Clay Mathematics Proceedings Volume 14, 2011

Koszul Cohomology and Applications to Moduli

Marian Aprodu and Gavril Farkas

1. Introduction

One of the driving problems in the theory of algebraic curves in the past two decades has been Green’s Conjecture on syzygies of canonical curves. Initially formulated by M. Green [Gr84a], it is a deceptively simple vanishing statement

concerning Koszul cohomology groups of canonical bundles of curves: If C ⟶^{|K_{C}|} ℙ^{g−1} is a smooth canonically embedded curve of genus g and K_{i,j}(C, K_{C}) are the Koszul cohomology groups of the canonical bundle on C, Green’s Conjecture predicts the equivalence

(1)  K_{p,2}(C, K_{C}) = 0 ⟺ p < Cliff(C),

where Cliff(C) := min{deg(L)−2r(L) : L ∈ Pic(C), h^{i}(C, L) ≥ 2, i = 0, 1} denotes the Clifford index of C. The main attraction of Green’s Conjecture is that it links the extrinsic geometry of C encapsulated in Cliff(C) and all the linear series 𝔤_{d}^{r} on C, to the intrinsic geometry (equations) of the canonical embedding. In particular, quite remarkably, it shows that one can read the Clifford index of any curve off the equations of its canonical embedding. Hence in some sense, a curve has no other interesting line bundle apart from the canonical bundle and its powers^{1}.

One implication in (1), namely that K_{p,2}(C, K_{C}) ≠ 0 for p ≥ Cliff(C), having been immediately established in [GL84], see also Theorem 2.4 in this survey, the converse, that is, the vanishing statement

K_{p,2}(C, K_{C}) = 0 for p < Cliff(C),

attracted a great deal of effort and resisted proof despite an amazing number of attempts and techniques devised to prove it, see [GL84], [Sch86], [Sch91], [Ein87], [Ei92], [PR88], [Tei02], [V93]. The major breakthrough came around 2002 when Voisin [V02], [V05], using specialization to curves on K3 surfaces, proved that Green’s Conjecture holds for a general curve [C] ∈ M_{g}:

\begin{matrix}c \\ ◯ \end{matrix} 2011 Marian Aprodu and Gavril Farkas

The authors would like to thank the Max Planck Institut für Mathematik Bonn for hospitality during the preparation of this work and the referee for a close reading. Research of the first author partially supported by a PN-II-ID-PCE-2008-2 grant, PCE-2228 (contract no. 502/2009).

Research of the second author partially supported by an Alfred P. Sloan Fellowship.

1 ”The canonical bundle is not called canonical for nothing”- Joe Harris

Theorem 1.1. For a general curve [C] ∈ M_{2p+3} we have that K_{p,2}(C, K_{C}) = 0. For a general curve [C] ∈ M_{2p+4} we have that K_{p,2}(C, K_{C}) = 0. It follows that Green’s Conjecture holds for general curves of any genus.

Combining the results of Voisin with those of Hirschowitz and Ramanan [HR98], one finds that Green’s Conjecture is true for every smooth curve [C] ∈ M_{2p+3} of maximal gonality gon(C) = p + 3. This turns out to be a remarkably strong result. For instance, via a specialization argument, Green’s Conjecture for arbitrary curves of maximal gonality implies Green’s Conjecture for general curves of genus g and arbitrary gonality 2 ≤ d ≤ [g/2] + 2. One has the following more precise result cf. [Ap05], see Theorem 4.5 for a slightly different proof:

Theorem 1.2. We fix integers 2 ≤ d ≤ [g/2] + 1. For any smooth d-gonal curve [C] ∈ M_{g} satisfying the condition

dim W_{g−d+2}^{1}(C) ≤ g − 2d + 2,

we have that K_{d−3,2}(C, K_{C}) = 0. In particular C satisfies Green’s Conjecture.

Dimension theorems from Brill-Noether theory due to Martens, Mumford, and Keem, cf. [ACGH85], indicate precisely when the condition appearing in the statement of Theorem 1.2 is verified. In particular, Theorem 1.2 proves Green’s Conjecture for general d-gonal curves of genus g for any possible gonality 2 ≤ d ≤ [g/2] + 2 and offers an alternate, unitary proof of classical results of Noether, Enriques-Babbage-Petri as well as of more recent results due to Schreyer and Voisin. It also implies the following new result which can be viewed as a proof of statement (1) for 6-gonal curves. We refer to Subsection 4.1 for details:

Theorem 1.3. For any curve C with Cliff(C) ≥ 4, we have K_{3,2}(C, K_{C}) = 0. In particular, Green’s Conjecture holds for arbitrary 6-gonal curves.

