we set again d := rs + r and g := rs + s, so that ρ(g, r, d) = 0. Like in

[F2] we define the Gieseker-Petri locus

GP_{g,d}^{r} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such that μ_{0}(L) is not injective }.

Theorem 1.6. For d = rs + r and g = rs + s, the class of the Gieseker-Petri divisor in \overline{M}_{g} is given by the formula:

\overline{GP}_{g,d}^{r} ≡ \frac{c_{r}(s − 1)r}{(r + s + 1)(rs + s − 2)(rs + s − 1)}(aλ − b_{0}δ_{0} − b_{1}δ_{1} − ∑_{j=2}^{[g/2]} b_{j}δ_{j}),

where c_{r} is an explicitly given constant defined in Lemma 2.6,

a = r^{2}s^{2}(4s + r + rs + 10) + s^{2}(5rs + 24r + 2s + 15) + 21s + 26rs + 7r^{2}s + 2r + 2,

b_{0} = \frac{s(s + 1)(r + 1)(r + 2)(rs + s + 4)}{6},

b_{1} = (rs + s − 1)(3rs^{2} + 2s^{2} + r^{2}s^{2} + 7s + 6rs + r^{2}s + 2r + 2), and b_{j} ≥ b_{1} for j ≥ 2. In particular we have the following expression for the slope:

s(\overline{GP}_{g,d}^{r}) = 6 + \frac{12}{g + 1} + \frac{6(s + r + 1)(rs + s − 2)(rs + s − 1)}{s(s + 1)(r + 1)(r + 2)(rs + s + 4)(rs + s + 1)}.

Theorem 1.6 shows that the Gieseker-Petri divisors satisfy the Slope Conjecture, that is, s(\overline{GP}_{g,d}^{r}) ≥ 6 + 12/(g + 1). This is consistent with Proposition 2.2 from [FP] stating that any effective divisor on \overline{M}_{g} violating the Slope Conjecture would have to contain the locus \overline{K}_{g} ⊂ \overline{M}_{g} of curves lying on K3 surfaces and with Lazarsfeld’s result (cf. [Laz]) that a general [C] ∈ \overline{K}_{g} satisfies Petri’s Theorem. For s = 2, Theorem 1.6 specializes to Eisenbud and Harris’s computation originally used to show that \overline{M}_{g} is of general type for large even genus (cf. [EH3], Theorem 2):

Corollary 1.7. For g = 2r + 2, the Gieseker-Petri divisor GP_{2r+2,3r}^{r} can be interpreted as the branch locus of the generically finite map σ : 𝔊_{3r}^{r} → M_{2r+2} from the Hurwitz stack 𝔊_{3r}^{r} = 𝔊_{r+2}^{1} of covers of degree r + 2 and one has the following expression for its class:

\overline{GP}_{2r+2,3r}^{r} ≡ c_{r}(\frac{6r^{2} + 25r + 20}{2r + 1}λ − \frac{(r + 1)(r + 2)}{2r + 1}δ_{0} − (3r + 4)δ_{1} − ∑_{j=2}^{r+1} b_{j}δ_{j}),

where b_{j} > 1 for j ≥ 2.

In Section 4 we describe five different ways of constructing Koszul divisors on \overline{M}_{g,n}. The direct analogue of Theorem 1.1 in the pointed case is the following statement:

Theorem 1.8. Fix positive integers g and i such that

n := \frac{2g + i + 1}{2} + \frac{\sqrt{(i + 1)^{2} + 4ig + 8g}}{2}

is an integer. Then the locus

𝔖𝔶𝔷_{g,n} := {[C, x_{1}, …, x_{n}] ∈ M_{g,n} : K_{i,2}(C, O_{C}(x_{1} + ⋯ + x_{n})) ≠ 0}

is a divisor on M_{g,n}, and the class of its compactification is given by the formula:

\overline{𝔖𝔶𝔷}_{g,n} ≡ \frac{1}{n − g − i}\binom{n − g − 1}{i}(−(n+g−1)λ+(3g−n+i+1) ∑_{j=1}^{n} ψ_{j}−0·δ_{irr}− ∑_{j,t≥0} ∑_{|S|=t} b_{j:t}δ_{j:S}),

where b_{j:t} > 1 are explicitly determined coefficients.

Another infinite sequence of interesting divisors on \overline{M}_{g,n} can be obtained by using the Gaussian-Wahl map associated to a line bundle on a curve. Recall that if L is a line bundle on a curve C, the Wahl map

ψ_{L} : ∧^{2}H^{0}(C, L) → H^{0}(C, K_{C} ⊗ L^{⊗2})

is defined by the formula ψ_{L}(f ∧ g) := f · dg − g · df. The Gaussian ψ_{L} measures deformations of the cone over the curve C embedded in projective space by the linear system |L| and it is known that if C lies on a K3 surface then the Wahl map ψ_{K_{C}} cannot be surjective (cf. [Wa]). Furthermore, the divisor \overline{Z}_{10,0} on \overline{M}_{10} can be viewed as the global degeneracy locus corresponding to the Wahl map for canonical curve of genus 10 (see [FP] for details and further references). If [C, x_{1}, …, x_{n}] ∈ M_{g,n}, we set Γ := x_{1} + ⋯ + x_{n} ∈ C_{n} for the divisor of marked points.

Theorem 1.9. Fix an integer g such that

n := \frac{2g + 3 + \sqrt{24g + 1}}{2}

is an integer. Then the locus

𝔚𝔞𝔥𝔩_{g,n} := {[C, x_{1}, …, x_{n}] ∈ M_{g,n} : ψ_{Γ} : ∧^{2}H^{0}(O_{C}(Γ)) → H^{0}(K_{C}⊗O_{C}(2Γ)) is degenerate}

is a divisor on M_{g,n} and its compactification has the following class:

\overline{𝔚𝔞𝔥𝔩}_{g,n} = −(n − g − 1)λ + (n − g − 1) ∑_{j=1}^{n} ψ_{j} − δ_{irr} − ∑_{j,t≥0} b_{j:t} ∑_{|S|=t} δ_{j:S},

where b_{j:t} > 1 are explicitly determined coefficients.

Note that although the divisors \overline{𝔖𝔶𝔷}_{g,n} and \overline{𝔚𝔞𝔥𝔩}_{g,n} live on \overline{M}_{g,n}’s for some very

particular choices of n, using the forgetful and clutching maps \overline{M}_{g,n} → \overline{M}_{g,n−1} and \overline{M}_{i,n_{1}} × \overline{M}_{g−i,n_{2}} → \overline{M}_{g,n_{1}+n_{2}} one immediately has explicit Koszul divisors on \overline{M}_{g,n} for all g and n.

Among other syzygetic ways of producing divisors on \overline{M}_{g,n} we single the one using the Minimal Resolution Conjecture (cf. Theorem 4.2), which can be thought of as a generalization of the divisor of higher Weierstrass points and which is especially useful in the case of a large number of marked points. An immediate application of the calculations in Section 4 is the following result about the Kodaira type of \overline{M}_{g,n}:

Theorem 1.10. For integers g = 4, …, 21, the moduli space \overline{M}_{g,n} is of general type for all n ≥ f(g) where f(g) is described in the following table.

\begin{matrix}g & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ f(g) & 16 & 15 & 16 & 15 & 14 & 13 & 11 & 12 & 13 & 11 & 10 & 10 & 9 & 9 & 9 & 7 & 6 & 4 \end{matrix}

This result represents an improvement of Logan’s Theorem 5.1 for g = 4 − 6, 10, 14 − 16, 18 − 22, the entries for the remaining values of g being those from [Log].

Acknowledgment: I am grateful to Sean Keel for many discussions over the years on topics related to this circle of ideas.

2. Constructing divisors of small slope using syzygies

For a projective variety X and a line bundle L on X we denote by K_{i,j}(X, L) the Koszul cohomology group obtained from the complex

∧^{i+1}H^{0}(L) ⊗ H^{0}(L^{⊗(j−1)}) ⟶ ∧^{i}H^{0}(L) ⊗ H^{0}(L^{⊗j}) ⟶ ∧^{i−1}H^{0}(L) ⊗ H^{0}(L^{⊗(j+1)}),

where the maps are the Koszul differentials (cf. [Gr]). Assume L is globally generated and M_{L} is the vector bundle on X defined by the exact sequence

0 → M_{L} → H^{0}(L) ⊗ O_{X} → L → 0.

A simple argument using the exact sequences

0 ⟶ ∧^{a}M_{L} ⊗ L^{⊗b} → ∧^{a}H^{0}(L) ⊗ L^{⊗b} ⟶ ∧^{a−1}M_{L} ⊗ L^{⊗(b+1)} ⟶ 0

for various a and b, shows that there is an identification

(1)  K_{a,b}(X, L) = \frac{H^{0}(∧^{a}M_{L} ⊗ L^{⊗b})}{Image\{∧^{a+1}H^{0}(L) ⊗ H^{0}(L^{⊗(b−1)})\}} .

From now on we fix integers i ≥ 0 and s ≥ 1 and set

r := 2s + si + i, g := rs + s, and d := rs + r.

We introduce the open substack M_{g}^{0} of M_{g} corresponding to curves [C] ∈ M_{g} such that W_{d−1}^{r}(C) = ∅ and W_{d}^{r+1}(C) = ∅. Then codim(M_{g} − M_{g}^{0}, M_{g}) ≥ 2. We denote by 𝔓𝔦𝔠^{d} the degree d Picard stack over M_{g} (precisely, the étale sheafification of the Picard functor). In particular if Pic_{M_{g}}^{d} is the coarse moduli space associated to 𝔓𝔦𝔠^{d}, then for any M_{g}-scheme T → M_{g} originating from a family of genus g curves X → T, the fibre product T ×_{M_{g}} Pic_{M_{g}}^{d} is the relative Picard algebraic space Pic_{X /T}^{d}. We denote by 𝔊_{d}^{r} ⊂ 𝔓𝔦𝔠^{d} the stack parameterizing pairs [C, L] with [C] ∈ M_{g} and L ∈ W_{d}^{r}(C) and by σ : 𝔊_{d}^{r} → M_{g} the natural projection. Since ρ(g, r, d) = 0, by general Brill-Noether theory, the general curve of genus g will have finitely many 𝔤_{d}^{r}’s and there exists a unique irreducible component of 𝔊_{d}^{r} which maps onto M_{g}. Moreover, the image of any component of 𝔊_{d}^{r} having dimension ≥ 3g − 2 is a substack of codimension ≥ 2 in M_{g} (cf. Corollary 2.5), thus one can ignore these extraneous components of 𝔊_{d}^{r} when doing divisor class calculations on \overline{M}_{g}.

We shall define a determinantal substack of 𝔊_{d}^{r} consisting of those pairs [C, L] satisfying the condition K_{i,2}(C, L) ≠ 0. We denote by π : M_{g,1}^{0} → M_{g}^{0} the universal curve and by L a universal Poincaré bundle on the fibre product M_{g,1}^{0} ×_{M_{g}^{0}} 𝔊_{d}^{r} (In the case such anL does not exist, we pass to an étale surjection Σ → 𝔊_{d}^{r} such that Σ is a scheme and M_{g,1}^{0} ×_{M_{g}^{0}} Σ admits a Poincaré bundle and we carry out the construction at this level. In the end our construction does not depend on the choice of Σ, see also [Est2], Section 6.2). If p_{1} : M_{g,1}^{0} ×_{M_{g}^{0}} 𝔊_{d}^{r} → M_{g,1}^{0} and p_{2} : M_{g,1}^{0} ×_{M_{g}^{0}} 𝔊_{d}^{r} → 𝔊_{d}^{r} are the natural projections, then E := p_{2∗}(L^{∨}) is a vector bundle of rank r + 1 and there is a tautological embedding of the pullback of the universal curve M_{g,1}^{0} ×_{M_{g}^{0}} 𝔊_{d}^{r} into the projective bundle u : P(E) → 𝔊_{d}^{r}. We define the vector bundle F on P(E) by the sequence

0 ⟶ F ⟶ u^{∗}(E) ⟶ O_{P(E)}(1) ⟶ 0,

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

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

If C →^{|L|} P^{r} is the map corresponding to a point [C, L] ∈ 𝔊_{d}^{r}, then

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

and there is a vector bundle morphism ϕ : A → B given by restriction. Grauert’s Theorem guarantees that both A and B are vector bundles over 𝔊_{d}^{r} and their ranks are

rank(A) = (i + 1)\binom{r + 2}{i + 2} and rank(B) = \binom{r}{i}(−\frac{id}{r} + 2d + 1 − g)

(We use that M_{L} is a stable vector bundle, see [F1], Proposition 2.1 and this implies that H^{1}(∧^{i}M_{L} ⊗ L^{⊗2}) = 0, hence rank(B) can be computed from Riemann-Roch). Because of the way we chose g, r and d we can see that rank(A) = rank(B).

While the construction of A and B clearly depends on the choice of the Poincaré bundle L (and of Σ), it is easy to check that the vector bundle Hom_{O_{𝔊^{r}_{d}}}(A, B) on 𝔊_{d}^{r}

as well as the morphism ϕ ∈ H^{0}(𝔊_{d}^{r}, Hom_{O_{𝔊^{r}_{d}}}(A, B)) are independent of such choices. More precisely, let us denote by Ξ the collection of pairs α := (π_{α}, L_{α}) where π_{α} : Σ_{α} → 𝔊_{d}^{r} is an étale surjective morphism from a scheme Σ_{α} and L_{α} is a Poincaré bundle on p_{2,α} : M_{g,1}^{0} ×_{M_{g}^{0}} Σ_{α} → Σ_{α}. Recall that if Σ → 𝔊_{d}^{r} is an étale surjection from a scheme and L and L′ are two Poincaré bundles on p_{2} : M_{g,1}^{0} ×_{M_{g}^{0}} Σ → Σ, then the sheaf N := p_{2∗}Hom(L, L′) is invertible and there is a canonical isomorphism L ⊗ p_{2}^{∗}N ≅ L′. For every α ∈ Ξ we construct the morphism between vector bundles of the same rank ϕ_{α} : A_{α} → B_{α} over Σ_{α} as above. Then since a straightforward cocycle condition is met, we find that there exists a vector bundle Hom_{O_{𝔊^{r}_{d}}}(A, B) on 𝔊_{d}^{r} together with a

section ϕ ∈ H^{0}(𝔊_{d}^{r}, Hom_{O_{𝔊^{r}_{d}}}(A, B)) such that for every α = (π_{α}, L_{α}) ∈ Ξ we have that π_{α}^{∗}(Hom_{O_{𝔊^{r}_{d}}}(A, B)) = Hom_{O_{Σα}} (A_{α}, B_{α}) and π_{α}^{∗}(ϕ) = ϕ_{α}.

Theorem 2.1. The cycle U_{g,i} := {(C, L) ∈ 𝔊_{d}^{r} : K_{i,2}(C, L) ≠ 0} is the degeneracy locus of vector bundle map ϕ : A → B over 𝔊_{d}^{r}.

Proof. Along the same lines as the proof of Proposition 2.5 in [F1]. □

Thus Z_{g,i} := σ_{∗}(U_{g,i}) is a virtual divisor on M_{g} when g = s(2s + si + i + 1).

Remark 2.2. Using (1) it is easy to prove that for every (C, L) ∈ 𝔊_{d}^{r} one have the vanishing of Koszul cohomology groups K_{a,0}(C, L) = 0 for all a ≥ 1 and K_{a,b}(C, L) = 0 for all b ≥ 3. Thus the only non-trivial Koszul type conditions one could impose on 𝔊_{d}^{r} involve the groups K_{a,1}(C, L) and K_{a,2}(C, L). Because M_{L} is a stable vector bundle on C, it is straightforward to show using (1) that

dim K_{i,2}(C, L) − dim K_{i+1,1}(C, L) = \binom{r}{i}(2d − \frac{id}{r} + 1 − g) − (i + 1)\binom{r + 2}{i + 2}.

For our choices of g, r and d, it follows that dim K_{i+1,1}(C, L) = dim K_{i,2}(C, L), hence U_{g,i} can also be defined as the locus where K_{i+1,1}(C, L) fails to vanish. This shows that, at least in the case of curves, there are no other Koszul divisors except U_{g,i}.

To prove Theorem 1.1 we shall extend the determinantal structure of Z_{g,i} over a substack of \overline{M}_{g} whose complement has codimension ≥ 2. We denote by M̃_{g} := M_{g}^{0} ∪ (∪_{j=0}^{[g/2]}Δ_{j}^{0}) the locally closed substack of \overline{M}_{g} obtained by adding to M_{g}^{0} the open subsets Δ_{j}^{0} ⊂ Δ_{j} for 1 ≤ j ≤ [g/2] 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_{d}^{r} → M_{g}^{0} to a proper, generically finite map

σ : 𝔊̃_{d}^{r} → M̃_{g}

by letting 𝔊̃_{d}^{r} be the space of limit 𝔤_{d}^{r}’s on the curves from M̃_{g} which are all tree-like (see [EH1], Theorem 3.4 for the construction of the variety of limit linear series and also [Oss] for a more functorial approach which in the case ρ(g, r, d) = 0 leads to the Eisenbud-Harris space). Strictly speaking, Eisenbud and Harris have only constructed the space of refined limit 𝔤_{d}^{r}’s. Using the observation that when ρ(g, r, d) = 0 every crude non-refined limit 𝔤_{d}^{r} on a curve of compact type C∪_{y}D, where [C] ∈ M_{j} and [D] ∈ M_{g−j} can be canonically interpreted as a refined limit 𝔤_{d}^{r} on the pre-stable curve C ∪_{y_{1}} P^{1} ∪_{y_{2}} D obtained from C ∪_{y} D by inserting a single P^{1} at the node y, their construction can be easily adapted to cover the case of crude 𝔤_{d}^{r}’s as well. Note that since all limit 𝔤_{d}^{r}’s are dimensionally proper (cf. [EH1], Corollary 3.7), every limit linear series from 𝔊̃_{d}^{r} is smoothable.

To compute the class [\overline{Z}_{g,i}], we intersect \overline{Z}_{g,i} with test curves in the boundary

of \overline{M}_{g} which are defined as follows: we fix a Brill-Noether general curve C of genus g − 1, a general point q ∈ C and a general elliptic curve E. We define two 1-parameter families

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

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

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

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

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

We review the notation used in the theory of limit linear series (see [EH1]). If X is a tree-like 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_{i}^{l_{Y}} (y)}_{i=0,...,r} the vanishing sequence of l at y, by {α_{i}^{l_{Y}} (y) = a_{i}^{l_{Y}} (y) − i}_{i=0,...,r} the ramification sequence and by ρ(l_{Y} , y) := ρ(g, r, d) − ∑_{i=0}^{r} α_{i}^{l_{Y}} (y) the adjusted Brill-Noether number with respect to y.