Theorem 1.1 can also be applied to solve various related problems. For instance, using precisely Theorem 1.1, the Green-Lazarsfeld Gonality Conjecture [GL86] was verified for general d-gonal curves, for any 2 ≤ d ≤ (g +2)/2, cf. [ApV03], [Ap05]. In a few words, this conjecture states that the gonality of any curve can be read off the Koszul cohomology with values in line bundles of large degree, such as the powers of the canonical bundle. We shall review all these results in Subsection 4.2.

Apart from surveying the progress made on Green’s and the Gonality Conjectures, we discuss a number of new conjectures for syzygies of line bundles on curves. Some of these conjectures have already appeared in print (e.g. the Prym-Green Conjecture [FaL08], or the syzygy conjecture for special line bundles on general curves [Fa06a]), whereas others like the Minimal Syzygy Conjecture are new and have never been formulated before.

For instance we propose the following refinement of the Green-Lazarsfeld Gonality Conjecture [GL86]:

Conjecture 1.4. Let C be a general curve of genus g = 2d − 2 ≥ 6 and η ∈ Pic^{0}(C) a general line bundle. Then K_{d−4,2}(C, K_{C} ⊗ η) = 0.

Conjecture 1.4 is the sharpest vanishing statement one can make for general line bundles of degree 2g − 2 on a curve of genus g. Since

dim K_{d−4,2}(C, K_{C} ⊗ η) = dim K_{d−3,1}(C, K_{C} ⊗ η),

it follows that the failure locus of Conjecture 1.4 is a virtual divisor in the universal degree 0 Picard variety 𝔓𝔦𝔠_{g}^{0} → M_{g}. Thus it predicts that the non-vanishing locus

{[C, η] ∈ 𝔓𝔦𝔠_{g}^{0} : K_{d−4,2}(C, K_{C} ⊗ η) ≠ 0}

is an ”honest” divisor on 𝔓𝔦𝔠_{g}^{0}. Conjecture 1.4 is also sharp in the sense that from the Green-Lazarsfeld Non-Vanishing Theorem 2.8 it follows that

K_{d−2,1}(C, K_{C} ⊗ η) ≠ 0.

Similarly, always K_{d−3,2}(C, K_{C} ⊗ η) ≠ 0 for all [C, η] ∈ 𝔓𝔦𝔠_{g}^{0}. A yet stronger conjecture is the following vanishing statement for l-roots of trivial bundles on curves:

Conjecture 1.5. Let C be a general curve of genus g = 2d − 2 ≥ 6. Then for every prime l and every line bundle η ∈ Pic^{0}(C) − {O_{C}} satisfying η^{⊗l} = O_{C}, we have that K_{d−4,2}(C, K_{C} ⊗ η) = 0.

In order to prove Conjecture 1.5 it suffices to exhibit a single pair [C, η] as above, for which K_{C} ⊗ η ∈ Pic^{2g−2}(C) satisfies property (N_{d−4}). The case most studied so far is that of level l = 2, when one recovers the Prym-Green Conjecture [FaL08] which has been checked using Macaulay2 for g ≤ 14. The Prym-Green Conjecture is a subtle statement which for small values of g is equivalent to the Prym-Torelli Theorem, again see [FaL08]. Since the condition K_{d−4,2}(C, K_{C}⊗η) ≠ 0 is divisorial in moduli, Conjecture 1.5 is of great help in the study of the birational geometry of the compactification \overline{R}_{g,l} := \overline{M}_{g}(Bℤ_{l}) of the moduli space R_{g,l} classifying pairs [C, η], where [C] ∈ M_{g} and η ∈ Pic^{0}(C) satisfies η^{⊗l} = O_{C}.

Concerning both Conjectures 1.4 and 1.5, it is an open problem to find an analogue of the Clifford index of the curve, in the sense that the classical Green Conjecture is not only a Koszul cohomology vanishing statement but also allows one to read off the Clifford index from a non-vanishing statement for K_{p,2}(C, K_{C}). It is an interesting open question to find a Prym-Clifford index playing the same role as the original Cliff(C) in (1) and to describe it in terms of the corresponding Prym varieties: Is there a geometric characterization of those Prym varieties P_{g}([C, η]) ∈ A_{g−1} corresponding to pairs [C, η] ∈ R_{g} with K_{p,2}(C, K_{C} ⊗ η) ≠ 0?