Proposition 2.3. (1) Let C_{y}^{1} = C ∪_{y} E be an element of Δ_{1}^{0}. If (l_{C}, l_{E}) is a limit 𝔤_{d}^{r} on C_{y}^{1}, then V_{C} = H^{0}(L_{C}) and L_{C} ∈ W_{d}^{r}(C) has a cusp at y. If y ∈ C is a general point, then l_{E} = (O_{E}(dy), (d − r − 1)y + |(r + 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_{d}^{r}(C) such that ρ(L_{C}, y) = −1, then l_{E}(−(d − r − 2)y) is a 𝔤_{r+2}^{r} with vanishing sequence at y being ≥ (0, 2, 3, …, r, r + 2). Moreover, at the level of 1-cycles we have the identification σ^{∗}(C^{1}) ≡ X + ν T, where

X := {(y, L) ∈ C × W_{d}^{r}(C) : h^{0}(C, L(−2y)) ≥ r},

T ≅ P(H^{0}(O_{E}((r+2)y))/H^{0}(O_{E}(ry))) is the curve consisting of 𝔤_{r+2}^{r}’s on E with vanishing ≥ (0, 2, …, r, r + 2) at the fixed point y ∈ E and ν is an explicitly known positive integer.

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

Y := {(y, L) ∈ C × W_{d}^{r}(C) : h^{0}(C, L(−y − q)) ≥ r}.

Proof. Part (1) is similar to the proof of Proposition 3.3 from [F1] and we omit the details. For part (2), we claim that for any limit 𝔤_{d}^{r} on a curve C_{y}^{0} where y ∈ C, the underlying torsion free sheaf is actually locally free. Indeed, otherwise the underlying sheaf would be of the form ν_{∗}(L), where ν : C → C_{0}^{y} is the normalization map and L ∈ W_{d−1}^{r}(C). But [C] ∈ M_{g−1} is assumed to be Brill-Noether general, hence W_{d−1}^{r}(C) = ∅. □

Throughout this paper we routinely use basic facts from Schubert calculus which we briefly recall. If G(r, d) denotes the Grassmannian of r-planes in P^{d} and

ℂ^{d+1} = V_{0} ⊃ V_{1} ⊃ ⋯ ⊃ V_{r+1} = 0

is a decreasing flag, then for any Schubert index 0 ≤ α_{0} ≤ … ≤ α_{r} ≤ d − r, we define the Schubert cycle

σ_{(α_{0},...,α_{r})} := {Λ ∈ G(r, d) : dim(Λ ∩ V_{α_{i}+i}) ≥ r + 1 − i, for i = 0, …, r}.

(This differs slightly from the standard notation from e.g. [FuPr], but it seems better suited for dealing with ramification sequences of linear series). Often we use the fact that if (α_{0}, …, α_{r}) is a Schubert index and g is an integer such that rg + ∑_{i=0}^{r} α_{i} = (r + 1)(d − r), then there is an identity in H^{∗}(G(r, d)):

(3)  σ_{(α_{0},...,α_{r})} · σ_{(0,1,...,1)}^{g} = g!\frac{∏_{i<j}(α_{j} − α_{i} + j − i)}{∏_{i=0}^{r}(g − d + i + α_{i} + r)!}.

Proposition 2.4. Let [C] ∈ M_{j} be a general curve with g−2 ≥ j ≥ [g/2] and C^{j} ⊂ Δ_{j} ⊂ \overline{M}_{g} the associated test curve of type (j, g − j). Then one has the following equality of 1-cycles in 𝔊̃_{d}^{r}:

σ^{∗}(C^{j}) = ∑_{(α_{0},...,α_{r})∈P_{1}} N_{g−j,α} · X_{j,α} + ∑_{(β_{0},...,β_{r})∈P_{2}} M_{j,β} · Y_{g−j,β} + ∑_{(β_{0},...,β_{r})∈P_{3}} Q_{g−j,β} · U_{j,β},

where we introduce the following notations: P_{1} := {(0 ≤ α_{0} ≤ … ≤ α_{r} ≤ s) : ∑_{i=0}^{r} α_{i} = j},

P_{2} := {(0 ≤ β_{0} ≤ … ≤ β_{r} ≤ s + 1) : ∑_{i=0}^{r} β_{i} = j + 1, β_{r−1} ≤ s},
P_{3} := {(0 = β_{0} < β_{1} ≤ … ≤ β_{r} ≤ s + 1) : ∑_{i=0}^{r} β_{i} = r + 1 + j},
M_{j,β} := σ_{(0,1,...,1)}^{j} · σ_{(β_{0},...,β_{r})} ∈ H^{∗}(G(r, r + j)) forβ ∈ P_{2},
N_{g−j,α} := σ_{(0,1,...,1)}^{g−j} · σ_{(j−α_{r},...,j−α_{0})} ∈ H^{∗}(G(r, d)) forα ∈ P_{1},
Q_{g−j,β} := σ_{(0,1,...,1)}^{g−j} · σ_{(j+1−β_{r},...,j+1−β_{1},j+1)} ∈ H^{∗}(G(r, d)) for β ∈ P_{3},

X_{j,α} := {(y, L_{C}) ∈ C × Pic^{r+j}(C) : α_{i}^{L_{C}}(y) ≥ α_{i} fori = 0 … r}, α ∈ P_{1},
Y_{g−j,β} := {l_{D} ∈ G_{d}^{r}(D) : α_{i}^{l_{D}}(y) ≥ j − β_{r−i} fori = 0 … r}, β ∈ P_{2},
U_{j,β} := {(y, l_{C}) ∈ C × G_{r+j+1}^{r}(C) : α^{l_{C}}(y) ≥ (0, β_{1}, …, β_{r})}, forβ ∈ P_{3}.

Proof. Suppose that l = (l_{C}, l_{D}) is a limit 𝔤_{d}^{r} on C ∪_{y} D. It is easy to see that the generic point of any component of σ^{∗}(C^{j}) corresponds to a refined limit 𝔤_{d}^{r}, so we may assume that l is refined as well. If (α_{0}, …, α_{r}) is the ramification sequence of l_{C} at y, then the condition that [D, y] ∈ M_{g−j,1} carries a 𝔤_{d}^{r} with ramification sequence at y being at least (d − r − α_{r}, …, d − r − α_{0}), is that σ_{(0,1,...,1)}^{g−j} · σ_{(d−r−α_{r},...,d−r−α_{0})} ≠ 0 ∈ H^{∗}(G(r, d)). Using the Littlewood-Richardson rule, we find that this implies that α_{r} ≤ rs + s − j. A similar reasoning can be used for C. Degenerating C to a stable curve consisting of a rational spine and j elliptic tails, we obtain that if there exists a point y ∈ C and a 𝔤_{d}^{r} with ramification sequence (α_{0}, …, α_{r}) at y, then either y specializes to a point on the rational spine in which case we find the condition σ_{(0,1,...,1)}^{j} · σ_{(α_{0},...,α_{r})} ≠ 0 ∈ H^{∗}(G(r, d)) which implies that α_{0} ≥ rs − j, or else, y specializes to a point on one of the elliptic tails in which case we find that there must exist two indices 0 ≤ e < f ≤ r with α_{e} ≥ α_{e−1} + 1 and α_{f} ≥ α_{f−1} + 1, such that σ_{(0,1,...,1)}^{j−1} · σ_{(α_{0}+1,α_{e−1}+1,α_{e},α_{e+1}+1,...,α_{f−1}+1,α_{f} ,α_{f+1}+1,...,α_{r}+1)} ≠ 0. This last condition leads to the inequality α_{0} ≥ max{0, rs − j − 1}.

Suppose we are in the first case, that is, α_{0} ≥ rs − j and moreover ρ(l_{C}, y) = ρ(l_{D}, y) = 0, which is the situation which occurs for a generic choice of y ∈ C. Then l_{C}(−(rs − j)y) = |L_{C}|, where L_{C} ∈ Pic^{r+j}(C) with α_{r}^{L_{C}}(y) ≤ s and ∑_{i=0}^{r} α_{i}^{L_{C}}(y) = j, that is, (L_{C}, y) ∈ X_{j,α}. If α_{0} ≥ rs − j but now ρ(l_{C}, y) = −1 and ρ(l_{D}, y) ≤ 1, then {α_{i}^{L_{C}}(y) − (rs − j)}_{i=0...r} must be one of the partitions from the set P_{2}. Choosing such a partition, we have M_{j,β} choices for the C-aspect, while l_{D} ∈ Y_{g−j,α^{L_{C}} (y)−(rs−j)}. Finally let us assume that we are in the case α_{0}^{l_{C}}(y) = rs−j−1. Then necessarily α_{1}^{l_{C}}(y) ≥ rs−j, ρ(l_{C}, y) = ρ(l_{D}, y) = 0 and l_{C}(−(rs−j −1)y) ∈ U_{j,β}, where β_{i} := α_{i}^{l_{C}}(y)−(rs−j −1) for i = 0 … r. This accounts for the third sum in σ^{∗}(C^{j}). Arguing along the lines of [EH2], Lemma 3.4, 𝔊̃_{d}^{r} is smooth along σ^{∗}(C^{j}) and since all limit 𝔤_{d}^{r} described in this proof are smoothable, we obtain that the claimed formula holds at the level of 1-cycles (including multiplicities). □

The next corollary shows that ”ghost” components of 𝔊̃_{d}^{r} having dimension > 3g − 3, do not matter in the calculation of [σ_{∗}(G_{i,2} − H_{i,2})].

Corollary 2.5. In the case ρ(g, r, d) = 0, every irreducible component of Z of 𝔊̃_{d}^{r} such that dim(Z) ≥ 3g − 2 has the property that dim σ(Z) ≤ 3g − 5.

Proof. For a general [C] ∈ M_{g}, the scheme W_{d}^{r}(C) is reduced and 0-dimensional, thus every component of 𝔊̃_{d}^{r} mapping dominantly onto M̃_{g} must have dimension 3g − 3. Suppose that Z is a component of dimension at least 3g − 2 such that σ(Z) is a divisor on M̃_{g}. Then for any [C] ∈ σ(Z) we have that dim σ^{−1}([C]) ≥ 2. Since this property does not hold along any of the curves σ^{∗}(C^{j}) for any 0 ≤ j ≤ [g/2] (see Proposition 2.4), it follows that σ(Z) is disjoint from the test curves C^{j} ⊂ \overline{M}_{g} for all j ≥ 0. This implies then that [σ(Z)] = 0 ∈ Pic(M̃_{g}), hence σ(Z) = 0 (use that the Satake compactification of M_{g} has boundary of codimension 2). This is a contradiction. □

Let C be a Brill-Noether general curve of genus g − 1 (recall that g = rs + s and d = rs + r). Then dim W_{d}^{r}(C) = r and it is easy to see that C carries no 𝔤_{d−1}^{r}’s or 𝔤_{d}^{r+1}’s, hence every L ∈ W_{d}^{r}(C) corresponds to a complete and base point free linear series. We denote by L a Poincaré bundle on C × Pic^{d}(C) and by π_{1} : C × Pic^{d}(C) → C and π_{2} : C × Pic^{d}(C) → Pic^{d}(C) the projections. We define the cohomology class η = π_{1}^{∗}([point]) ∈ H^{2}(C × Pic^{d}(C)), and if δ_{1}, …, δ_{2g} ∈ H^{1}(C, ℤ) ≅ H^{1}(Pic^{d}(C), ℤ) is a symplectic basis, then we set

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

We have the formula (cf. [ACGH], p. 335) c_{1}(L) = dη + γ, corresponding to the Hodge decomposition of c_{1}(L). We also record that γ^{3} = γη = 0, η^{2} = 0 and γ^{2} = −2ηπ_{2}^{∗}(θ). On W_{d}^{r}(C) we have the tautological rank r + 1 vector bundle E := (π_{2})_{∗}(L_{|C×W_{d}^{r}(C)}). The Chern numbers of E can be computed using the Harris-Tu formula (cf. [HT]): if we write ∑_{i=0}^{r} c_{i}(E^{∨}) = (1 + x_{1}) ⋯ (1 + x_{r+1}), then for every class ζ ∈ H^{∗}(Pic^{d}(C), ℤ) one has the formula^{1}

x_{1}^{i_{1}} ⋯ x_{r+1}^{i_{r+1}} ζ = det(\frac{θ^{g−1+r−d+i_{j}−j+l}}{(g − 1 + r − d + i_{j} − j + l)!})_{1≤j,l≤r+1} ζ.

If we use the expression of the Vandermonde determinant, we get the formula

det(\frac{1}{(a_{j} + l − 1)!})_{1≤j,l≤r+1} = \frac{Π_{j>l} (a_{l} − a_{j})}{Π_{j=1}^{r+1} (a_{j} + r)!}.

By repeatedly applying this formula we compute all the intersection numbers on W_{d}^{r}(C) which we shall need:

Lemma 2.6. If c_{i} := c_{i}(E^{∨}) we have the following identities in H^{∗}(W_{d}^{r}(C), ℤ):

(1) c_{r−1}θ = \frac{r(s+1)}{2}c_{r}

(2) c_{r−2}θ^{2} = \frac{r(r−1)(s+1)(s+2)}{6}c_{r}

(3) c_{r−2}c_{1}θ = \frac{r(s+1)}{2}(1 + \frac{(r−2)(r+2)(s+2)}{3(s+r+1)})c_{r}

(4) c_{r−1}c_{1} = (1 + \frac{(r−1)(r+2)(s+1)}{2(s+r+1)})c_{r}

(5) c_{r} = \frac{1! 2!···(r−1)! (r+1)!}{(s−1)! (s+1)! (s+2)!···(s+r)!}θ^{g−1} .

We point out that the constant c_{r} equals the number of linear series 𝔤_{d}^{r} on a general curve of genus g (note that ρ(g, r, d) = 0). In Section 3 we shall use the following result:

1There is a confusing sign error in the formula (1.4) in [HT]: the formula is correct as it is appears in [HT], if the x_{j}’s denote the Chern roots of the dual of the kernel bundle.

Lemma 2.7. If [C] ∈ M_{g−1}, then one has the following identity in H^{∗}(W_{d}^{r}(C), ℤ):

c_{1}(R^{1}π_{2∗}(L_{|C×W_{d}^{r}(C)})) = θ − c_{1}(E^{∨}).

Proof. Let us recall how one can obtain a determinantal structure on W_{d}^{r}(C). Once we fix a divisor D ∈ C_{e} of degree e >> 0, W_{d}^{r}(C) is the degeneracy locus of rank d+e−g+1−r of the vector bundle map (π_{2})_{∗}(L⊗O(π_{1}^{∗}D)) → (π_{2})_{∗}(L⊗O(π_{1}^{∗}D)_{|π_{1}^{∗}(D)}). Consequently, we have an exact sequence of vector bundles over W_{d}^{r}(C):

0 ⟶ E ⟶ (π_{2})_{∗}(L⊗O(π_{1}^{∗}D)) ⟶ (π_{2})_{∗}(L⊗O(π_{1}^{∗}D)_{|π_{1}^{∗}D}) ⟶ R^{1}π_{2∗}(L_{|C×W_{d}^{r}(C)}) ⟶ 0,

from which the claim follows by using that (π_{2})_{∗}(L⊗O(π^{∗}D)_{|π_{1}^{∗}D}) is numerically trivial while c_{t}((π_{2})_{∗}(L ⊗ O(π_{1}^{∗}D))) = e^{−θ} (cf. [EH3] or [ACGH]). □

For integers 0 ≤ a ≤ r and b ≥ 2 we shall define vector bundles G_{a,b} and H_{a,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}) ⊂ 𝔊̃_{d}^{r} which over the locus corresponding to smooth curves have fibres

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

for each (C, L) ∈ 𝔊_{d}^{r} giving a map C →^{|L|} P^{r}. Clearly G_{i,2|𝔊_{d}^{r}} = B and H_{i,2|𝔊_{d}^{r}} = A, where A and B are the vector bundles introduced in Theorem 2.1. Partially extending these bundles over the boundary of 𝔊̃_{d}^{r} will enable us to compute the λ, δ_{0} and δ_{1} coefficients of \overline{Z}_{g,i} and determine the slope s(\overline{Z}_{g,i}).

Proposition 2.8. For each b ≥ 2 there exists a vector bundle G_{0,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}) ⊂ 𝔊̃_{d}^{r} having rank bd + 1 − g whose fibres admit the following description:

• For (C, L) ∈ 𝔊_{d}^{r}, 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 ∈ W_{d}^{r}(C) has a cusp at y ∈ C, 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 ∈ W_{d}^{r}(C) is such that

h^{0}(C, L(−y − q)) = h^{0}(L) − 1, 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. Very similar to Proposition 3.9 in [F1]. □

Having defined the vector bundles G_{0,b} we now define inductively all vector bundles 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.

To define H_{a,b} is even easier. We set H_{0,b} := Sym^{b}G_{0,1} for all b ≥ 1 and we define H_{a,b} inductively via the exact sequence

(5)  0 ⟶ H_{a,b} ⟶ ∧^{a}H_{0,1} ⊗ Sym^{b}H_{0,1} ⟶ H_{a−1,b+1} ⟶ 0.

The surjectivity of the right map in (5) is obvious, whereas to prove that d_{a,b} is surjective, one argues like in [F1], Proposition 3.10. There is a natural vector bundle morphism ϕ_{a,b} : H_{a,b} → G_{a,b}. Moreover rank(H_{i,2}) = rank(G_{i,2}) and the degeneracy locus of ϕ_{i,2} is the codimension one compactification of Z_{g,i} over M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}.

We prove a technical result we shall use later for extending the bundles G_{0,b} with b ≥ 2 over the boundary of 𝔊_{d}^{r}. It can be interpreted as saying that on a suitably general curve, the ramification points of a linear series are distinct from those of its higher order powers.

Proposition 2.9. Fix integers s ≥ 2, r ≥ 2s and a partition 0 ≤ β_{0} ≤ β_{1} ≤ ⋯ ≤ β_{r} ≤ s such that ∑_{i=0}^{r} β_{i} = γ. Let (D, y) be a general pointed curve of genus γ ≥ 3s. Then for every line bundle L_{D} ∈ Pic^{γ+r}(D) satisfying the conditions α_{i}^{L_{D}}(y) = β_{i} for 0 ≤ i ≤ r, we have that

H^{0}(D, K_{D} ⊗ L_{D}^{⊗(−2)} ⊗ O_{D}(ay)) = 0, for alla ≤ 2(r + s).

Proof. Clearly it suffices to prove the theorem in the case a = 2(r + s). We degenerate (D, y) to a stable curve E_{0} ∪ … ∪ E_{γ−1}, consisting of a string of elliptic curves such that E_{i−1} ∩ E_{i} = {p_{i}} for 1 ≤ i ≤ γ − 1. Moreover, we assume that y = p_{0} specializes to a point lying on E_{0} and that the differences p_{i} − p_{i−1} ∈ Pic^{0}(E_{i−1}) are not torsion for all 1 ≤ i ≤ γ −1. We assume by contradiction that H^{0}(D, K_{D} ⊗L_{D}^{⊗(−2)}⊗O_{D}(2(r+s)y)) ≠ 0 for some L_{D} ∈ Pic^{γ+r}(D) and denote by L_{E_{i}} ∈ Pic^{γ+r}(E_{i}) the E_{i}-aspect of the induced limit linear series 𝔤_{r+γ}^{r} on ∪_{i=0}^{γ−1}E_{i} satisfying the ramification conditions α_{t}^{L_{E_{0}}} (p_{0}) = α_{t} for 0 ≤ t ≤ r. Fix an integer 1 ≤ i ≤ γ − 1. By the additivity of the Brill-Noether number, we have that ρ(L_{E_{i−1}}, p_{i}, p_{i−1}) = 0 and there exists an integer 0 ≤ k ≤ r such

that α_{t}^{L_{E_{i}}} (p_{i}) = α_{t}^{L_{E_{i−1}}} (p_{i−1}) + 1 for t ≠ k while α_{k}^{L_{E_{i}}} (p_{i}) = α_{k}^{L_{E_{i−1}}} (p_{i−1}). In particular,

L_{E_{i−1}} = O_{E_{i−1}}((α_{k}^{L_{E_{i−1}}} (p_{i−1}) + k) · p_{i−1} + (γ + r − k − α_{k}^{L_{E_{i−1}}} (p_{i−1})) · p_{i}),

that is L_{E_{i−1}} corresponds to a divisor supported only at the points p_{i−1} and p_{i}. Our assumption implies that for all 0 ≤ i ≤ γ − 1 there exist sections

0 ≠ ρ_{i} ∈ H^{0}(E_{i}, O_{E_{i}}(2(r + s + i)p_{i} + 2(γ − i − 1)p_{i+1}) ⊗ L_{E_{i}}^{⊗(−2)}))

satisfying the compatibility conditions

ord_{p_{i}}(ρ_{i}) ≥ ord_{p_{i−1}}(ρ_{i−1}) and ord_{p_{i}}(ρ_{i−1}) + ord_{p_{i}}(ρ_{i}) = 2s − 2.

We reach a contradiction once we show that

ord_{p_{γ−1}}(ρ_{γ−1}) > 2s − 2 = deg(O_{E_{γ−1}}(2(r + s + γ − 1)p_{γ−1} ⊗ L_{E_{γ−1}}^{⊗(−2)}))

which gives that ρ_{γ−1} = 0. Assume now that ord_{p_{i+1}}(ρ_{i+1}) = ord_{p_{i}}(ρ_{i}) for some 0 ≤ i ≤ γ − 2. Then ord_{p_{i}}(ρ_{i}) + ord_{p_{i+1}}(ρ_{i}) = 2s − 2, hence the section ρ_{i} vanishes only at p_{i} and p_{i+1} ∈ E_{i} and ord_{p_{i}}(ρ_{i}) = 2b for some integer b ≥ 0. We must have that L_{E_{i}} = O_{E_{i}}((r+s+i−b)·p_{i}+(γ−i−s−b)·p_{i+1}) (we use that p_{i+1}−p_{i} ∈ Pic^{0}(E_{i}) is not torsion). In particular r + s + i − b is one entry in the vanishing sequence a^{L_{E_{i}}} (p_{i}) and the vanishing sequence a^{L_{E_{i+1}}} (p_{i+1}) is obtained from a^{L_{i}}(p_{i}) by raising all entries by 1, except for r+s+i−b which remains unmodified. Obviously then, the number r+s+i+1−b cannot appear in the vanishing sequence a^{L_{E_{i+1}}} (p_{i+1}). But this implies that ord_{p_{i+2}}(ρ_{i+2}) ≥ ord_{p_{i+1}}(ρ_{i+1})+1. This argument shows that as we trace the non-decreasing sequence of vanishing orders {ord_{p_{i}}(ρ_{i})}_{i=0}^{γ−1} along any group of 3 consecutive components E_{i−1}, E_{i} and E_{i+1}, we will find at least 2 along which ord_{p_{i}}(ρ_{i}) jumps. Since γ ≥ 3s, we find that ord_{p_{γ−1}}(ρ_{γ−1}) > 2s − 2 and this brings about a contradiction. □

Next we extend G_{a,b} and H_{a,b} over the divisors σ^{−1}(Δ_{j}^{0}) for [g/2] ≤ j ≤ g − 2.

Proposition 2.10. (1) For g = rs + s, d = rs + r and b ≥ 1, there exists a vector bundle G_{0,b} defined over 𝔊̃_{d}^{r}, extending the already constructed vector bundle G_{0,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}) and such that if t = (C ∪_{y} D, l_{C}, l_{D}) ∈ σ^{−1}(Δ_{j}^{0}), where g(C) = j ≥ [s(r + 1)/2], g(D) = g − j ≥ 2 and (l_{C}, l_{D}) is a limit 𝔤_{d}^{r} on C ∪_{y} D, then

G_{0,b}(t) = H^{0}(C ∪_{y} D, L^{⊗b}),

with

L := (L_{C} = l_{C}(−(rs − j)y), L_{D} = l_{D}(−(j + r)y)) ∈ Pic^{j+r}(C) × Pic^{rs−j}(D),

in the case s(r + 1)/2 ≤ j ≤ s(r − 1) = g − 2s, and

L := (L_{C} = l_{C}(−(g − j + r)y), L_{D} = l_{D}(−(j − s)y)) ∈ Pic^{j−s}(C) × Pic^{rs+s−j+r}(D),

in the case s(r − 1) < j ≤ s(r + 1) − 2.

(2) For each 0 ≤ a ≤ r, b ≥ 1 there exists a vector bundle H_{a,b} over 𝔊̃_{d}^{r} restricting to the already defined vector bundle H_{a,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}), such that H_{0,b} = Sym^{b}(G_{0,1}) for all b ≥ 1 and which also has the property that the exact sequences (5) remain exact over 𝔊̃_{d}^{r}.

(3) For each 0 ≤ a ≤ r, b ≥ 1, there exists a torsion free sheaf G_{a,b} over 𝔊̃_{d}^{r} that restricts to the vector bundle G_{a,b} over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}), which for a = 0 agrees with the vector bundle G_{0,b} defined above, and which has the property that the vector bundle morphisms ϕ_{a,b} defined over σ^{−1}(M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0}) extend to morphisms ϕ_{a,b} : H_{a,b} → G_{a,b} over 𝔊̃_{d}^{r}.

Proof. We start with an arbitrary point t = (C ∪_{y} D, l_{C}, l_{D}) ∈ σ^{∗}(C^{j}) where we assume first that [g/2] ≤ j ≤ rs − s. We set L_{C} := l_{C}(−(rs − j)y) ∈ Pic^{r+j}(C) and L_{D} := l_{D}(−(r+j)y) ∈ Pic^{rs−j}(D). If L = (L_{C}, L_{D}) ∈ Pic^{rs+r}(C ∪_{y} D), the essential observation is that because [C] ∈ M_{j} and [D, y] ∈ M_{g−j,1} are Brill-Noether general, we always have that rs + r − 1 ≤ a_{0}^{l_{C}}(y) + a_{r}^{l_{D}}(y) ≤ rs + r, hence h^{0}(L_{C}) ≥ r, h^{0}(L_{D}) ≤ 1 and h^{0}(C ∪_{y} D, L) = r + 1 (see Proposition 2.4). If p : M̃_{g,1} ×M̃_{g} 𝔊̃_{d}^{r} → 𝔊̃_{d}^{r} is the universal

curve over 𝔊̃_{d}^{r}, we denote by P a Poincaré bundle of relative degree d = rs + r enjoying the following properties:

(1) For each [g/2] ≤ j ≤ g − 2, P_{|p^{−1}(σ^{−1}(Δ_{j}^{0}))} parameterizes line bundles of bide-

gree (r + j, rs − j) on curves of type C ∪_{y} D where g(C) = j and g(D) = g − j.

(2) If τ_{j} : σ^{−1}(Δ_{j}^{0}) → M̃_{g,1} ×M̃_{g} 𝔊̃_{d}^{r} denotes the section which assigns the single

node corresponding to every curve from σ^{−1}(Δ_{j}^{0}), then τ_{j}^{∗}(P) = O_{σ^{−1}(Δ_{j}^{0})}.

Note that since the divisors σ^{−1}(Δ_{i}^{0}) and σ^{−1}(Δ_{j}^{0}) are disjoint for [g/2] ≤ i <

j ≤ g − 2, the construction can be carried out over a fixed divisor Δ_{j}^{0} at a time. Since h^{0}(p^{−1}(t), P_{|p^{−1}(t)}) = h^{0}(C ∪_{y} D, L) = r + 1 for each t ∈ 𝔊̃_{d}^{r}, by Grauert’s Theorem, G_{0,1} := p_{∗}(P) is a locally free sheaf which satisfies our first requirement. For b ≥ 2 we define G_{0,b} = p_{∗}(P^{⊗b}). Based on degree considerations we have that H^{1}(L_{C}^{⊗b}) = H^{1}(L_{C}^{⊗b} ⊗ O_{C}(−y)) = 0. Using Proposition 2.9 we see that H^{1}(L_{D}^{⊗b}) = 0, hence h^{0}(C ∪ D, L^{⊗b}) = h^{0}(L_{C}^{⊗b}) + h^{0}(L_{D}^{⊗b}) − 1 = bd + 1 − g. Grauert’s Theorem implies that G_{0,b} is locally free for all b.

In the remaining case when rs − s + 1 ≤ j ≤ g − 2, that is, 2 ≤ g(D) ≤ s − 1, we define L_{C} := l_{C}(−(rs + s + r − j)y) ∈ Pic^{j−s}(C) and L_{D} := l_{D}(−(j − s)y) ∈ Pic^{rs+s−j+r}(D). Then h^{0}(L_{C}) ≤ 1, h^{0}(L_{D}) ≥ r and h^{0}(C ∪_{y} D, L) = r + 1. Proposition 2.9 gives again that h^{1}(L_{C}^{⊗b}) = 0 for all b ≥ 2, hence h^{0}(C ∪ D, L^{⊗b}) = bd + 1 − g. This time , we denote by P the Poincaré bundle parametrizing line bundles of bidegree (j − s, rs + s + r − j) on curves of type C ∪_{y} D and then G_{0,b} := p_{∗}(P^{⊗b}) is locally free in this case too because of Grauert’s Theorem.

To define G_{a,b} for a ≥ 1, we introduce the sheaf M := Ker{p^{∗}(p_{∗}(P)) → P} and then we set G_{a,b} := p_{∗}(∧^{a}M ⊗ P^{⊗b}). The morphism ϕ_{0,b} is simply the natural map Sym^{b}p_{∗}(P) → p_{∗}(P^{⊗b}), and to define these maps for a ≥ 1 we use that the vector bundles H_{a,b} fit into exact sequences of type (5) and then proceed inductively. □

Remark 2.11. Just like in the case of the vector bundles A and B defined initially over 𝔊_{d}^{r} (cf. Section 2), the sheaves H_{a,b}, G_{a,b} depend on the choice of a Poincaré bundle, whereas Hom _{O\overline{𝔊}_{d}^{r}} (H_{a,b}, G_{a,b}) and ϕ_{a,b} ∈ H^{0}(\overline{𝔊}_{d}^{r}, Hom _{O\overline{𝔊}_{d}^{r}} (H_{a,b}, G_{a,b})) are independent

of such a choice. Moreover, since the projection p : M̃_{g,1} ×M̃_{g} 𝔊̃_{d}^{r} → 𝔊̃_{d}^{r} has a canonical section over each divisor σ^{−1}(Δ_{j}^{0}) where [g/2] ≤ j ≤ g − 1, it is possible to choose the Poincaré bundle P_{|p^{−1}(σ^{−1}(Δ_{j}^{0}))} in an unambiguous way (which is precisely what we did in the proof of Proposition 2.10) and then H_{a,b|σ^{−1}(Δ_{j}^{0})} and G_{a,b|σ^{−1}(Δ_{j}^{0})} are unambiguously defined as vector bundles over σ^{−1}(Δ_{j}^{0}). This is of course a minor point which plays no role in the calculation of σ_{∗}(c_{1}(G_{i,2} − H_{i,2})) ∈ A^{1}(M̃_{g}).

We now determine the class of the curves X and Y defined in Proposition 2.3:

Proposition 2.12. Let C be a Brill-Noether general curve of genus g − 1 and q ∈ C a general point. We denote by π_{2} : C × W_{d}^{r}(C) → W_{d}^{r}(C) the projection and set c_{i} := (π_{2})^{∗}(c_{i}(E^{∨})).

(1) The class of the curve X = {(y, L) ∈ C × W_{d}^{r}(C) : h^{0}(C, L(−2y)) ≥ r} is given by

[X] = c_{r} + c_{r−1}(2γ + (2d + 2g − 4)η) − 6c_{r−2} ηθ.

(2) The class of the curve Y = {(y, L) ∈ C × W_{d}^{r}(C) : h^{0}(C, L(−y − q)) ≥ r} is given by

[Y] = c_{r} + c_{r−1}(γ + (d − 1)η) − 2c_{r−2} ηθ.

Proof. We realize both X and Y as degeneracy loci over C × W_{d}^{r}(C) and compute their classes using the Thom-Porteous formula. For each (y, L) ∈ C × W_{d}^{r}(C) we have a natural map

H^{0}(C, L_{|2y})^{∨} → H^{0}(C, L)^{∨}

which globalizes to a vector bundle map ζ : J_{1}(L)^{∨} → (π_{2})^{∗}(E^{∨}). Clearly X = Z_{1}(ζ), hence

[X] = [\frac{c_{t}(π_{2}^{∗}(E^{∨}))}{c_{t}(J_{1}(L)^{∨})}]_{r} .

From the exact sequence defining the jet bundle of L

0 ⟶ π_{1}^{∗}(K_{C}) ⊗ L → J_{1}(L) → L → 0

we obtain that c_{t}(J_{1}(L^{∨}))^{−1} = 1 + 2γ + 2dη + (2g − 4)η − 6ηθ, which quickly leads to the desired expression for [X]. The calculation of [Y] is entirely similar and we skip it. □

We also need the following intersection theoretic result:

Lemma 2.13. For each j ≥ 2 we have the following formulas:

(1) c_{1}(G_{0,j |X}) = −j^{2}θ − (2g − 4)η − j(dη + γ).

(2) c_{1}(G_{0,j |Y}) = −j^{2}θ + η.

Proof. We observe that for all j ≥ 2, H^{1}(L^{⊗j}) = 0, hence (π_{2})_{∗}(L^{⊗j}) is a vector bundle over Pic^{d}(C). Riemann-Roch applied to the map π_{2} : C × Pic^{d}(C) → Pic^{d}(C) yields c_{1}((π_{2})_{∗}(L^{⊗j})) = −j^{2}θ. If we denote by u, v : C × C × Pic^{d}(C) → C × Pic^{d}(C) the two projections and by Δ ⊂ C × C × Pic^{d}(C) the diagonal, we have the following exact sequences

0 ⟶ u_{∗}(v^{∗}(L^{⊗j}) ⊗ I_{Δ}^{2})_{|X} ⟶ G_{0,j|X} ⟶ L_{|X}^{⊗j} ⟶ 0

and

0 ⟶ u_{∗}(v^{∗}(L^{⊗j}) ⊗ I_{Δ}^{2}) ⟶ (π_{2})_{∗}(L^{⊗j}) ⟶ J_{1}(L^{⊗j}) ⟶ 0

(and an entirely similar situation for G_{0,j|Y}) from which both claims follow easily. □

Now we are in a position to prove Theorem 1.1: Proof of Theorem 1.1. 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

(6)  σ_{∗}(G_{i,2} − H_{i,2}) = Aλ − B_{0} δ_{0} − B_{1} δ_{1} − ⋯ − B_{[g/2]} δ_{[g/2]} ∈ Pic(\overline{M}_{g}),

where λ, δ_{0}, …, δ_{[g/2]} 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_{g−1,1} 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 (5). We take [C ∪_{q} E] ∈ \overline{M}_{g} to be an arbitrary curve from R, where E is an elliptic curve. The pointed curve [C, q] being Brill-Noether general, limit 𝔤_{d}^{r}’s on C ∪_{q} E are in 1 : 1 correspondence with linear series L ∈ W_{d}^{r}(C) having a cusp at q (This is a statement independent of the j-invariant of E, in particular, it also holds for the 12 rational nodal curves in the pencil). Furthermore, the fibre of G_{0,b|σ^{∗}(Δ_{1}^{0})} over each point from σ^{∗}(R) consists of the global sections of the genus g − 1 aspect of the limit 𝔤_{d}^{r} and the claim now follows.

Now we determine explicitly the coefficients A, B_{0} and B_{1}. We fix a general curve [C, q] ∈ M_{g−1,1} and construct the test curves C^{1} ⊂ Δ_{1} and C^{0} ⊂ Δ_{0}. Using the notation from Proposition 2.3, we write 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})) = (2g − 2)B_{0} − B_{1} and C^{1} · σ_{∗}(c_{1}(G_{i,2} − H_{i,2})) = (2g − 4)B_{1},

while we already know that A − 12B_{0} + B_{1} = 0. Next we use the relations

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

+ ∑_{l=0}^{i}(−1)^{l}((l + 2)(rs + r) + 1 − rs − s)\binom{r}{i − l − 1}c_{1}(G_{0,1}), and

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

= ∑_{l=0}^{i}(−1)^{l}(\binom{r}{i − l − 1}\binom{r + l + 2}{l + 2} + \binom{r + 1}{i − l}\binom{r + l + 2}{r + 1})c_{1}(G_{0,1}) =

= \binom{2s + is + i}{i}(s + 1)(i + 2)c_{1}(G_{0,1}),

which when restricted to X and Y, enable us (also using Lemma 2.13), to obtain explicit expressions for c_{1}(G_{i,2} − H_{i,2})_{|X} and c_{1}(G_{i,2} − H_{i,2})_{|Y} in terms of the classes η, θ, γ and c_{1} = π_{2}^{∗}(c_{1}(E^{∨})). Intersecting these classes with [X] and [Y] and using Lemma 2.6, we finally 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. □

Next we prove that when i = 0, we can get a formula for the slope of \overline{Z}_{s(2s+1),0}: precisely we show that if we write σ_{∗}(c_{1}(G_{0,2} − H_{0,2})) = Aλ − B_{0}δ_{0} − ⋯ − B_{[g/2]}δ_{[g/2]}, then B_{j} ≥ B_{0} for all j ≥ 1. In particular, s(σ_{∗}(c_{1}(G_{0,2} − H_{0,2}))) = A/B_{0} which has already been computed in Theorem 1.1. We note that the proof uses in an essential way the divisor class calculation from Theorem 4.6.

Proof of Theorem 1.4. Using the convention B_{g−j} = B_{j} for g/2 ≤ j ≤ g − 1, we show that B_{j} ≥ B_{0} only when s(2s + 1)/2 ≤ j ≤ s(2s − 1). The case 2s^{2} ≤ j ≤ s(2s + 1) − 1 is dealt with in a similar fashion. To compute B_{j} we intersect the class σ_{∗}(c_{1}(G_{0,2} − H_{0,2})) with C^{j}. Then we use that [Y_{g−j,β}] · c_{1}(G_{0,2} − H_{0,2}) = [U_{j,γ}] · c_{1}(G_{0,2} − H_{0,2}) = 0, for all β ∈ P_{2}, γ ∈ P_{3}, to obtain that

(2j − 2)B_{j} = σ^{∗}(C^{j}) · c_{1}(G_{0,2} − H_{0,2}) = ∑_{(α_{0},...,α_{r})∈P_{1}} N_{g−j,α}([X_{j,α}] · c_{1}(G_{0,2} − H_{0,2})).