Another recent development on syzygies of curves came from a completely different direction, with the realization that loci in the moduli space M_{g} consisting of curves having exceptional syzygies can be used effectively to answer questions about the birational geometry of the moduli space \overline{M}_{g} of stable curves of genus g, cf. [FaPo05], [Fa06a], [Fa06b], in particular, to produce infinite series of effective divisors on \overline{M}_{g} violating the Harris-Morrison Slope Conjecture [HaM90]. We recall that the slope s(D) of an effective divisor D on \overline{M}_{g} is defined as the smallest rational number a/b with a, b ≥ 0, such that the class aλ−b(δ_{0}+⋯+δ_{[g/2]})−[D] ∈ Pic(\overline{M}_{g}) is an effective ℚ-combination of boundary divisors. The Slope Conjecture [HaM90] predicts a lower bound for the slope of effective divisors on \overline{M}_{g},

s(\overline{M}_{g}) := inf_{D∈Eff(\overline{M}_{g})} s(D) ≥ 6 + \frac{12}{g + 1},

with equality precisely when g + 1 is composite; the quantity 6 + 12/(g + 1) is the slope of the Brill-Noether divisors on \overline{M}_{g}, in case such divisors exist. A first counterexample to the Slope Conjecture was found in [FaPo05]: The locus

K_{10} := {[C] ∈ M_{g} : C lies on a K3 surface}

can be interpreted as being set-theoretically equal to the locus of curves [C] ∈ M_{10} carrying a linear series L ∈ W_{12}^{4}(C) such that the multiplication map

ν_{2}(L) : Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})

is not an isomorphism, or equivalently K_{0,2}(C, L) ≠ 0. The main advantage of this Koszul-theoretic description is that it provides a characterization of the K3 divisor K_{10} in a way that makes no reference to K3 surfaces and can be easily generalized to other genera. Using this characterization one shows that s(\overline{K}_{10}) = 7 < 78/11, that is, \overline{K}_{10} ∈ Eff(\overline{M}_{10}) is a counterexample to the Slope Conjecture.

Koszul cohomology provides an effective way of constructing cycles on M_{g}. Under suitable numerical conditions, loci of the type

Z_{g,2} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such that K_{p,2}(C, L) ≠ 0}

are virtual divisors on M_{g}, that is, degeneracy loci of morphisms between vector bundles of the same rank over M_{g}. The problem of extending these vector bundes over \overline{M}_{g} and computing the virtual classes of the resulting degeneracy loci is in general daunting, but has been solved successfully in the case ρ(g, r, d) = 0, cf. [Fa06b]. Suitable vanishing statements of the Koszul cohomology for general curves (e.g. Conjectures 1.4, 5.4) show that, when applicable, these virtual Koszul divisors are actual divisors and they are quite useful in specific problems such as the Slope Conjecture or showing that certain moduli spaces of curves (with or without level structure) are of general type, see [Fa08], [FaL08]. A picturesque application of the Koszul technique in the study of parameter spaces is the following result about the birational type of the moduli space of Prym varieties \overline{R}_{g} = \overline{R}_{g,2}, see [FaL08]:

Theorem 1.6. The moduli space \overline{R}_{g} is of general type for g ≥ 13 and g ≠ 15.

The proof of Theorem 1.6 depends on the parity of g. For g = 2d − 2, it boils down to calculating the class of the compactification in \overline{R}_{g} of the failure locus of the Prym-Green Conjecture, that is, of the locus

{[C, η] ∈ R_{2d−2} : K_{d−4,2}(C, K_{C} ⊗ η) ≠ 0}.

For odd g = 2d − 1, one computes the class of a ”mixed” Koszul cohomology locus in \overline{R}_{g} defined in terms of Koszul cohomology groups of K_{C} with values in K_{C} ⊗ η.