We fix a Schubert index (α_{0} ≤ … ≤ α_{r}) ∈ P_{1} and denote by π_{1} : X_{j,α} → C and π_{2} : X_{j,α} → Pic^{r+j}(C) the two projection maps. As before, L is the Poincaré bundle on C × Pic^{r+j}(C). There is an isomorphism of bundles G_{0,1|X_{j,α}} = π_{2}^{∗}((π_{2})_{∗}(L))_{|X_{j,α}} obtained by globalizing the projection isomorphism at the level of spaces of sections H^{0}(C ∪_{y} D, L) ≅ H^{0}(C, L_{C}) valid for each point (y, L_{C}) ∈ X_{j,α}. (We recall that L = (L_{C}, L_{D}) ∈ Pic^{2s+j}(C) × Pic^{2s^{2}−j}(D)). For b ≥ 2, we have a surjective morphism of vector bundles G_{0,b|X_{j,α}} ↠ π_{2}^{∗}((π_{2})_{∗}(L^{⊗b}))_{|X_{j,α}} whose kernel is a trivial bundle along

X_{j,α}. Thus one has that c_{1}(G_{0,b|X_{j,α}}) = −b^{2}θ_{|X_{j,α}} and c_{1}(H_{0,2|X_{j,α}}) = c_{1}(Sym^{2}G_{0,1|X_{j,α}}) = −(2s + 2)θ_{|X_{j,α}}, therefore

(7)  (2j − 2)B_{j} = (2s − 2) ∑_{(α_{0},...,α_{r})∈P_{1}} N_{g−j,α} ([X_{j,α}] · θ).

The class of the curve X_{j,α} can be computed using the generalized Giambelli formula (cf. [FuPr], pg. 15-17) as follows: If J_{α_{r}+r−1}(L) ↠ ⋯ ↠ J_{α_{i}+i−1}(L) ↠ ⋯ ↠ J_{α_{0}−1}(L) is the flag of jet bundles corresponding to the ramification sequence (α_{0}, …, α_{r}), then

X_{j,α} = {(y, L) ∈ C×Pic^{r+j}(C) : rk{π_{2}^{∗}((π_{2})_{∗}(L))(y, L) → J_{α_{i}+i−1}(L)(y, L)} ≤ i for all i}

and then [X_{j,α}] is given by the determinant of the (α_{r} × α_{r})-matrix having entries

a_{ik} = c_{r+1−l+k−i}(\frac{π_{2}^{∗}((π_{2})_{∗}(L))}{J_{α_{l}+l−1}(L)}), for all α_{l−1} ≤ i ≤ α_{l}, 0 ≤ l ≤ r and 1 ≤ j ≤ α_{r}.

Since

c_{i}(\frac{π_{2}^{∗}((π_{2})_{∗}(L))}{J_{a−1}(L)}) = \frac{θ^{i}}{i!} + \frac{θ^{i−1}}{(i − 1)!}(aγ + a(j + r)η + a(a − 1)(j − 1)η) − \frac{θ^{i−1}}{(i − 2)!}a(a + 1)η,

clearly [X_{j,α}] is a linear combination of θ^{j}, θ^{j−1}η and θ^{j−1}γ in H^{2j}(C × Pic^{r+j}(C)). The intersection number [X_{j,α}] · θ can be interpreted as the number of line bundles L_{C} ∈ Pic^{r+j}(C) satisfying the condition α_{i}^{L_{C}}(y) ≥ α_{i} for i = 0, …, r at an unspecified point y ∈ C, and which, moreover, are also ramified at a fixed point q ∈ C, that is, a_{r}^{L_{C}}(q) ≥ r + 1.

Using this interpretation, the quantity ∑_{α∈P_{1}} N_{g−j,α}([X_{g−j,α}]·θ) can be expressed as the intersection number C̃_{j} · \overline{𝔏𝔦𝔫}_{d}^{r}(1) over the moduli space \overline{M}_{g,1}. Here 𝔏𝔦𝔫_{d}^{r}(1) is the divisor on M_{g,1} consisting of pointed curves [C, q] such that there exists L ∈ W_{d}^{r}(C) with h^{0}(C, L ⊗ O_{C}(−(r + 1)q) ≥ 1, while C̃_{j} = \{C ∪_{y} D, q\}_{y∈C} ⊂ Δ_{j:1} ⊂ \overline{M}_{g,1} is the test curve obtained by varying the point of attachment y on the genus j component, while the marked point q ∈ C remains fixed. The class of \overline{𝔏𝔦𝔫}_{d}^{r}(1) is computed in the course of the proof of Theorem 4.6 and one has

\overline{𝔏𝔦𝔫}_{d}^{r}(1) ≡ μ((g + 3)λ − \frac{g + 1}{6}δ_{irr} − ∑_{j=1}^{g−1} δ_{j:1}) + ν(−λ + ψ − ∑_{j=1}^{g−1} \binom{g − j + 1}{2}δ_{j:1}),

where

ν = \frac{r(r + 2)}{(rs + s − 1)(rs + s + 1)} and μ = \frac{r(r + 1)(r + 2)(s − 1)(s + 1)(rs + s + 4)}{2(s + r + 1)(rs + s − 2)(rs + s − 1)(rs + s + 1)}.

Since C̃_{j} · ψ = 1, C̃_{j} · δ_{g−j:1} = 1 (the only point of intersection corresponds to y = q ∈ C), C̃_{j} · δ_{j:1} = −(2j − 1), while C̃_{j} · λ = C̃_{j} · δ_{i:1} = 0 for i ≠ j, g − j, we can compute that

\frac{B_{j}}{c_{r}} = \frac{s − 1}{(j − 1)c_{r}} ∑_{α∈P_{1}} N_{g−j,α}([X_{j,α}] · θ) = \frac{s − 1}{(j − 1)c_{r}} C̃_{j} · \overline{𝔏𝔦𝔫}_{d}^{r}(1) =

= \frac{4(s − 1)j(2js^{3} + js^{2} − 2js − 2j + 4s^{3} + 4s^{2} − 3s)(2s^{2} + s − j)}{(2s^{2} + s − 2)(3s + 1)(2s − 1)(j − 1)} ≥

≥ \frac{B_{0}}{c_{r}} = \frac{s(8s^{6} − 8s^{5} − 2s^{4} + s^{2} + 11s + 2)}{3(2s^{2} + s − 2)(3s + 1)(2s − 1)} .

This finishes the proof and shows that s(σ_{∗}(G_{0,2} − H_{0,2})) = A/B_{0}. □

As we have already pointed out, Theorem 1.1 produces only virtual divisors on \overline{M}_{g} of slope less than 6+12/(g+1). To get actual divisors one has to show that the vector bundle map ϕ : H_{i,2} → G_{i,2} is generically non-degenerate. We carry this out in the case i = 0 and we produce for the first time an infinite sequence of genuine counterexamples to the Slope Conjecture.

Proof of Theorem 1.5. From Brill-Noether theory one knows that there exists a unique component of 𝔊̃_{d}^{r} which maps onto M̃_{g}. Moreover, if (C, L) ∈ 𝔊_{d}^{r} is such that L ∈ W_{d}^{r}(C) − W_{d}^{r+1}(C) corresponds to an embedding C ⊂ P^{r}, then a sufficient condition for the smoothness of 𝔊_{d}^{r} at [C, L] is that H^{1}(N_{C/P^{r}}) = 0, and then, the differential (dσ)_{[C,L]} is surjective if and only if the Petri map μ_{0}(C) : H^{0}(L) ⊗ H^{0}(K_{C} ⊗ L^{∨}) → H^{0}(K_{C}) is injective (see e.g. [AC2]). In our situation, it is then enough to produce a Brill-Noether-Petri general smooth curve C ⊂ P^{2s} having degree 2s(s+1) and genus s(2s+1) such that C does not sit on any quadrics, that is H^{0}(I_{C/P^{2s}}(2)) = H^{1}(I_{C/P^{2s}}(2)) = 0. We carry this out inductively: For each 0 ≤ a ≤ s, we construct a smooth non-degenerate curve C_{a} ⊂ P^{s+a} with deg(C_{a}) = \binom{s+a+1}{2} + a and g(C_{a}) = \binom{s+a+1}{2} + a − s, h^{1}(C_{a}, O_{C_{a}}(1)) = a (or equivalently, h^{0}(C_{a}, O_{C_{a}}(1)) = s+a+1), and such that (1) C_{a} satisfies the Petri Theorem (in particular one has that H^{1}(C_{a}, N_{C_{a}/P^{s+a}}) = 0), and (2) the multiplication map μ_{2} : Sym^{2}H^{0}(C_{a}, O_{C_{a}}(1)) → H^{0}(C_{a}, O_{C_{a}}(2)) is surjective (or equivalently, an isomorphism).

To construct C_{0} ⊂ P^{s} we consider the White surface S = Bl_{{p_{1},...,p_{δ}}}(P^{2}) ⊂ P^{s} obtained by blowing-up P^{2} at general points p_{1}, …, p_{δ} ∈ P^{2} where δ = \binom{s+1}{2}, and embedding it via the linear system |sh − ∑_{i=1}^{δ} E_{p_{i}}|. Here h is the class of a line on P^{2}. It is known that S ⊂ P^{s} is a projectively Cohen-Macaulay surface and its ideal is generated by the (3 × 3)-minors of a certain (3 × s)-matrix of linear forms (see e.g. [GG] even though these surfaces have been studied in the classical literature by T.G. Room in [R]). The Betti diagram of S ⊂ P^{s} is the same as that of the ideal of (3 × 3)-minors of a (3 × s)-matrix of indeterminates. In particular, we have that H^{i}(I_{S/P^{s}}(2)) = 0 for i = 0, 1. On S we consider a generic smooth curve C ≡ (s + 1)h − ∑_{i=1}^{δ} E_{p_{i}}. We find that the embedded curve C ⊂ S ⊂ P^{s} has deg(C) = \binom{s+1}{2} and g(C) = \binom{s}{2}. From the exact sequence

0 ⟶ I_{S/P^{s}}(1) ⟶ I_{C/P^{s}}(1) ⟶ I_{C/S}(1) ⟶ 0,

using also that H^{1}(I_{S/P^{s}}(1)) = 0 and that H^{1}(I_{C/S}(1)) = 0 (e.g. by Riemann-Roch), we find that H^{1}(I_{C/P^{s}}(1)) = 0 and H^{1}(O_{C}(1)) = 0, hence h^{0}(O_{C}(1)) = s + 1. Furthermore, since H^{0}(I_{S/P^{s}}(2)) = H^{1}(I_{S/P^{s}}(2)) = 0, we obtain that H^{1}(I_{C/P^{s}}(2)) = 0. Finally, since H^{1}(O_{C}(1)) = 0, it follows trivially that H^{1}(N_{C/P^{s}}) = 0 and μ_{0}(C) is injective, being a map with source the trivial vector space. Even though [C] ∈ M_{g(C)} itself is not a Petri general curve, the map H_{C} → M_{g(C)} from the Hilbert scheme H_{C} of curves C′ ⊂ P^{s} with deg(C′) = deg(C) and g(C′) = g(C), is smooth and dominant around the point [C] ∈ H_{C}, hence a generic deformation [C_{0} ↪ P^{s}] ∈ H_{C} of [C ↪ P^{s}] will be Petri general and still satisfy the condition H^{1}(I_{C_{0}/P^{s}}(2)) = 0.

Assume now that we have constructed a Petri general curve C_{a} ⊂ P^{s+a} with all the desired properties. We pick general points p_{1}, …, p_{s+a+2} ∈ C_{a} with the property that if Δ := p_{1} + ⋯ + p_{s+a+2} ∈ Sym^{s+a+2}C_{a}, then the variety

T := {(M, V) ∈ W_{d(C_{a})+s+a+2}^{s+a+1}(C_{a}) : dim(V ∩ H^{0}(C_{a}, M ⊗ O_{C_{a}}(−Δ))) ≥ s + a + 1}

of linear series having an (s + a + 2)-fold point along Δ, has the expected dimension ρ(g(C_{a}), s + a + 1, d(C_{a}) + s + a + 2) − (s + a + 1)^{2}. We identify the projective space P^{s+a} containing C_{a} with a hyperplane H ⊂ P^{s+a+1} and choose a linearly normal elliptic curve E ⊂ P^{s+a+1} such that E ∩ H = {p_{1}, …, p_{s+a+2}}. The fact that such an E exists is an easy consequence of the vanishing H^{1}(N_{E/P^{s+a+1}}(−1)) = 0 for each elliptic curve E embedded by a complete linear series; the vanishing itself is a consequence of the fact that N_{E/P^{s+a+1}} is a poly-stable vector bundle (cf. [GL], Theorem 4.1), having the property that μ(N_{E/P^{s+a+1}}(−1)) > 1). We now set X := C_{a} ∪_{{p_{1},...,p_{s+a+2}}} E ↪ P^{s+a+1} and then deg(X) = p_{a}(X) + s. From the exact sequence

0 ⟶ O_{E}(−p_{1} − ⋯ − p_{s+a+2}) ⟶ O_{X} ⟶ O_{C_{a}} ⟶ 0,

we can write that h^{0}(O_{X}(1)) ≤ h^{0}(O_{C_{a}}(1)) + h^{0}(O_{E}) = s + a + 2, hence h^{0}(O_{X}(1)) = s + a + 2 and h^{1}(O_{X}(1)) = a + 1. One can also write the exact sequence

0 ⟶ I_{E/P^{s+a+1}}(1) ⟶ I_{X/P^{s+a+1}}(2) ⟶ I_{C_{a}/H}(2) ⟶ 0,

from which we obtain that H^{1}(I_{X/P^{s+a+1}}(2)) = 0, hence by a dimension count we also get that H^{0}(I_{X/P^{s+a+1}}(2)) = 0, that is, X and a general deformation of X inside P^{s+a+1} lie on no quadrics.

We now show that X ↪ P^{s+a+1} can be deformed to an embedding of a smooth curve C_{a+1} in P^{s+a+1} such that H^{1}(N_{C_{a+1}/P^{s+a+1}}) = 0. We choose an (s + a + 2)- dimensional subspace H^{0}(O_{C_{a}}(1)) ⊂ V ⊂ H^{0}(O_{C_{a}}(1) ⊗ O_{C_{a}}(Δ)) which gives a map f : C_{a} → P^{s+a+1} such that f(p_{1}) = ⋯ = f(p_{s+a+2}) = p. If we denote by P̃^{s+a+1} the blow-up of P^{s+a+1} at p, by choosing V suitably we may assume that f lifts to an embedding f̃: C_{a} ↪ P̃^{s+a+1} which projected from p gives rise to the original embedding C_{a} ↪ H. We consider another copy of P^{s+a+1} which we denote by P_{1}^{s+a+1} and we denote by Z the scheme obtained by gluing P_{1}^{s+a+1} and P̃^{s+a+1} along H, where we identify the exceptional divisor of P̃^{s+a+1} with H ⊂ P^{s+a+1} via the projection from p. There is a natural map h : Z → P^{s+a+1} which on P_{1}^{s+a+1} is the identity while on P̃^{s+a+1} is the projection from p. Via this map, the inclusion X ↪ P^{s+a+1} lifts to an embedding X ↪ Z. Note that Z is a degeneration of P^{s+a+1} something which can be seen by blowing-up the codimension 2 subscheme H × {0} of P^{s+a+1} × P^{1}. If we denote by X the total space of the blow-up and by ϵ : X → P^{1} the projection onto the second component, then for t ≠ 0 we have that ϵ^{−1}(t) = P^{s+a+1}, whereas ϵ^{−1}(0) = P∪ E, where Pis the strict transform of P^{s+a+1} × {0} which is isomorphic to P^{s+a+1}, while E = P(O_{H} ⊕ O_{H}(1)) is the exceptional divisor, which is isomorphic to P^{s+a+1} blown-up at a point. In the special fibre, Pand E are joined along a divisor which is H inside P.

Next we write down the standard exact sequences of normal bundles

0 ⟶ N_{E/P^{s+a+1}} ⊗ O_{E}(−Δ) ⟶ N_{X/Z} ⟶ N_{C_{a}/P̃^{s+a+1}} ⟶ 0

(the right hand side map is restriction to the component C_{a} of X), and

0 ⟶ O_{C_{a}}(1) ⊗ O_{C_{a}}(2Δ) ⟶ N_{C_{a}/P̃^{s+a+1}} ⟶ N_{C_{a}/H} ⟶ 0,

from which it easily follows that H^{1}(N_{X/Z}) = 0 (Use the hypothesis H^{1}(N_{C_{a}/P^{s+a}}) = 0 and that H^{1}(N_{E/P^{s+a+1}} ⊗ O_{E}(−Δ)) = 0 because N_{E/P^{s+a+1}} is semi-stable). Thus the space of deformations of X in Z is unobstructed of dimension h^{0}(N_{X/Z}). On the other hand, by general theory the space T_{(X,Z)}^{1} of infinitesimal deformations of the pair (X, Z) has dimension at least χ(N_{X/Z}) + 1 = h^{0}(N_{X/Z}) + 1. This shows that there exists a deformation of (X, Z) in which Z deforms non-trivially. But dim(T_{Z}^{1}) = 1, that is, the only possible deformation of Z is the smoothing to P^{s+a+1} previously described, and in this deformation the map X ↪ Z will deform to an embedding C_{a+1} ↪ P^{s+a+1} of a smooth curve, which proves our claim. We are left with showing that the dimension estimate

(8)  dim(W_{d(C_{a+1})}^{s+a+1}(C_{a+1})) = ρ(g(C_{a+1}), s + a + 1, d(C_{a+1}))

holds. Assuming that (8) has been proved, since the condition H^{1}(N_{C_{a+1}/P^{s+a+1}}) = 0 guarantees the local smoothness of the scheme 𝔊_{d(C_{a+1})}^{s+a+1}, it follows that the morphism 𝔊_{d(C_{a+1})}^{s+a+1} → M_{g(C_{a+1})} is dominant in a neighbourhood of the point [C_{a+1} ↪ P^{s+a+1}]. Therefore the curve C_{a+1} ⊂ P^{s+a+1} can be chosen to be Petri general as well, which enables us to continue the induction.

We return to proving (8) and denote by U the versal deformation space of [X] ∈ \overline{M}_{g(C_{a+1})} and by ϕ : C → U the universal family such that ϕ^{−1}(0) = X, where 0 ∈ U. Then in a way similar to [EH1], Theorem 3.3, one can construct a quasi-projective variety σ : 𝔊̃_{d(C_{a+1})}^{s+a+1} → U of limit linear series such that for points u ∈ U with C_{u} = ϕ^{−1}(u) smooth, we have that σ^{−1}(u) = G_{d(C_{a+1})}^{s+a+1}(C_{u}), whereas σ^{−1}(0) consists of the following data: an underlying line bundle L on X together with linear series {L_{a}, V_{a} ∈ G(s+a+2, H^{0}(X, L_{a}))} and {L_{E}, V_{E} ∈ G(s+a+2, H^{0}(X, L_{E}))} such that the following conditions are satisfied (see also [Est], Theorem 1):

(1) The line bundles L_{a} and L_{E} on X are suitable twists of L by multiples of the divisor Δ: precisely there exists an integer l such that L_{a|C_{a}} = L_{E|C_{a}} ⊗ O_{C_{a}}(l · Δ) and L_{a|E} = L_{E|E} ⊗ O_{E}(−l · Δ). Moreover deg(L_{a|C_{a}}) + deg(L_{E|E}) = deg(C_{a+1}) + l(s + a + 2).

(2) The restriction maps V_{a} → H^{0}(C_{a}, L_{a|C_{a}}) and V_{E} → H^{0}(E, L_{E|E}) are both injective.

(3) The restriction maps V_{a} → H^{0}(E, L_{a|E}) and V_{E} → H^{0}(C_{a}, L_{E|C_{a}}) are both nonzero.

(4) If l is the integer defined above and (a_{0} ≤ … ≤ a_{s+a+1}) denotes the vanishing sequence of (L_{a|C_{a}}, V_{a+1}) with respect to the divisor Δ ∈ Sym^{s+a+2}C_{a} while (b_{0} ≤ ⋯ ≤ b_{s+a+1}) denotes the vanishing sequence of (L_{E|E}, V_{E}) with respect to Δ ∈ Sym^{s+a+2}E, then we have the inequalities a_{i} +b_{s+a+1−i} ≥ l for all indices 0 ≤ i ≤ s + a + 1 (see also [Est], Proposition 6).

By construction we have the dimension estimate dim(G) ≥ dim(U) + ρ(g(C_{a+1}), s + a + 1, d(C_{a+1})), (see also [EH1]), thus in order to prove (8) it suffices to show that

dim(σ^{−1}(0)) = ρ(g(C_{a+1}), s + a + 1, d(C_{a+1})) = ρ(g(C_{a}), s + a, d(C_{a})) − a

(here by dimension we mean the smallest dimension of an irreducible component). It is now easy to describe the fibre σ^{−1}(0) in a neighbourhood of the point corresponding to a smoothing of the embedding X ↪ P^{s+a+1}: If {L_{a}, V_{a}}, {L_{E}, V_{E}} is a limit linear series on X, then the aspect corresponding to E is just a very ample line bundle on X giving the embedding into P^{s+a+1}, that is, L_{E|C_{a}} ∈ W_{d(C_{a})}^{s+a}(C_{a}) and L_{E|E} = O_{E}(Δ), whereas the aspect corresponding to C_{a} is described by L_{a|E} = O_{E} and L_{a|C_{a}} = O_{C_{a}}(Δ) ⊗ L_{E|C_{a}} (and in particular l = 1). The only possibility for the vanishing sequences of the E and C_{a} aspects is that (a_{0}, …, a_{s+a+1}) = (0, 1, …, 1) and (b_{0}, …, b_{s+a+1}) = (0, …, 0, 1). This shows that locally, σ^{−1}(0) is isomorphic to the variety of line bundles L ∈ Pic^{d(C_{a+1})}(X) such that L_{|E} = O_{E}, h^{0}(X, L) ≥ s + a + 2 and h^{0}(C_{a}, L_{|C_{a}}(−Δ)) ≥ s + a + 1 (Loosely speaking this is the subscheme consisting of those L_{a} ∈ W_{d(C_{a})}^{s+a}(C_{a}) for which there exists a section τ ∈ P(H^{0}(L_{a} ⊗ O(Δ))/H^{0}(L_{a})) which glues to the unique section of the trivial bundle O_{E} at the points of attachment p_{1}, …, p_{s+a+2}). Thus locally σ^{−1}(0) is a (ℂ^{∗})^{s+a+1}-bundle over the subvariety T of W_{d(C_{a+1})}^{s+a+1} having an (s + a + 2)-fold point along the divisor Δ, and by our inductive hypothesis we know that dim(T) = dim(W_{d(C_{a+1})}^{s+a+1}(C_{a})) − (s + a + 1)^{2}. It follows that

dim(σ^{−1}(0)) = dim(T) + s + a + 1 = ρ(g(C_{a+1}), s + a + 1, d(C_{a+1})),

and this finishes the proof. □

Remark 2.14. It is natural to wonder whether (8) could not be proved more easily by showing directly that the Petri map μ_{0}(X) : H^{0}(O_{X}(1)) ⊗ H^{0}(ω_{X}(−1)) → H^{0}(ω_{X}) is injective. Indeed Theorem 1.3 from [CR] seems to imply this to be the case based on the inductive hypothesis that μ_{0}(C_{a}) is injective whereas μ_{0}(E) is injective for trivial reasons. That claim is incorrect: from the exact sequence 0 ⟶ ω_{E} ⟶ ω_{X} ⟶ ω_{C_{a}}(Δ) ⟶ 0, we find the isomorphism H^{0}(ω_{X}(−1)) = H^{0}(ω_{C_{a}}(−1) ⊗ O_{C_{a}}(Δ)) and then a simple analysis shows that H^{0}(O_{E})⊗H^{0}(ω_{C_{a}}(−1)) ⊂ H^{0}(O_{X}(1))⊗H^{0}(ω_{X}(−1)) is an a-dimensional subspace lying entirely inside Ker(μ_{0}(X)).

3. The class of the Gieseker-Petri divisors

In this section we prove Theorem 1.6. We use the same strategy as in the previous section and we intersect \overline{GP}_{g,d}^{r} with the test curves C^{0}, C^{1} and C^{j} for [g/2] ≤ j ≤ g − 2. Recall that we have constructed a rank r + 1 vector bundle G_{0,1} over the variety 𝔊̃_{d}^{r} (cf. Proposition 2.10). As usual, we denote by 𝔼 the Hodge bundle over \overline{M}_{g}.

Proposition 3.1. There exists a rank s vector bundle N over 𝔊̃_{d}^{r} together with a morphism G_{0,1} ⊗ N → σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1})) of vector bundles of the same rank over 𝔊̃_{d}^{r} such that the fibres of N admit the following description:

• If (C, L) ∈ 𝔊_{d}^{r}, then N (C, L) = H^{0}(C, K_{C} ⊗ L^{∨}).

• If t = (C∪_{y}E, L_{C}, l_{E}) ∈ σ^{−1}(Δ_{1}^{0}), where L ∈ W_{d}^{r}(C) is such that h^{0}(L⊗O_{C}(−2y)) =

r, then N (t) = H^{0}(C, K_{C} ⊗ L_{C}^{∨} ⊗ O_{C}(2y)).

• If t = (C/y ∼ q, L) ∈ σ^{−1}(Δ_{0}^{0}), where y, q ∈ C and L ∈ W_{d}^{r}(C) is a linear series such that h^{0}(L ⊗ O_{C}(−y − q)) = h^{0}(L) − 1, then N (t) = H^{0}(C, K_{C} ⊗ L^{∨} ⊗ O_{C}(y + q)).

• If t = (C ∪_{y} D, l_{C}, l_{D}) ∈ σ^{−1}(Δ_{j}^{0}) where [g/2] ≤ j ≤ g − 2, g(C) = j, g(D) = g − j, then N (t) = H^{0}(C ∪_{y} D, ω_{C∪D} ⊗L^{∨}), where L = (l_{C}(−(rs−j)y), l_{D}(−(j +r)y)) ∈

Pic^{j+r}(C) × Pic^{rs−j}(D).

Note that over 𝔊_{d}^{r} the morphism G_{0,1} ⊗ N → σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1})) is simply the Petri

multiplication map. We start the proof of Theorem 1.6 by expanding [\overline{GP}_{g,d}^{r}] in Pic(\overline{M}_{g}):

\overline{GP}_{g,d}^{r} ≡ aλ − b_{0}δ_{0} − ⋯ − b_{[g/2]}δ_{[g/2]}.

We show that the coefficients a, b_{0} and b_{1} as well as s(\overline{GP}_{g,d}^{r}) can be read off from the vector bundle map G_{0,1} ⊗ N → σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1})). Even though this bundle map is degenerate along the boundary components contained in σ^{∗}(Δ_{j}^{0}) with j ≥ 2, we can show that it is generically non-degenerate along σ^{∗}(Δ_{0}^{0}) and σ^{∗}(Δ_{1}^{0}) which ultimately suffices to compute s(\overline{GP}_{g,d}^{r}).

Proposition 3.2. One has the relation a − 12b_{0} + b_{1} = 0. Moreover, one has the identity

\overline{GP}_{g,d}^{r} ≡ σ_{∗}(c_{1}(σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1})) − c_{1}(G_{0,1} ⊗ N)) + ∑_{j=2}^{[g/2]} d_{j}δ_{j},

where d_{j} ≥ 0.

Proof. It is enough to show that if [C, y] ∈ M_{g−1,1} is a general pointed curve, then for every L ∈ W_{d}^{r}(C) satisfying h^{0}(L ⊗ O_{C}(−2y)) = r, the multiplication map

μ_{0}(L, y) : H^{0}(L) ⊗ H^{0}(K_{C} ⊗ L^{∨} ⊗ O_{C}(2y)) → H^{0}(K_{C} ⊗ O_{C}(2y))

is an isomorphism. This shows that the morphism G_{0,1} ⊗ N → σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1})) is nondegenerate along each component of the divisor σ^{−1}(Δ_{1}^{0}) and the conclusion follows. To show that μ_{0}(L, y) is an isomorphism, we use a variation of the degeneration considered by Eisenbud and Harris to prove the Gieseker-Petri Theorem (cf. [EH4]). Precisely, we consider a 1-dimensional family π : C → B of generically smooth pointed curves of genus g − 1 with a section τ : B → C, degenerating to a curve of compact type C_{0} consisting of a string of rational components and g elliptic components E_{1}, …, E_{g} such that the stable model of C_{0} is E_{1} ∪_{p_{1}} E_{2} ∪_{p_{2}} E_{3} ∪ ⋯ ∪_{p_{g−1}} E_{g−1}. We assume moreover that the marked point specializes to a point p_{0} ∈ E_{1} and we choose our degeneration general enough such that p_{i} − p_{i−1} ∈ Pic^{0}(E_{i}) is not a torsion point for all 1 ≤ i ≤ g − 1. By contradiction, we assume that for a general t ∈ B there exists L_{t} ∈ W_{d}^{r}(π^{−1}(t)) with h^{0}(π^{−1}(t), L_{t} ⊗O(−2τ(t))) = r, such that μ_{0}(L_{t}, τ(t)) has non-trivial kernel. For 1 ≤ i ≤ g we denote by L^{i} ∈ Pic^{d}(C_{0}) the limit line bundle of the L_{t}’s having the property that deg(L_{|E_{j}}^{i}) = 0 for i ≠ j, hence deg(L_{|E_{i}}^{i}) = d. Similarly, we define M^{i} ∈ Pic^{2g−2−d}(C_{0}) to be the limit when t → 0 of K_{π^{−1}(t)} ⊗ L_{t}^{∨} ⊗ O_{π^{−1}(t)}(2τ(t)) uniquely characterized by the property deg(M_{|E_{j}}^{i}) = 0 for i ≠ j and deg(M_{|E_{i}}^{i}) = 2g − 2 − d. We denote by {(L_{|E_{i}}^{i}, V_{i} ∈ G(r + 1, H^{0}(E_{i}, L_{|E_{i}})))} and by {(M_{|E_{i}}^{i}, W_{i} ∈ G(r + 1, H^{0}(E_{i}, M_{|E_{i}})))} the limit linear series on C_{0} corresponding to L_{t} and K_{π^{−1}(t)} ⊗ L_{t}^{∨} respectively as t → 0. Reasoning along the lines of [EH4] or [F2], Proposition 5.2, for each 1 ≤ i ≤ g we find non-trivial elements

ρ_{i} ∈ Ker{V_{i} ⊗ W_{i} → H^{0}(E_{i}, L^{i} ⊗ M_{|E_{i}}^{i})}

satisfying ord_{p_{i}}(ρ_{i+1}) ≥ ord_{p_{i−1}}(ρ_{i}) + 2 for 1 ≤ i ≤ g − 1. Since both V_{1} and W_{1} have a cusp at p_{0} ∈ E_{1}, it follows that ord_{p_{1}}(ρ_{1}) ≥ 2, hence ord_{p_{g−1}}(ρ_{g}) ≥ 2g − 2 = deg(L_{|E_{g}}^{g}) + deg(M_{|E_{g}}^{g}), which is a contradiction because ρ_{g−1} ∈ H^{0}(E_{g}, L_{|E_{g}}^{g}) ⊗ H^{0}(E_{g}, M_{|E_{g}}^{g}) being an element in the kernel of the multiplication map must be a tensor of rank at least

4. □

Proposition 3.3. If c_{r} is the constant defined in Lemma 2.6, then the δ_{1} coefficient in the expression of [\overline{GP}_{g,d}^{r}] is given by:

b_{1} = c_{r} \frac{r(s − 1)}{(s + r + 1)(rs + s − 2)}(3rs^{2} + 2s^{2} + r^{2}s^{2} + 7s + 6rs + r^{2}s + 2r + 2).

Proof. We fix a general curve C of genus g − 1 and consider the associated test curve C^{1} ⊂ Δ_{1}. We view the curve X ⊂ C × W_{d}^{r}(C) defined in Proposition 2.12, as sitting inside 𝔊̃_{d}^{r}. Then the projection π_{1} : X → C is the restriction of σ : 𝔊̃_{d}^{r} → M̃_{g} once we identify C with C_{1} (Note that the degree of π_{1} is precisely c_{r}). One can write the relation (2g − 4)B_{1} = C^{1} · \overline{GP}_{g,d}^{r} = c_{1}(σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1}))_{|X}) − c_{1}(G_{0,1 |X} ⊗ N_{|X}) and we are going to compute each term in this expression.

The restriction 𝔼⊗OM̃_{g}(δ_{1})_{|C^{1}} is identified with the vector bundle (p_{2})_{∗}(p_{1}^{∗}(K_{C})⊗ O(2Δ)), where p_{1}, p_{2} : C×C → C are the two projections and Δ ⊂ C×C is the diagonal. Using Grothendieck-Riemann-Roch for the map p_{2}, we find that

c_{1}(𝔼 ⊗ OM̃_{g}(δ_{1})_{|C^{1}}) = c_{1}((p_{2})_{!}(p_{1}^{∗}(K_{C}) ⊗ O(2Δ))) = −2g + 4,

hence c_{1}(σ^{∗}(𝔼) ⊗ OM̃_{g}(δ_{1}))_{|X} = −(2g − 4)c_{r} (remember that we have set c_{i} = c_{1}(E^{∨})).

The fibre N_{|X}(y, L) is identified with H^{0}(K_{C} ⊗ L^{∨}(2y)) = H^{1}(L ⊗ O(−2y))^{∨}. Keeping in mind that we have introduced the vector bundle map ζ in Proposition 2.12, we have an exact sequence over X

0 ⟶ Ker(ζ)^{∨} ⟶ N_{|X}^{∨} ⟶ π_{2}^{∗}(R^{1}π_{2∗}(L_{|C×W_{d}^{r}(C)})) ⟶ 0,

globalizing the cohomology exact sequence for each (y, L) ∈ C × W_{d}^{r}(C)

⋯ ⟶ H^{0}(L) ⟶^{ζ^{∨}} H^{0}(L_{|2y}) ⟶ H^{1}(L(−2y)) ⟶ H^{1}(L) ⟶ 0.

Hence c_{1}(N_{|X}^{∨}) = θ − c_{1}(E^{∨}) + c_{1}(Ker(ζ)^{∨}). Using Proposition 2.12 we can write that

C^{1} · \overline{GP}_{g,d}^{r} = −(2g − 4)c_{r}η − (r + 1 − s)c_{1} · [X] + (r + 1)c_{1}(Ker(ζ)^{∨}) =
−(2g − 4)c_{r}η − (r + 1 − s)((2d + 2g − 4)c_{1}c_{r−1}η − 6c_{1}c_{r−2}θη)+
+(r + 1)((2d + 2g − 4)c_{r−1}θη − 6c_{r−2}θ^{2}η) + (r + 1)c_{1}(Ker(ζ)^{∨}).

To compute the Chern number c_{1}(Ker(ζ)^{∨}) we use once more [HT] and we find the following relation in H^{top}(C × W_{d}^{r}(C)):

c_{1}(Ker(ζ)^{∨}) = c_{r+1}(\frac{π_{2}^{∗}(E^{∨})}{J_{1}(L^{∨})}) = (2d + 2g − 4)c_{r}η − 6ηθc_{r−1}.

Combining the last two relations and then applying Lemma 2.6 we obtain the formula for b_{1}. □

Proposition 3.4. The δ_{0} coefficient in the expression of [\overline{GP}_{g,d}^{r}] is given by:

b_{0} = c_{r} \frac{r(r + 1)(r + 2)(s − 1)s(s + 1)(rs + s + 4)}{6(r + s + 1)(rs + s − 2)(rs + s − 1)}.

Proof. We pick a general curve C of genus g − 1 and consider the test curve C^{0} ⊂ Δ_{0}. Similarly to the proof of Proposition 3.3 we view the projection π_{1} : Y → C as the restriction of σ : 𝔊̃_{d}^{r} → M̃_{g} over C^{0}. Then one has the relation

(2g − 2)b_{0} − b_{1} = C^{0} · \overline{GP}_{g,d}^{r} = c_{1}(σ^{∗}(𝔼 ⊗ OM̃_{g}(δ_{1}))_{|Y}) − c_{1}(G_{0,1|Y} ⊗ N_{|Y}).

The Hodge bundle 𝔼 ⊗ OM̃_{g}(δ_{1})_{|C^{0}} is identified with (p_{2})_{∗}(p_{1}^{∗}(K_{C}) ⊗ O(Δ + Γ_{q})), where Γ_{q} = {q}×C, and it is easy to compute that c_{1}(σ^{∗}(𝔼⊗OM̃_{g}(δ_{1}))_{|Y}) = c_{r}. If we denote by υ the vector bundle morphism over Y which globalizes the maps H^{0}(L_{|y+q})^{∨} → H^{0}(L)^{∨} for each (y, L) ∈ Y, we obtain an exact sequence of vector bundles over Y

0 ⟶ Ker(υ)^{∨} ⟶ N_{|Y}^{∨} ⟶ π_{2}^{∗}(R^{1}π_{2∗}(L_{|C×W_{d}^{r}(C)})) ⟶ 0,

from which we can compute c_{1}(N_{|Y}^{∨}) if we use [HT] which in this case reads

c_{1}(Ker(υ^{∨})) = c_{r+1}(\frac{π_{2}^{∗}(E^{∨})}{F}),

where F is the vector bundle on C × W_{d}^{r}(C) with fibre F(y, L) = H^{0}(L_{|y+q})^{∨}. Finally, we write

C^{0} · \overline{GP}_{g,d}^{r} = c_{r}η + ((r + 1)(θ − c_{1} + c_{1})) · [Y] + (r + 1)(c_{r}(d − 1) − 2c_{r−1}θ)η,

which eventually leads to the stated formula. □

To finish the proof of Theorem 1.6 it suffices to show that for [g/2] ≤ j ≤ g − 2, the coefficient of δ_{j} in the expression of σ_{∗}(c_{1}(σ^{∗}(𝔼) ⊗ OM̃_{g}(δ_{1})) − c_{1}(G_{0,1} ⊗ N)) always exceeds the coefficient of δ_{0}, which equals b_{0} and was computed in Proposition 3.4. This is a calculation along the lines of the proof of Theorem 1.4. To keep the length of this paper under control, we skip the details.