The outline of the paper is as follows. In Section 2 we review the definition of Koszul cohomology as introduced by M. Green [Gr84a] and discuss basic facts. In Section 3 we recall the construction of (virtual) Koszul cycles on \overline{M}_{g} following [Fa06a] and [Fa06b] and explain how their cohomology classes can be calculated. In Section 4 we discuss a number of conjectures on syzygies of curves, starting with Green’s Conjecture and the Gonality Conjecture and continuing with the Prym-Green Conjecture. We end by proposing in Section 5 a strong version of the Maximal Rank Conjecture.

Some results stated in [Ap04], [Ap05] are discussed here in greater detail. Other results are new (see Theorems 4.9 and 4.16).

2. Koszul cohomology

2.1. Syzygies. Let V be an n-dimensional complex vector space, S := S(V) the symmetric algebra of V, and X ⊂ ℙV^{∨} := Proj(S) a non-degenerate subvariety, and denote by S(X) the homogeneous coordinate ring of X. To the embedding of X in ℙV^{∨}, one associates the Hilbert function, defined by

h_{X}(d) := dim_{ℂ} (S(X)_{d})

for any positive integer d. A remarkable property of h_{X} is its polynomial behavior for large values of d. It is a consequence of the existence of a graded minimal resolution of the S-module S(X), which is an exact sequence

0 → E_{s} → … → E_{2} → E_{1} → S → S(X) → 0

with

E_{p} = ⨁_{j>p} S(−j)^{β_{pj}(X)}.

The Hilbert function of X is then given by

(2)  h_{X}(d) = ∑_{p,j}(−1)^{p}β_{pj}(X) ( \begin{matrix}n + d − j \\ n \end{matrix} ) ,

where

( \begin{matrix}t \\ n \end{matrix} ) = \frac{t(t − 1) ⋯ (t − n + 1)}{n !}, for any t ∈ R,

and note that the expression on the right-hand-side is polynomial for large d. This reasoning leads naturally to the definition of syzygies of X, which are the graded components of the graded S-modules E_{p}. The integers

β_{pj}(X) = dim_{ℂ}Tor^{j}(S(X), ℂ)_{p}

are called the graded Betti numbers of X and determine completely the Hilbert function, according to formula (2). Sometimes we also write b_{i,j}(X) := β_{i,i+j}(X).

One main difficulty in developing syzygy theory was to find effective geometric methods for computing these invariants. In the eighties, M. Green and R. Lazarsfeld published a series of papers, [Gr84a], [Gr84b], [GL84], [GL86] that shed a new light on syzygies. Contrary to the classical point of view, they look at integral closures of the homogeneous coordinates rings, rather than at the rings themselves. This approach, using intensively the language of Koszul cohomology, led to a number of beautiful geometrical results with numerous applications in classical algebraic geometry as well as moduli theory.

2.2. Definition of Koszul cohomology. Throughout this paper, we follow M. Green’s approach to Koszul cohomology [Gr84a]. The general setup is the following. Suppose X is a complex projective variety, L ∈ Pic(X) a line bundle, F is a coherent sheaf on X, and p, q ∈ ℤ. The canonical contraction map

⋀^{p+1} H^{0}(X, L) ⊗ H^{0}(X, L)^{∨} → ⋀^{p} H^{0}(X, L),

acting on tensors as

(s_{0} ∧ ⋯ ∧ s_{p}) ⊗ σ ↦ ∑_{i=0}^{p}(−1)^{i}σ(s_{i})(s_{0} ∧ ⋯ ∧ ŝ_{i} ∧ ⋯ ∧ s_{p}),

and the multiplication map

H^{0}(X, L) ⊗ H^{0}(X, F ⊗ L^{q−1}) → H^{0}(X, F ⊗ L^{q}),

define together a map

⋀^{p+1} H^{0}(X, L) ⊗ H^{0}(X, F ⊗ L^{q−1}) → ⋀^{p} H^{0}(X, L) ⊗ H^{0}(X, F ⊗ L^{q}).

In this way, we obtain a complex (called the Koszul complex)

∧^{p+1}H^{0}(L)⊗H^{0}(F⊗L^{q−1}) → ∧^{p}H^{0}(L)⊗H^{0}(F⊗L^{q}) → ∧^{p−1}H^{0}(L)⊗H^{0}(F⊗L^{q+1}),

whose cohomology at the middle-term is denoted by K_{p,q}(X, F, L). In the particular case F ≅ O_{X}, to ease the notation, one drops O_{X} and writes directly K_{p,q}(X, L) for Koszul cohomology.