4. Five ways of constructing Koszul divisors for pointed curves

In this section we construct Koszul divisors on moduli spaces of pointed curves. As an application we improve Logan’s results on which \overline{M}_{g,n}’s are of general type.

We start by recalling a few things about divisor classes on \overline{M}_{g,n}. For 0 ≤ i ≤ g and a set of indices S ⊂ {1, …, n}, the boundary divisor Δ_{i:S} corresponds to the closure of the locus of nodal curves C_{1} ∪ C_{2}, with C_{1} smooth of genus i, C_{2} smooth of genus g − i, and such that the marked points sitting on C_{1} are precisely those labeled by S. We also introduce the divisor Δ_{irr} consisting of irreducible pointed curves with one node. We denote by δ_{i:S} ∈ Pic(\overline{M}_{g,n}) the class of Δ_{i:S} and by δ_{irr} that of Δ_{irr}. For each 1 ≤ i ≤ n we define the tautological class ψ_{i} := c_{1}(𝕃_{i}), where 𝕃_{i} is the line bundle on \overline{M}_{g,n} with fibre 𝕃_{i}([C, x_{1}, …, x_{n}]) = T_{x_{i}}^{∨}(C) over each point [C, x_{1}, …, x_{n}] ∈ \overline{M}_{g,n}. It is well known that when g ≥ 3, the Hodge class λ, the boundaries δ_{irr} and δ_{i:S}, and the tautological classes ψ_{i} for 1 ≤ i ≤ n, freely generate Pic(\overline{M}_{g,n}).

4.1. Divisors defined in terms of the Minimal Resolution Conjecture.

We fix integers g, r ≥ 1 and 0 ≤ i ≤ g and set n := (2r + 1)(g − 1) − 2i. We define a divisor on M_{g,n} consisting of smooth pointed curves (C, x_{1}, …, x_{n}) such that the points

x_{1}, …, x_{n} fail the Minimal Resolution Conjecture for the canonical curve C ↪^{|K_{C}|} P^{g−1} (see [FMP] for background on MRC). Precisely we define the locus

𝔐𝔯𝔠_{g,i}^{r} := {[C, x_{1}, …, x_{n}] ∈ M_{g,n} : h^{1}(C, ∧^{i}M_{K_{C}} ⊗ K_{C}^{⊗(r+1)} ⊗ O_{C}(−x_{1} − ⋯ − x_{n})) ≥ 1}.

If we denote by Γ := x_{1} + ⋯ + x_{n} ∈ C_{n}, by Serre duality, the condition appearing in the definition of 𝔐𝔯𝔠_{g,i}^{r} is equivalent to

h^{0}(C, ∧^{i}M_{K_{C}}^{∨} ⊗ O_{C}(Γ) ⊗ K_{C}^{⊗(−r)}) ≥ 1 ⟺ O_{C}(Γ) ⊗ K_{C}^{⊗(−r)} ∈ Θ_{∧^{i}M_{K_{C}}^{∨}} ,

where we recall that for a stable vector bundle E on C having slope ν(E) = ν ∈ ℤ, its theta divisor is the determinantal locus

Θ_{E} := {η ∈ Pic^{g−μ−1}(C) : h^{0}(C, E ⊗ η) ≥ 1}.

The main result from [FMP] gives an identification Θ_{∧^{i}M_{K_{C}}^{∨}} = C_{g−i−1} − C_{i}, where the right hand side is one of the difference varieties associated to C. Thus one has an alternative description of points in 𝔐𝔯𝔠_{g,i}^{r}: a point (C, x_{1}, …, x_{n}) ∈ 𝔐𝔯𝔠_{g,i}^{r} if and only if there exists D ∈ C_{i} such that h^{0}(C, O_{C}(Γ + D) ⊗ K_{C}^{⊗(−r)}) ≥ 1.

First we equip 𝔐𝔯𝔠_{g,i}^{r} with a determinantal scheme structure. We consider the following cartesian diagram of stacks

X ⟶^{q} M_{g,n}
↓^{f}  ↓^{π}
C_{g} ⟶^{p} M_{g}

in which all the morphisms are smooth and p (hence also q) is proper. We denote by ω_{p} ∈ Pic(C_{g}) the relative dualizing sheaf of the universal curve p : C_{g} → M_{g} and by 𝔼 := p_{∗}(ω_{p}) the Hodge bundle. We define the vector bundle M over C_{g} having rank g − 1 as the kernel of the evaluation map p^{∗}𝔼 ⟶ ω_{p}. Thus for every [C] ∈ M_{g}, we have M|_{p^{−1}([C])} ≃ M_{K_{C}}. For each 1 ≤ j ≤ n we have a section q_{j} : M_{g,n} ⟶ X of q given by q_{j}([C, x_{1}, …, x_{n}]) = ([C, x_{1}, …, x_{n}], x_{j}) ∈ X and we set E_{j} := Im(q_{j}), hence E_{j} is a relative divisor over M_{g,n}.

For integers 0 ≤ a ≤ i, b ≥ r + 2 and (a, b) = (0, r + 1) we define the vector bundle

A_{a,b} := q_{∗}(f^{∗}(∧^{a}M ⊗ ω_{p}^{⊗b}) ⊗ O_{X} (− ∑_{j=1}^{n} E_{j})),

hence A_{a,b}([C, x_{1}, …, x_{n}]) = H^{0}(∧^{a}M_{K_{C}} ⊗K_{C}^{⊗b} ⊗O_{C}(−Γ)). To prove that A_{a,b} is locally free over M_{g,n}, we use the fact that M_{K_{C}} is a semi-stable vector bundle over C and that μ(∧^{a}M_{K_{C}} ⊗ K_{C}^{⊗b}(−Γ)) > 2g − 1, hence

R^{1}q_{∗}(f^{∗}(∧^{a}M ⊗ ω_{p}^{⊗b}) ⊗ O_{X} (− ∑_{j=1}^{n} E_{j})) = 0

whenever b ≥ r + 2. To reach the same conclusion in the case of the sheaf A_{0,r+1}, we use that H^{1}(K_{C}^{⊗(r+1)}(−Γ))^{∨} = H^{0}(O_{C}(Γ) ⊗ K_{C}^{⊗(−r)}) = 0, if Γ ∈ C_{n} lies outside a subset of codimension ≥ 3. Furthermore there is a vector bundle map

ϕ : π^{∗}(∧^{i}𝔼) ⊗ A_{0,r+1} → A_{i−1,r+2}

which over each point [C, x_{1}, …, x_{n}] ∈ M_{g,n} corresponds to the natural map

ϕ(C, Γ) : ∧^{i}H^{0}(K_{C}) ⊗ H^{0}(K_{C}^{⊗(r+1)} ⊗ O_{C}(−Γ)) → H^{0}(∧^{i−1}M_{K_{C}} ⊗ K_{C}^{⊗(r+2)} ⊗ O_{C}(−Γ)).

Note that rank(A_{i−1,r+2}) = rank(π^{∗}(∧^{i}𝔼) ⊗ A_{0,r+1}) = 2i\binom{g}{i} and a simple argument using the exact sequence 0 ⟶ ∧^{i}M_{K_{C}} ⟶ ∧^{i}H^{0}(K_{C}) ⊗ O_{C} ⟶ ∧^{i−1}M_{K_{C}} ⊗ K_{C} ⟶ 0 shows that 𝔐𝔯𝔠_{g,i}^{r} is the degeneracy locus of the map ϕ.

Proposition 4.1. The vector bundle morphism ϕ : π^{∗}(∧^{i}𝔼)⊗A_{0,r+1} → A_{i−1,r+2} is generically non-degenerate. It follows that 𝔐𝔯𝔠_{g,i}^{r} is a divisor on M_{g,n}.

Proof. We show that ϕ is generically non-degenerate over the pull-back π^{∗}(H_{g}) of the hyperelliptic locus. We fix a hyperelliptic curve C of genus g and we denote by L ∈ W_{2}^{1}(C) its hyperelliptic involution. By writing down the Euler sequence on P^{1} one shows that M_{K_{C}} = (L^{∨})^{⊕(g−1)}, hence the condition H^{1}(∧^{i}M_{K_{C}} ⊗K_{C}^{⊗(r+1)}⊗O_{C}(−Γ)) = 0 is equivalent to H^{0}(O_{C}(Γ) ⊗ L^{⊗(i−r(g−1))}) = 0. This however is obvious because when Γ ∈ C_{n} is a general divisor of degree n then O_{C}(Γ)⊗L^{⊗(i−r(g−1))} is a general line bundle of degree g − 1, therefore it has no global sections. □

The main result here is the computation of the class of \overline{𝔐𝔯𝔠}_{g,i}^{r}:

Theorem 4.2. When n = (2r + 1)(g − 1) − 2i, the locus 𝔐𝔯𝔠_{g,i}^{r} is a divisor on M_{g,n} and the class of its compactification in \overline{M}_{g,n} is given by the following formula:

\overline{𝔐𝔯𝔠}_{g,i}^{r} ≡ \frac{1}{g − 1}\binom{g − 1}{i}(aλ + c ∑_{j=1}^{n} ψ_{j} − b_{irr}δ_{irr} − ∑_{j,s≥0,} b_{j:s} ∑_{|S|=s} δ_{j:S}),

where

c = rg + g − i − r − 1, b_{irr} = −\frac{1}{g − 2}(\binom{r + 1}{2}(g − 1)(g − 2) + i(i + 1 + 2r − rg − g)),

a = −\frac{1}{g − 2}((g − 1)(g − 2)(6r^{2} + 6r + 1) + i(24r + 10i + 10 − 10g − 12rg)),

KOSZUL DIVISORS ON MODULI SPACES OF CURVES 29

b_{0:s} = \binom{s + 1}{2}(g − 1) + s(rg − r) − si, andb_{j:s} ≥ b_{0:s} forj ≥ 1.

Remark 4.3. For i = 0, n = (2r + 1)(g − 1), the divisor \overline{𝔐𝔯𝔠}_{g,0}^{r} specializes to the locus of points [C, x_{1}, …, x_{(2r+1)(g−1)}] ∈ M_{g,(2r+1)(g−1)} such that ∑_{j=1}^{(2r+1)(g−1)} x_{j} ∈ |K_{C}^{⊗r}| and Theorem 4.2 gives that:

\overline{𝔐𝔯𝔠}_{g,0}^{r} ≡ −(6r^{2}+6r+1)λ+(r+1)( ∑_{j=1}^{(2r+1)(g−1)} ψ_{j})+\binom{r + 1}{2}δ_{irr} −(2r+3) ∑_{|S|=2} δ_{0:S} −· · · .

By letting all the marked points coalesce and using the standard formulas for pushing forward products of tautological classes (cf. e.g. [FMP] or [Log], Theorem 2.8), Theorem 4.2 offers a quick way of computing the class of the closure of the locus W^{r+1} of (r + 1)- Weierstrass points in \overline{M}_{g,1} which is the main result of [CF].

First we determine the class of the locus 𝔐𝔯𝔠_{g,i}^{r} over the interior M_{g,n}. In order to do this, we first recall a few well-known intersection theory relations (see e.g. [HM]): Lemma 4.4. If q : X → M_{g,n} is the morphism defined earlier, one has the following identities:

(i) q_{∗}(f^{∗}c_{1}(ω_{p})^{2}) = 12λ. (ii) q_{∗}(q^{∗}λ · f^{∗}c_{1}(ω_{p})) = (2g − 2)λ. (iii) q_{∗}q^{∗}(λ^{2}) = 0. (iv) q_{∗}(c_{1}(E_{j}) · q^{∗}λ) = λ.

(v) q_{∗}(c_{1}(E_{j}) · f^{∗}c_{1}(ω_{p})) = ψ_{j}. (vi) q_{∗}q^{∗}c_{2}(π^{∗}(𝔼)) = 0. (vii) q_{∗}(c_{1}(E_{j})^{2}) = −ψ_{j}. Proposition 4.5. If a and b are the numbers defined in the statement of Theorem 4.2, we have the following relation in Pic(M_{g,n}):

𝔐𝔯𝔠_{g,i}^{r} ≡ \frac{1}{g − 1}\binom{g − 1}{i}(aλ + b ∑_{j=1}^{n} ψ_{j}).

Proof. Since ϕ is generically non-degenerate (cf. Proposition 4.1), we have the identity 𝔐𝔯𝔠_{g,i}^{r} ≡ c_{1}(A_{i−1,r+2}) − c_{1}(π^{∗}(∧^{i}𝔼) ⊗ A_{0,r+1}). To compute these Chern classes we use Grothendieck-Riemann-Roch applied to the proper map q. For simplicity we set 𝔇 := ∑_{j=1}^{n} E_{j} and F := f^{∗}(∧^{i−1}M ⊗ ω_{p}^{⊗(r+2)}) ⊗ O_{X} (−𝔇)). Then we have that

c_{1}(A_{i−1,r+2}) = q_{∗}[(\binom{g − 1}{i − 1}+c_{1}(F)+\frac{c_{1}^{2}(F) − 2c_{2}(F)}{2}+⋯)·(1−\frac{f^{∗}c_{1}(ω_{p})}{2}+\frac{f^{∗}c_{1}^{2}(ω_{p})}{12})+…)].

Using that c_{1}(M) = p^{∗}(λ) − c_{1}(ω_{p}), one obtains that

c_{1}(F) = \binom{g − 2}{i − 2}q^{∗}(λ) + ((r + 2)\binom{g − 1}{i − 1} − \binom{g − 2}{i − 2})f^{∗}c_{1}(ω_{p}) − \binom{g − 1}{i − 1}c_{1}(𝔇).

We also use the identity

c_{2}(F) = c_{2}(f^{∗} ∧^{i−1} M) + (\binom{g − 1}{i − 1} − 1)c_{1}(f^{∗} ∧^{i−1} M) · ((r + 2)f^{∗}c_{1}(ω_{p}) − c_{1}(𝔇))+

+\frac{1}{2}\binom{g − 1}{i − 1}(\binom{g − 1}{i − 1} − 1)((r + 2)f^{∗}c_{1}(ω_{p}) − c_{1}(𝔇))^{2},

which together with the formula c_{2}(M) = c_{1}^{2}(ω_{p}) − c_{1}(ω_{p}) · p^{∗}(λ) and Lemma 4.4, enable us to compute c_{1}(A_{i−1,r+2}). In a similar fashion, we obtain from Grothendieck-Riemann-Roch applied to the map q, that

c_{1}(A_{0,r+1}) = (6r^{2} + 6r + 1)λ − (r + 1) ∑_{j=1}^{n} ψ_{j}

and finally

c_{1}(π^{∗} ∧^{i} 𝔼 ⊗ A_{0,r+1}) = \binom{g}{i}c_{1}(A_{0,r+1}) + ((2r + 1)(g − 1) − r)\binom{g − 1}{i − 1}λ,

which quickly leads to the stated formula. □

To compute the remaining coefficients in [\overline{𝔐𝔯𝔠}_{g,i}^{r}] we extend the vector bundles A_{a,b} to sheaves over \overline{M}_{g,n} as follows. We denote by q : \overline{M}_{g,n+1} → \overline{M}_{g,n} the projection dropping the (n + 1)-st marked point and by π : \overline{M}_{g,n} → \overline{M}_{g} the forgetful map. We introduce the following twist of the Hodge bundle on \overline{M}_{g,n}:

H := q_{∗}(ω_{q} ⊗ O_{\overline{M}_{g,n+1}}( ∑_{[g/2]≤j≤g−1} ∑_{|S|≤n}(g − j)Δ_{j:S})).

(In other words, the fibre of H over a pointed curve from π^{∗}(Δ_{j}) where [g/2] ≤ j ≤ g, is the space of global sections of the genus j-aspect of the limit 𝔤_{2g−2}^{g−1} corresponding to the canonical linear series). We then define

M := Ker\{q^{∗}(H) → O_{\overline{M}_{g,n+1}}( ∑_{[g/2]≤j≤g−1} ∑_{|S|≤n}(g − j)Δ_{j:s})\}.

Furthermore, for each pair of integers 0 ≤ a ≤ i, b ≥ r +2 or (a, b) = (0, r +1), we define

A_{a,b} := q_{∗}(∧^{a}M⊗ω_{q}^{⊗b}⊗O_{\overline{M}_{g,n+1}}( ∑_{[g/2]≤j≤g−1} ∑_{|S|≤n}((2b−1)(g−j)−b)Δ_{j:S}−∑_{j=1}^{n} Δ_{0:j,n+1})).

(Obviously, this is an extension of the definition of A_{a,b} over M_{g,n}). The twists were chosen in such a way that we have exact sequences of the type

0 ⟶ A_{a,b} ⟶ ∧^{a}H ⊗ A_{0,b} ⟶ A_{a−1,b+1} ⟶ 0,

at least in a dense open subset inside π^{−1}(M_{g} ∪Δ_{0} ∪Δ_{1}). Also, there exists a morphism ϕ : ∧^{i}H⊗A_{0,r+1} → A_{i−1,r+2} which over M_{g,n} restricts to the vector bundle map defined in Proposition 4.1.

Proof of Theorem 4.2. We expand the class of \overline{𝔐𝔯𝔠}_{g,i}^{r} in Pic(\overline{M}_{g,n}):

\overline{𝔐𝔯𝔠}_{g:i}^{r} ≡ Aλ + B ∑_{j=1}^{n} ψ_{j} − B_{irr}δ_{irr} − ∑_{j,s≥0} B_{j:s} ∑_{|S|=j} δ_{j:S}.

We have already determined the values of A and B. One can write down the following relation in Pic(\overline{M}_{g,n}):

(9)  c_{1}(A_{i−1,r+2} − ∧^{i}H ⊗ A_{0,r+1}) = [\overline{𝔐𝔯𝔠}_{g,i}^{r}] + ∑_{j,s≥0} d_{j,s} ∑_{|S|=s} δ_{j:S},

where d_{j,s} is the multiplicity of the divisor Δ_{j:S} in the degeneracy locus of ϕ. By intersecting both sides of (9) with test curves in \overline{M}_{g,n}, sometimes we are able to show that ϕ is generically non-degenerate along Δ_{j:S} (that is, d_{j:s} = 0), and then we explicitly determine the value of B_{j:S} in Theorem 4.2, otherwise we only get lower bounds on B_{j:S}. We are only going to explain in detail the case of the coefficient B_{irr} the remaining ones being somewhat similar.

We define a test curve in the boundary of \overline{M}_{g,n} as follows. If [C, q, x_{1}, …, x_{n}] ∈ M_{g−1,n+1} is a general pointed curve, then we define the 1-dimensional family

C_{n}^{0} := \{[C/y ∼ q, x_{1}, …, x_{n}]\}_{y∈C} ⊂ Δ_{irr} ⊂ \overline{M}_{g,n}.

The fibre of this family when the variable point y ∈ C hits the fixed marked point x_{i} for 1 ≤ i ≤ n is the pointed curve (C̃_{x_{i}} := C ∪_{x_{1},q} P^{1}, x̃_{1}, x_{2}, …, x_{n}), where x̃_{1} ∈ P^{1} (here we regard x_{1}, x̃_{1}, q ∈ P^{1} as three distinct points). One has the identities

C_{n}^{0}·δ_{irr} = −2g+2, C_{n}^{0}·δ_{1:∅} = 1, C_{n}^{0}·ψ_{i} = 1 for 1 ≤ i ≤ n, C_{n}^{0}·λ = C_{n}^{0}·δ_{i:S} = 0 for (i, S) ≠ (1, ∅).