Here are some samples of direct applications of Koszul cohomology:

Example 2.1. If L is ample, then L is normally generated if and only if K_{0,q}(X, L) = 0 for all q ≥ 2. This fact follows directly from the definition.

Example 2.2. If L is globally generated with H^{1}(X, L) = 0, then

H^{1}(X, O_{X}) ≅ K_{h^{0}(L)−2,2}(X, L)

see [ApN08]. In particular, the genus of a curve can be read off (vanishing of) Koszul cohomology with values in non-special bundles. More generally, under suitable vanishing assumptions on L, all the groups H^{i}(X, O_{X}) can be computed in a similar way, and likewise H^{i}(X, F) for an arbitrary coherent sheaf F, cf. [ApN08].

Example 2.3. If L is very ample the Castelnuovo-Mumford regularity can be recovered from Koszul cohomology. Specifically, if F is a coherent sheaf on X, then

reg_{L}(F) = min{m : K_{p,m+1}(X, F, L) = 0, for all p}.

As a general principle, any invariant that involves multiplication maps is presumably related to Koszul cohomology.

Very surprisingly, Koszul cohomology interacts much closer with the geometry of the variety than might have been expected. This phenomenon was discovered by Green and Lazarsfeld [Gr84a, Appendix]:

Theorem 2.4 (Green-Lazarsfeld). Suppose X is a smooth variety and consider a decomposition L = L_{1} ⊗ L_{2} with h^{0}(X, L_{i}) = r_{i} + 1 ≥ 2 for i ∈ {1, 2}. Then K_{r_{1}+r_{2}−1,1}(X, L) ≠ 0.

In other words, non-trivial geometry implies non-trivial Koszul cohomology. We shall discuss the case of curves, which is the most relevant, and then some consequences in Section 4.

Many problems in the theory of syzygies involve vanishing/nonvanishing of Koszul cohomology. One useful definition is the following.

Definition 2.5. An ample line bundle L on a projective variety is said to satisfy the property (N_{p}) if and only if K_{i,q}(X, L) = 0 for all i ≤ p and q ≥ 2.

From the geometric viewpoint, the property (N_{p}) means that L is normally generated, the ideal of X in the corresponding embedding is generated by quadrics, and all the syzygies up to order p are linear. In many cases, for example canonical curves, or 2-regular varieties, the property (N_{p}) reduces to the single condition K_{p,2}(X, L) = 0, see e.g. [Ein87]. This phenomenon justifies the study of various loci given by the nonvanishing of K_{p,2}, see Section 4.

2.3. Kernel bundles. The proofs of the facts discussed in examples 2.2 and 2.3 use the kernel bundles description which is due to Lazarsfeld [La89]: Consider L a globally generated line bundle on the projective variety X, and set

M_{L} := Ker (H^{0}(X, L) ⊗ O_{X} ⟶^{ev} L) .

Note that M_{L} is a vector bundle on X of rank h^{0}(L) − 1. For any coherent sheaf F on X, and integer numbers p ≥ 0, and q ∈ ℤ, we have a short exact sequence on X

0 → ∧^{p+1}M_{L} ⊗ F ⊗ L^{q−1} → ∧^{p+1}H^{0}(L) ⊗ F ⊗ L^{q−1} → ∧^{p}M_{L} ⊗ F ⊗ L^{q} → 0.

Taking global sections, we remark that the Koszul differential factors through the map

∧^{p+1}H^{0}(X, L) ⊗ H^{0}(X, F ⊗ L^{q−1}) → H^{0}(X, ∧^{p}M_{L} ⊗ F ⊗ L^{q}),

hence we have the following characterization of Koszul cohomology, [La89]:

Theorem 2.6 (Lazarsfeld). Notation as above. We have

K_{p,q}(X, F, L) ≅ coker(∧^{p+1}H^{0}(L) ⊗ H^{0}(F ⊗ L^{q−1}) → H^{0}(∧^{p}M_{L} ⊗ F ⊗ L^{q}))

≅ ker(H^{1}(X, ∧^{p+1}M_{L} ⊗ F ⊗ L^{q−1}) → ∧^{p+1}H^{0}(L) ⊗ H^{1}(F ⊗ L^{q−1})).

Theorem 2.6 has some nice direct consequences. The first one is a duality theorem which was proved in [Gr84a].