By intersecting both sides of (9) with C_{n}^{0} one can write down the identity C_{n}^{0} · \overline{𝔐𝔯𝔠}_{g,i}^{r} = (2g − 2)B_{irr} + nB − B_{1:∅}. On the other hand one also has the relation

A − 12B_{irr} + B_{1:∅} = 0,

reflecting the fact that \overline{𝔐𝔯𝔠}_{g,i}^{r} is physically disjoint from the curve \{[C∪_{q}R, x_{1}, x_{2}, …, x_{n}]\}_{R} obtained by attaching to a fixed Brill-Noether general curve [C, q, x_{1}, …, x_{n}] ∈ M_{g−1,n+1} a pencil of plane cubics in which R denotes a generic member. Thus determining B_{irr} and B_{1:∅} boils down to (i) showing that ϕ is generically non-degenerate along C_{n}^{0} and (ii) estimating the intersection number C_{n}^{0}·c_{1}(A_{i−1,r+2}−∧^{i}H⊗A_{0,r+1}). By local analysis one can see that for 1 ≤ l ≤ i − 1 there are exact sequences of bundles over C_{n}^{0}

0 ⟶ A_{i−l,r+l+1|C_{n}^{0}} ⟶ ∧^{i−l}H ⊗ A_{0,r+l+1|C_{n}^{0}} ⟶ A_{i−l−1,r+l+2|C_{n}^{0}} ⟶ 0,

therefore we can write the identities

C_{n}^{0} · c_{1}(A_{i−1,r+2}) = ∑_{l=1}^{i}(−1)^{l−1}c_{1}(∧^{i−l}H_{|C_{n}^{0}} ⊗ A_{0,r+l+1|C_{n}^{0}}) =

∑_{l=1}^{i}(−1)^{l−1}(\binom{g − 1}{i − l − 1}((2r + 2l + 1)(g − 1) − n)c_{1}(H_{|C_{n}^{0}}) + \binom{g}{i − l}c_{1}(A_{0,r+l+1|C_{n}^{0}})).

Next we describe the vector bundle A_{0,j|C_{n}^{0}}. We identify C_{n}^{0} with C via the map

C ∋ y ↦ [C/y ∼ q, x_{1}, …, x_{n}] ∈ \overline{M}_{g,n}

and denote by p_{1}, p_{2} : C × C → C the two projections, by Δ ⊂ C × C the diagonal and set Γ_{q} := {q} × C ⊂ C × C. Then for every j ≥ r we have the following exact sequence of vector bundles on C:

0 ⟶ (p_{2})_{∗}(p_{1}^{∗}(K_{C}^{⊗j}) ⊗ O_{C×C}((j − 1)Δ + (j − 1)Γ_{q} − ∑_{j=1}^{n}{x_{j}} × C)) ⟶ A_{0,j|C_{n}^{0}} ⟶

⟶ (p_{2})_{∗}(p_{1}^{∗}(K_{C}^{⊗j})⊗O_{C×C}(jΔ+jΓ_{q} −∑_{j=1}^{n}{x_{j}}×C)⊗O_{Γ_{q}})⊗O_{C}(−x_{1} −⋯−x_{n}) ⟶ 0,

which quickly leads to the formula

c_{1}(A_{0,j|C_{n}^{0}}) = 1 + 2j − 2jrg − 2jg − j^{2} + j^{2}g + 2jr + 2ji.

Since one also has that c_{1}(H_{|C_{n}^{0}}) = 1 (use that H(y) = H^{0}(K_{C} ⊗ O_{C}(y + q)) for each y ∈ C_{n}^{0} under the identification described above), we obtain a formula for C_{n}^{0} · \overline{𝔐𝔯𝔠}_{g,i}^{r} and ultimately a formula for B_{irr}. Dealing with the other coefficients B_{j:S} is similar in general.

□

4.2. Divisors defined by imposing linear conditions on marked points.

Here we present another general construction that produces families of effective divisors on \overline{M}_{g,n}. Like in Section 2, we pick integers g, r, d ≥ 1 such that ρ(g, r, d) = 0, therefore we can write d = rs+r and g = rs+s for some integer s ≥ 1. We set n := r+1 and define the following divisor on M_{g,n}:

𝔏𝔦𝔫_{d}^{r} := {[C, x_{1}, …, x_{r+1}] : ∃L ∈ W_{d}^{r}(C) such that h^{0}(L ⊗ O_{C}(−x_{1} − ⋯ − x_{r+1})) ≥ 1}.

We recall that we have introduced the number

N := c_{r} = g!\frac{1! 2! … r!}{(g − d + r)! … (g − d + 2r)!},

which counts linear series 𝔤_{d}^{r} on a general curve of genus g (cf. [ACGH]). Our main result is the computation of the class [\overline{𝔏𝔦𝔫}_{d}^{r}]:

Theorem 4.6. Fix integers r, s ≥ 1 and set d := rs + r, g := rs + s. Then \overline{𝔏𝔦𝔫}_{d}^{r} is an effective divisor on \overline{M}_{g,n} and we have the following formula for its class in Pic(\overline{M}_{g,r+1}):

\overline{𝔏𝔦𝔫}_{d}^{r} ≡ \frac{rc_{r}}{rs + s − 1}(aλ + c ∑_{j=1}^{r+1} ψ_{j} − b_{irr}δ_{irr} − ∑_{j,t≥0} b_{j:t} ∑_{|S|=t} δ_{j:S}),

where

a = \frac{(r + 2)(r^{2}s^{3} − r^{2}s + 2rs^{3} + 6rs^{2} − 2rs − 8r + s^{3} + 6s^{2} + 3s − 8)}{2(s + r + 1)(rs + s − 2)},

c = \frac{s + 1}{2}, b_{irr} = \frac{(s − 1)(s + 1)(r + 1)(r + 2)(rs + s + 4)}{12(s + r + 1)(rs + s − 2)},

b_{j:0} = \frac{j(r + 2)(rs(s^{2} − 1)(r + 2) + s(s^{2} − 2j − 3) + (r + 1)(3s^{2} − js^{2} + 2j − 2))}{2(r + s + 1)(rs + s − 2)} forj ≥ 1

b_{0:t} = \frac{t}{2r} (trs + ts − t + r − s + 1) for2 ≤ t ≤ r + 1,

b_{1:t} = \binom{t − 1}{2}\frac{rs + s − 1}{r} + \frac{(s − 1)(s + 1)(r + 1)(r^{3}s + 3r^{2}s − 2s + 4)}{2r(r + s + 1)(rs + s − 2)}for allt ≥ 1

andb_{j:t} ≥ b_{0:t} for allj ≥ 2.

Remark 4.7. For s = 1 and r = g −1, Theorem 4.6 specializes to Logan’s formula for the class of the divisor \overline{𝔏𝔦𝔫}_{2g−2}^{g−1} of points [C, x_{1}, …, x_{g}] ∈ M_{g,g} with h^{0}(O_{C}(x_{1} + ⋯ + x_{g})) ≥ 2. We have the formula (cf. [Log], Theorem 5.4):

\overline{𝔏𝔦𝔫}_{2g−2}^{g−1} ≡ −λ + ∑_{j=1}^{g} ψ_{j} − 0 · δ_{irr} − ∑_{t=2}^{g} \binom{t + 1}{2} ∑_{|S|=t} δ_{0:S} − ∑_{t=1}^{g} \binom{t}{2} ∑_{|S|=t} δ_{1:S} − · · · .

In the next case, s = 2, g = 2r + 2 and d = 3r we get a new divisor on \overline{M}_{2r+2,r+1} and our formula reads

(10) \overline{𝔏𝔦𝔫}_{3r}^{r} ≡ \frac{1}{2(2r + 1)}((3r + 5)(r + 2)λ + 3r ∑_{j=1}^{r+1} ψ_{j} − \binom{r + 2}{2}δ_{irr} − ∑_{|S|=2} δ_{0:S} − ⋯).

Proof of Theorem 4.6. Proving that 𝔏𝔦𝔫_{d}^{r} is a divisor on M_{g,r+1} follows immediately from Brill-Noether theory: a general [C] ∈ M_{g} has precisely N linear series L ∈ W_{d}^{r}(C) and each of them is base point free and satisfies h^{0}(L) = r + 1 and 𝔏𝔦𝔫_{d}^{r} consists of those (r+1)-tuples of points on C which are not in general linear position with respect to some L ∈ W_{d}^{r}(C). We start the calculation of the class of \overline{𝔏𝔦𝔫}_{d}^{r} by determining the coefficients of λ, δ_{irr} and ψ_{j}. To do this we use the observation that if π_{n} : \overline{M}_{g,n} → \overline{M}_{g,n−1} is the projection forgetting the marked point labeled by n and D is any divisor class on \overline{M}_{g,n}, then for distinct labels i, j ≠ n, the λ, δ_{irr} and ψ_{j} coefficients of the divisors D on \overline{M}_{g,n} and (π_{n})_{∗}(D · δ_{0:in}) on \overline{M}_{g,n−1} are the same (see [FMP], Prop. 4.4). The divisor (π_{n})_{∗}(D · δ_{0:in}) can be thought of as the locus of those points [C, x_{1}, …, x_{n}] ∈ D where the points x_{i} and x_{n} are allowed to come together. Using this observation repeatedly, we obtain that the divisor \overline{𝔏𝔦𝔫}_{d}^{r}(1) on \overline{M}_{g,1} obtained by letting all points x_{1}, …, x_{n} come together, has the same λ and δ_{irr} coefficients as 𝔏𝔦𝔫_{d}^{r}. But clearly

𝔏𝔦𝔫_{d}^{r}(1) = {[C, x] ∈ M_{g,1} : ∃L ∈ W_{d}^{r}(C) such that h^{0}(L ⊗ O_{C}(−(r + 1)x)) ≥ 1},

that is, 𝔏𝔦𝔫_{d}^{r}(1) is generically the locus of ramification points in one of the finitely many linear series 𝔤_{d}^{r} on a given curve of genus g. By applying Theorem 4.1 from [EH2], we obtain that the class of \overline{𝔏𝔦𝔫}_{d}^{r}(1) can be written as a combination \overline{𝔏𝔦𝔫}_{d}^{r}(1) ≡ μ·BN +ν ·W, where

BN := (g + 3)λ − \frac{g + 1}{6}δ_{irr} − ∑_{j=1}^{g−1} j(g − j)δ_{j:1}

is the pull-back from \overline{M}_{g} of the class of the Brill-Noether divisor and

W := −λ + \binom{g + 1}{2}ψ − ∑_{j=1}^{g−1} \binom{g − j + 1}{2}δ_{j:1}

is the class of the Weierstrass divisor. To determine the coefficients μ and ν we use two test curves inside \overline{M}_{g,1}. First we fix a genus g curve C and we let the marked point vary along C. If we denote this curve by C̄ ⊂ \overline{M}_{g,1}, then the only generator of Pic(\overline{M}_{g,1}) which has non-zero intersection number with C̄ is ψ, and C̄ · ψ = 2g − 2. On the other hand C̄ · \overline{𝔏𝔦𝔫}_{d}^{r}(1) is the total number of ramification points on all 𝔤_{d}^{r}’s on C. This number is N(r + 1)(d + r(g − 1)) (see e.g. [EH1], pg. 345), which shows that ν = N(r + 1)(d + r(g − 1))/((g − 1)g(g + 1)). To compute μ we use a second test curve constructed as follows: we fix a 2-pointed elliptic curve [E, x, y] ∈ M_{1,2} such that the class x − y ∈ Pic^{0}(E) is not torsion, and a fixed general curve [C] ∈ M_{g−1}. We define the family C̄_{1} := {(C ∪_{y} E, x)}_{y∈C} (that is, the point of attachment varies on C). The only generator of Pic(\overline{M}_{g,1}) meeting C̄_{1} non-trivially is δ_{1:1} = δ_{g−1:∅}, in which case C̄_{1} · δ_{1;1} = −2g + 4. The calculation of C̄_{1} · \overline{𝔏𝔦𝔫}_{d}^{r}(1) is a standard exercise in the theory of limit linear series. Suppose l = {l_{C}, l_{E}} is a limit 𝔤_{d}^{r} on C ∪_{y} E such that a_{r}^{l_{E}}(x) ≥ r + 1. Then because the class x − y ∈ Pic^{0}(E) is not torsion, we must have that ρ(l_{E}, x, y) = 0 and ρ(l_{C}, y) = −1. An easy calculation shows that we must also have a^{l_{E}}(x) = (0, 1, …, r − 1, r + 1) and a^{l_{C}}(y) = (0, 2, 3, …, r, r + 2), and moreover the aspect l_{E} is uniquely determined. Thus we have to count the number of points y ∈ C such that there exists L ∈ W_{d}^{r}(C) with the property that h^{0}(L(−2y)) ≥ r and h^{0}(L(−(r + 2)y)) ≥ 1.

To compute this number we further degenerate the curve C to a transverse union R ∪_{y_{1}} E_{1} ∪ … ∪_{y_{g−1}} E_{g−1} consisting of a smooth rational spine R and g − 1 elliptic tails E_{1}, …, E_{g−1}. Using Proposition 1.1 from [EH1] we see that the point y has to specialize to one of the tails E_{j}, and without loss of generality we assume that y ∈ E_{1} (all the intersection numbers we compute will be multiplied by g − 1 to account for y lying on a different elliptic tail). Suppose now that l = {l_{R}, l_{E_{1}}, …, l_{E_{g−1}}} is a limit 𝔤_{d}^{r} on R ∪ E_{1} ∪ … ∪ E_{g−1} such that a^{l_{E_{1}}} (y) = (0, 2, 3, …, r, r + 2). Then ρ(l_{R}, y_{1}, …, y_{g−1}) = 0, ρ(l_{E_{j}}, x_{j}) = 0 for 2 ≤ j ≤ g − 1 and ρ(l_{E_{1}}, y_{1}, y) = −1. A close inspection shows that there are three numerical possibilities:

(α) a^{l_{R}}(y_{1}) = (0, 2, 4, 5, …, r, r + 1, r + 3) and then y_{1} − y ∈ Pic^{0}(E_{1})[2]. This contribution will be equal to 3(g − 1) multiplied by the number of 𝔤_{d}^{r}’s on R having ordinary cusps at g − 2 general points and vanishing (0, 2, 4, …, r + 1, r + 3) at another fixed point. By Schubert calculus this number equals the product of Schubert cycles σ_{(0,1,...,1)}^{g−2} · σ_{(0,1,2,...,2,3)} ∈ H^{∗}(G(r, d)).

(β) a^{l_{R}}(y_{1}) = (0, 3, 4, …, r +1, r +2), in which case y −y_{1} ∈ Pic^{0}(E_{1})[r +2]. The number we get in this situation is (g − 1)((r + 2)^{2} − 1)σ_{(0,1,...,1)}^{g−2} · σ_{(0,2,...,2)} ∈ H^{∗}(G(r, d)).

(γ) a^{l_{r}}(y) = (1, 2, 4, …, r + 1, r + 2) and then y − y_{1} ∈ Pic^{0}(E_{1})[r]. We obtain a final contribution of (g − 1)(r^{2} − 1)σ_{(0,1,...,1)}^{g−2} · σ_{(0,0,1,...,1)} ∈ H^{∗}(G(r, d − 1)). Adding all these together and using (3), we obtain that the total intersection number is

C̄_{1} · \overline{𝔏𝔦𝔫}_{d}^{r}(1) = \frac{N r(r + 1)(r + 2)(rs + 2s^{2} − 4 + s)}{s + r + 1},

which leads to

μ = \frac{Nr(r + 1)(r + 2)(s − 1)(s + 1)(rs + s + 4)}{2(s + r + 1)(rs + s − 2)(rs + s − 1)(rs + s + 1)}

and then the stated formulas for the λ and δ_{irr} coefficients. We also note that the δ_{j:∅} coefficient of \overline{𝔏𝔦𝔫}_{d}^{r} equals the δ_{j:∅} = δ_{g−j:1} coefficient of \overline{𝔏𝔦𝔫}_{d}^{r}(1) and this is equal to j(g −j)μ+j(j +1)ν/2 and we obtain the desired expression for b_{j:0}. Next we determine the coefficient c of the ψ_{j} classes. We introduce the divisor \overline{𝔏𝔦𝔫}_{d}^{r}(2) on \overline{M}_{g,2} obtained by letting x_{2}, …, x_{r+1} coincide while keeping x_{1} apart. Then

𝔏𝔦𝔫_{d}^{r}(2) = {[C, x_{1}, x_{2}] ∈ M_{g,2} : ∃L ∈ W_{d}^{r}(C) such that h^{0}(L ⊗ O_{C}(−x_{1} − rx_{2})) ≥ 1}

and we can write \overline{𝔏𝔦𝔫}_{d}^{r}(2) ≡ N(aλ + cψ_{1} + c_{2}ψ_{2} − e_{12}δ_{0:12} − ⋯) (that is, the λ and ψ_{1} coefficients coincide with those of \overline{𝔏𝔦𝔫}_{d}^{r}). We intersect \overline{𝔏𝔦𝔫}_{d}^{r}(2) with two curves in \overline{M}_{g,2}: consider a curve C of genus g and define C̃_{1} := \{[C, x_{1}, x_{2}]\}_{\{x_{2} moves on C\}} and C̃_{2} := \{[C, x_{1}, x_{2}]\}_{\{x_{1} moves on C\}}. Then C̃_{2} · \overline{𝔏𝔦𝔫}_{d}^{r}(2) = N(c_{2} + (2g − 1)c − e_{12}) = N(d − r) and C̃_{1} · \overline{𝔏𝔦𝔫}_{d}^{r}(2) = N((2g − 1)c_{2} + c − e_{12}) = N(r + 1)(d − 1 + (r − 1)(g − 1)). (The first identity is obvious, for the second, use that what we are counting is the total number of ramification points on all linear series L ⊗ O_{C}(−x_{1}), where L ∈ W_{d}^{r}(C) and x_{1} ∈ C is a fixed general point). We thus have a system of two equations in the unknowns c, c_{2} and e_{12}, but we can also use that e_{12} equals the ψ coefficient of \overline{𝔏𝔦𝔫}_{d}^{r}(1) = (π_{2})_{∗}(\overline{𝔏𝔦𝔫}_{d}^{r}(2)·δ_{0:12}), where π_{2} : \overline{M}_{g,2} → \overline{M}_{g,1} is the map forgetting the second point. Thus e_{12} = νg(g + 1)/2, which gives us enough relations to determine c. We note that in this way we also determine b_{0:2} = (2rs + r + s − 1)/(rs + s − 1).

To compute the coefficient b_{j:t} for 1 ≤ t ≤ r + 1 we consider another test curve defined as follows: we fix integers 1 ≤ j ≤ g − 1 and 1 ≤ t ≤ r + 1, together with general pointed curves [C, y, x_{2}, …, x_{t}] ∈ M_{j,t} and [Y, y, x_{t+1}, …, x_{r+1}] ∈ M_{g−j,r−t+2}. We define the test curve C̄_{j,s} := {C ∪_{y} Y, x_{1}, …, x_{t}, x_{t+1}, x_{r+1}}_{x_{1}∈C} (thus x_{1} is the moving point on the genus j component). Then we have the relation

(11)  C̄_{j,t} · \overline{𝔏𝔦𝔫}_{d}^{r} = (2j + 2t − 3)c − (t − 1)b_{0:2} + b_{j:t} − b_{j:t−1},

which can be used to compute b_{j:t} provided we know b_{j:t−1} (note that we have already computed b_{j:∅} for all j).

We now describe directly the intersection cycle C̄_{j,t} · \overline{𝔏𝔦𝔫}_{d}^{r}. Since [C, y] ∈ M_{j,1} and [Y, y] ∈ M_{g−j,1} are general, on the stable Y ∪_{y} C there will be precisely N limit 𝔤_{d}^{r}’s which can be described as follows: We first choose a Schubert (ramification) sequence max{0, j − t} ≤ α_{0} ≤ … α_{r} ≤ j such that ∑_{i=0}^{r} α_{i} = rj. Then we choose l_{Y} ∈ G_{d}^{r}(Y) having vanishing sequence (a^{l_{Y}} (y) = α_{i} + i)_{0≤i≤r}; in fact there will be σ_{(0,1,...,1)}^{g−j} · σ_{(α_{0},...,α_{r})} ∈ H^{∗}(G(r, d)) such linear series. On C we choose a complementary

linear series l_{C} ∈ G_{d}^{r}(C) with vanishing sequence (a_{i}^{l_{C}}(y) = rs + i − α_{i})_{0≤i≤r}; there are σ_{(0,1,...,1)}^{j} · σ_{(rs−α_{r},...,rs−α_{0})} ∈ H^{∗}(G(r, d)) choices. Every limit linear series on C ∪_{y} Y appears in this way and the intersection C̄_{j,t} · \overline{𝔏𝔦𝔫}_{d}^{r} is everywhere transverse (cf. [EH3]). We also have the identity

(12) N = ∑ ; j−t≤α_{0}≤...≤α_{r}≤j, ∑_{l=0}^{r} α_{l}=j(σ_{(0,1,...,1)}^{g−j} · σ_{(α_{0},...,α_{r})}) (σ_{(0,1,...,1)}^{j} · σ_{(rs−α_{r},...,rs−α_{0})}).

If l = {l_{C}, l_{Y} } is one of these N limit 𝔤_{d}^{r}’s corresponding to a sequence (α_{0} ≤ … ≤ α_{r}) as above, the condition that there exists x_{1} ∈ C such that the divisor x_{1} + ⋯ + x_{r+1} is the specialization of a linear divisor with respect to a 𝔤_{d}^{r} on a nearby smooth curve, can be translated as follows: there exist sections σ_{Y} ∈ |l_{Y} |, σ_{C} ∈ |l_{C}| such that div(σ_{Y}) ≥ x_{t+1} + ⋯ + x_{r+1} and div(σ_{C}) ≥ x_{1} + … + x_{t}; the sections σ_{C} and σ_{Y} being the limit linear series specializations of a single section on a nearby smooth curve, they must also satisfy the compatibility relation ord_{y}(σ_{Y}) + ord_{y}(σ_{C}) = rs + r. Because the fixed points x_{t+1}, …, x_{r+1} ∈ Y are general, they impose independent conditions on l_{Y} which quickly leads to the equalities ord_{y}(σ_{Y}) = a_{t−1}^{l_{Y}}(y) = α_{t−1} + t − 1, hence ord_{y}(σ_{C}) = a_{r−t+1}^{l_{C}}(y). Thus div(σ_{C}) ≥ a_{r−t+1}^{l_{C}}(y) + x_{2} + … + x_{b} and up to multiplication by scalars, the sections σ_{C} and σ_{Y} are unique with this property. For each σ_{C} we have precisely d − a_{r−t+1}^{l_{C}}(y) − (t − 1) = α_{t−1} choices for x_{1} ∈ C. Therefore

(13) C̄_{j,t} · \overline{𝔏𝔦𝔫}_{d}^{r} = ∑_{α_{0}≤···≤α_{r}} α_{t−1}(σ_{(0,1,...,1)}^{g−j} · σ_{(α_{0},...,α_{r})}) · (σ_{(0,1,...,1)}^{j} · σ_{(rs−α_{r},...,rs−α_{0})}).

For j = 0 the only sequence (α_{l})_{0≤l≤r} allowed is the sequence (0, …, 0) which shows that C̄_{0:t} ·\overline{𝔏𝔦𝔫}_{d}^{r} = 0 for all 3 ≤ t ≤ r +1. Since b_{0:2} has already been determined, applying (11) we obtain the stated formulas for b_{0:t}. Similarly, for j = 1 the only sequence allowed is (0, 1, …, 1) and then C̄_{1:t} · \overline{𝔏𝔦𝔫}_{d}^{r} = N for t ≥ 2, while C̄_{1:1} · \overline{𝔏𝔦𝔫}_{d}^{r} = 0; this allows us to determine b_{1:t} for all t. When j ≥ 2 for each sequence (α_{l})_{0≤l≤r} appearing in this sum, we have the inequalities rj = ∑_{l=0}^{r} α_{l} ≤ tα_{t−1} + (r + 1 − t)j, therefore α_{t−1} ≥ (t − 1)j/t and then C̄_{j,t} ·\overline{𝔏𝔦𝔫}_{d}^{r} ≥ N(t−1)j/t. To obtain the desired bound on b_{j:t} we use repeatedly

(11) and we can write

b_{j:t} − b_{j:0} = \binom{t}{2}b_{0:2} − (2bj + b^{2} − 2b)c + ∑_{l=1}^{t} C̄_{j,l} · \overline{𝔏𝔦𝔫}_{d}^{r}.

Using the previous inequality we can now check that b_{j:t} ≥ b_{j:0}.

□

4.3. The divisor of n-fold points.

We describe another way of constructing effective divisors on \overline{M}_{g,n}. Instead of looking at loci of points [C, x_{1}, …, x_{n}] ∈ M_{g,n} for which the points x_{1}, …, x_{n} become linearly dependent in a suitable embedding of C, we can consider the loci where the marked points give rise to an n-fold point on a suitable model of C. Given [C] ∈ M_{g} and a linear series l = (L, V) ∈ G_{d}^{r}(C), we say that the divisor Γ := x_{1} + ⋯ + x_{n} is an n-fold point for C and l if dim(V ∩ H^{0}(L ⊗ O_{C}(−Γ))) ≥ r.

Definition 4.8. Fix integers g, r, d, n ≥ 1 such that ρ(g, r, d) − r(n − 1) = −1. We define the locus of n-fold points in M_{g,n}

𝔑𝔣𝔬𝔩𝔡_{g,d}^{r} := {[C, x_{1}, …, x_{n}] ∈ M_{g,n} : ∃L ∈ W_{d}^{r}(C) with dim H^{0}(L(−x_{1} −⋯−x_{n})) = r}.

We have computed the class of 𝔑𝔣𝔬𝔩𝔡_{g,d}^{r} in the case r = 1. The calculation is along the same lines as that of the class of \overline{𝔏𝔦𝔫}_{d}^{r} in Theorem 4.6:

Theorem 4.9. Fix integers g ≥ 1 and n ≥ 0 such that d := (g + n)/2 ∈ ℤ. The class of the compactification of the divisor 𝔑𝔣𝔬𝔩𝔡_{g,d}^{1} of n-fold points on M_{g,n} is given by the formula:

\overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1} ≡ (\frac{10n}{g − 2}\binom{g − 2}{d − 1} − \frac{n}{g}\binom{g}{d})λ + \frac{n − 1}{g − 1}\binom{g − 1}{d − 1} ∑_{j=1}^{n} ψ_{j} − \frac{n}{g − 2}\binom{g − 2}{d − 1}δ_{irr}−

− ∑_{t≥2} \frac{t(n^{2} − g + tgn − tn)}{2(g − 1)(g − d)}\binom{g − 1}{d} ∑_{|S|=t} δ_{0:S} − · · · .

Proof. The coefficients of λ, δ_{irr} and ψ_{j} (1 ≤ j ≤ n) in the expansion of \overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1} equal the coefficients of λ, δ_{irr} and ψ respectively in the expansion of the divisor \overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1}(1) on \overline{M}_{g,1} obtained from \overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1} by letting the points x_{1}, …, x_{n} ∈ C coalesce. Clearly,

𝔑𝔣𝔬𝔩𝔡_{g,d}^{1}(1) := {[C, x] ∈ M_{g,1} : ∃L ∈ W_{d}^{1}(C) with h^{0}(C, L(−n · x)) ≥ 1}, and this is a ”pointed” Brill-Noether divisor on M_{g,1} in the sense of [EH2]. To compute the class of its compactification in \overline{M}_{g,1} once again we use [EH2], Theorem 4.1 and write \overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1}(1) ≡ μ · BN + ν · W, where the divisor classes BN and W have the same significance as in the proof of Theorem 4.6. By applying [Log], Theorem 4.5, we find that

μ = \frac{6n}{(g + 1)(g − 2)}\binom{g − 2}{d − 1} and ν = \frac{n(n − 1)(n + 1)}{g(g − 1)(g + 1)}\binom{g}{d}.

The remaining coefficients of [\overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1}] are determined by intersecting the locus \overline{𝔑𝔣𝔬𝔩𝔡}_{g,d}^{1} with the fibral test curves lying entirely in the boundary divisors of \overline{M}_{g,n}. The calculation is straightforward and relies on Section 3 from [Log]. We skip these details. □

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

In this section we prove Theorem 1.10. We treat each case individually but for each g we only work out the case of the minimal n = n(g) for which our methods show that \overline{M}_{g,n(g)} is of general type. From this it follows automatically that \overline{M}_{g,n} is of general type for all g ≥ g(n) (see [Log], Theorem 2.4). Proof of Theorem 1.10. [\overline{M}_{4,16}] and [\overline{M}_{6,16}]. We consider the divisor 𝔐𝔯𝔠_{4,0}^{2} on \overline{M}_{4,15} introduced in Theorem 4.2. We have seen that \overline{𝔐𝔯𝔠}_{4,0}^{2} ≡ −37λ + 3 ∑_{j=1}^{1} 5ψ_{j} + 3δ_{irr} − 7 ∑_{|S|=2} δ_{0:S} − · · · . We consider the maps π_{j} : \overline{M}_{4,16} → \overline{M}_{4,15} obtained by forgetting the marked point labeled by 1 ≤ j ≤ 16. Then there exists a constant α > 0 such that

∑_{j=1}^{16}(π_{j})^{∗}(\overline{𝔐𝔯𝔠}_{4,0}^{2}) ≡ α(−37λ + \frac{45}{16} ∑_{j=1}^{1} 6ψ_{j} + 3δ_{irr} − \frac{13}{2} ∑_{|S|=2} δ_{0:S} − ⋯).

The class of the Petri divisor on \overline{M}_{4} being (up to a > 0 constant) 17λ − 2δ_{irr} − ⋯, we obtain that K_{\overline{M}_{4,16}} is big, being a positive combination of ∑_{j=1}^{16}(π_{j})^{∗}(\overline{𝔐𝔯𝔠}_{4:0}^{2}), the pull-back of the Petri class, an ample class and boundary divisors. The same argument works in the case of \overline{M}_{6,16} except that we start with the divisor \overline{𝔐𝔯𝔠}_{6,0}^{1} on \overline{M}_{6,15} which is pulled back to \overline{M}_{6,16} in all possible ways. [\overline{M}_{5,15}]. On \overline{M}_{5,12} we have the identity of divisor classes

\overline{𝔐𝔯𝔠}_{5,0}^{1} ≡ −13λ + 2 ∑_{j=1}^{1} 2ψ_{j} + δ_{irr} − 5 ∑_{|S|=2} δ_{0:S} − · · · .

Pulling this class back to \overline{M}_{5,15} in all possible ways by forgetting sets of three marked points, we obtain that the class −66λ + \frac{42}{5} ∑_{j=1}^{15} ψ_{j} + 5δ_{irr} − ⋯ is effective on \overline{M}_{5,15}. Using the Brill-Noether class 8λ − δ_{irr} − ⋯ on \overline{M}_{5}, we get that K_{\overline{M}_{5,15}} is a big class.

[\overline{M}_{18,9}]. We use our divisor \overline{𝔏𝔦𝔫}_{24}^{8}: there is a positive constant α such that

α\overline{𝔏𝔦𝔫}_{24}^{8} ≡ 290λ + 24 ∑_{j=1}^{9} ψ_{j} − 45δ_{irr} − · · · .

On the other hand the class of the multiple of the Petri divisor \overline{GP}_{18,24}^{8} on \overline{M}_{18} is equal to \frac{302}{45}λ − δ_{0} − ∑_{j=1}^{9} b_{j}δ_{j}, where b_{j} > 1 for j ≥ 1. It follows that we can write K_{\overline{M}_{18,9}}

as a positive combination of multiples of \overline{𝔏𝔦𝔫}_{24}^{8}, π^{∗}(\overline{GP}_{18,24}^{8}), boundary divisors and an ample class on \overline{M}_{18,9}.

[\overline{M}_{19,7}] and [\overline{M}_{14,10}]. In these cases we use the divisors of n-fold points, \overline{𝔑𝔣𝔬𝔩𝔡}_{19,13}^{1} on \overline{M}_{19,7} and \overline{𝔑𝔣𝔬𝔩𝔡}_{14,12}^{1} on \overline{M}_{14,10} respectively. Using Theorem 4.9 we see that the canonical bundle of \overline{M}_{g,n} can be written as a positive combination of these divisors, the pull-back of the Brill-Noether divisor from \overline{M}_{19} and \overline{M}_{14} respectively, a suitable ample class and boundary divisors.

[\overline{M}_{15,10}]. We use a slightly different technique. On \overline{M}_{15,11} we have the divisor \overline{𝔑𝔣𝔬𝔩𝔡}_{15,13}^{1} of points [C, x_{1}, …, x_{11}] such that x_{1} +⋯+x_{11} appears in a fibre of a 𝔤_{13}^{1} on C. We push this divisor down to \overline{M}_{15,10} by letting two of the points x_{j} ∈ C coalesce, that is, we define

E := \frac{1}{11} ∑_{j=1}^{10}(π_{j})_{∗}(\overline{𝔑𝔣𝔬𝔩𝔡}_{15,13}^{1} · δ_{0:j,11}),

where π_{j} : \overline{M}_{15,11} → \overline{M}_{15,10} forgets the marked point labeled by j. It is easy to check using Theorem 4.9 that

E ≡ 33λ + \frac{396}{5} ∑_{j=1}^{10} ψ_{j} − 11δ_{irr} − ⋯ ( use that (π_{j})_{∗}(ψ_{j} · δ_{0:j,11}) = (π_{j})_{∗}(ψ_{11} · δ_{0:j,11}) = 0 ).

It turns out that K_{\overline{M}_{15,11}} is in the span of E, π^{∗}(\overline{M}_{15,14}^{3}), an ample class and boundaries.

[\overline{M}_{20,6}]. From [Log] Theorem 5.4, one knows that the class −λ+ \frac{22}{3} ∑_{j=1}^{6} ψ_{j}−0·δ_{irr}−⋯ is effective on \overline{M}_{20,6}. Next, if χ_{i,j} : \overline{M}_{20,6} → \overline{M}_{21} denotes the map which associates to a 6-pointed curve of genus 20 a nodal curve of genus 21 obtained by identifying the marked points labeled i and j, we also get that the class

∑_{i<j} χ_{i,j}^{∗}(\overline{Z}_{21,0}) ≡ c(\frac{2459}{377}λ + \frac{1}{3} ∑_{j=1}^{6} ψ_{j} − δ_{irr} − ⋯),

with c > 0, is also effective on \overline{M}_{20,6}. The conclusion now follows easily.

References

[AC1] 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.

[AC2] E. Arbarello and M. Cornalba, Su una congettura di Petri, Comment. Math. Helvetici 56 (1981), 1-38.

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

[CF] F. Cukierman and L.-Y. Fong, On higher Weierstrass points, Duke Mathematical Journal 62 (1991), 179-203.

[CR] M.C. Chang and Z. Ran, Deformations and smoothings of reducible curves, in: Algebraic Geometry, Bowdoin 1985, Proc. Symph. Pure Math. 46, 63-75.

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

[EH2] D. Eisenbud and J. Harris, Irreducibility of some families of linear series with Brill-Noether number −1, Annales Scient. École Normale Superieure 22 (1989), 33–53.

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

[EH4] D. Eisenbud and J. Harris, A simpler proof of the Gieseker-Petri theorem on special divisors, Inventiones Math. 74 (1983), 269-280.

[Est] E. Esteves, Linear systems and ramification points on reducible nodal curves, math.AG/9808069, Algebra Meeting (Rio de Janeiro 1996), Mat. Contemp. 14 (1998), 21-35.

[Est2] E. Esteves, Compactifying the relative Jacobian over families of reduced curves, Transactions American Mathematical Society, 353 (2001), 3045-3095.

[F1] G. Farkas, Syzygies of curves and the effective cone of \overline{M}_{g}, math.AG/0503498, Duke Mathematical Journal 135 (2006), 53-98.

[F2] G. Farkas, Gaussian maps, Gieseker-Petri loci and large theta-characteristics, J. reine angew. Mathematik 581 (2005), 151-173.

[F3] G. Farkas (with an appendix by K. Ludwig), The Kodaira dimension of the moduli space of Prym varieties, in preparation.

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

[FMP] G. Farkas, M. Mustaţă and M. Popa, Divisors on M_{g,g+1} and the minimal resolution conjecture for points on canonical curves, Annales Scient. École Normale Superieure 36 (2003), 558-581.

[FuPr] W. Fulton and P. Pragacz, Schubert varieties and degeneration loci, Springer Lecture Notes in Math. 1689 (1998).

[GG] A. Geramita and A. Gimigliano, Generators for the defining ideal of certain rational surfaces, Duke Mathematical Journal 62 (1991), 61-83.

[GL] M. Green and R. Lazarsfeld, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, in: Complex Projective Geometry, London Mathematical Society Lecture Notes 179, 149-156.

[Gr] M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geometry 19 (1984), 125-171.

[H1] J. Harris, On the Kodaira dimension of the moduli space of curves II: The even-genus case, Inventiones Math. 75 (1984), 437-466.

[HT] J. Harris and L. Tu, Chern numbers of kernel and cokernel bundles, Inventiones Math. 75 (1984), 467-475.

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

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

[Kh] D. Khosla, Moduli spaces of curves with linear series and the Slope Conjecture, preprint.

[Laz] R. Lazarsfeld, Brill-Noether-Petri without degenerations, J. Diff. Geometry 23 (1986), 299-307.

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

[Oss] B. Osserman, A limit linear series moduli scheme, Ann. Inst. Fourier (Grenoble) 56 (2004), 1165-1205.

[R] T.G. Room, The geometry of determinantal loci, Cambridge University Press, 1938.

[Ver] A. Verra, The unirationality of the moduli space of curves of genus 14 or lower, Compositio Math. 141 (2005), 1425-1444.

[V1] C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface, J. European Mathematical Society 4 (2002), 363-404.

[V2] C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163-1190.

[Wa] J. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Mathematical Journal 55 (1987), 843-871.

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

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