Theorem 2.7. Let L be a globally generated line bundle on a smooth projective variety X of dimension n. Set r := dim|L|. If

H^{i}(X, L^{q−i}) = H^{i}(X, L^{q−i+1}) = 0, i = 1, …, n − 1

then

K_{p,q}(X, L)^{∨} ≅ K_{r−n−p,n+1−q}(X, K_{X}, L).

Another consequence of Theorem 2.6, stated implicitly in [Fa06b] without proof, is the following:

Theorem 2.8. Let L be a non-special globally generated line bundle on a smooth curve C of genus g ≥ 2. Set d = deg(L), r = h^{0}(C, L) − 1, and consider 1 ≤ p ≤ r. Then

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

In particular, if

p < \frac{(d + 1 − g)(d − g)}{d} − 1,

then K_{p,1}(C, L) ≠ 0, and if

\frac{(d + 1 − g)(d − g)}{d} ≤ p ≤ d − g,

then K_{p−1,2}(C, L) ≠ 0.

Proof. Since we work with a spanned line bundle on a curve, Theorem 2.7 applies, hence we have

K_{p,1}(C, L) ≅ K_{r−p−1,1}(C, K_{C}, L)^{∨}

and

K_{p−1,2}(C, L) ≅ K_{r−p,0}(C, K_{C}, L)^{∨}.

Set, as usual, M_{L} = Ker{H^{0}(C, L) ⊗ O_{C} → L}, and consider the Koszul complex

(3) 0 → ∧^{r−p}H^{0}(C, L) ⊗ H^{0}(C, K_{C}) → H^{0}(C, ∧^{r−p−1}M_{L} ⊗ K_{C} ⊗ L) → 0.

Since L is non-special, K_{r−p,0}(C, K_{C}, L) is isomorphic to the kernel of the differential appearing in (3), hence the difference which we wish to compute coincides with the Euler characteristic of the complex (3).

Next we determine h^{0}(C, ∧^{r−p−1}M_{L} ⊗ K_{C} ⊗ L). Note that rk(M_{L}) = r and ∧^{r−p−1}M_{L} ⊗ L ≅ ∧^{p+1}M_{L}^{∨}. In particular, since H^{0}(C, ∧^{p+1}M_{L}) ≅ K_{p+1,0}(C, L) = 0, we obtain

h^{0}(C, ∧^{r−p−1}M_{L} ⊗ K_{C} ⊗ L) = −χ(C, ∧^{p+1}M_{L}).

Observe that

deg(∧^{p+1}M_{L}) = deg(M_{L})\binom{r − 1}{p} = −d\binom{r − 1}{p}

and

rk(∧^{p+1}M_{L}) = \binom{r}{p + 1}

From the Riemann-Roch Theorem it follows that

−χ(C, ∧^{p+1}M_{L}) = d\binom{r − 1}{p} + (g − 1)\binom{r}{p + 1}

and hence

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

The formula is obtained by replacing r by d − g. □

Remark 2.9. A full version of Theorem 2.8 for special line bundles can be obtained in a similar manner by adding alternating sums of other groups K_{p−i,i+1}. For example, if L^{⊗2} is non-special, then from the complex

0 → ∧^{r−p+1}H^{0}(C, L) ⊗ H^{0}(C, K_{C} ⊗ L^{−1}) → ∧^{r−p}H^{0}(C, L) ⊗ H^{0}(C, K_{C}) →

→ H^{0}(C, ∧^{r−p−1}M_{L} ⊗ K_{C} ⊗ L) → 0

we obtain the following conclusion:

dim K_{p,1}(C, L) − dim K_{p−1,2}(C, L) + dim K_{p−2,3}(C, L) =

= d\binom{r − 1}{p} + (g − 1)\binom{r}{p + 1} − g\binom{r + 1}{p + 1} + \binom{r + 1}{p}(r − d + g).

2.4. Hilbert schemes. Suppose X is a smooth variety, and consider L ∈ Pic(X). A novel description of the Koszul cohomology of X with values in L was provided in [V02] via the Hilbert scheme of points on X.

Denote by X^{[n]} the Hilbert scheme parameterizing zero-dimensional length n subschemes of X, let X_{curv}^{[n]} be the open subscheme parameterizing curvilinear length n subschemes, and let

Ξ_{n} ⊂ X_{curv}^{[n]} × X

be the incidence subscheme with projection maps p : Ξ_{n} → X and q : Ξ_{n} → X_{curv}^{[n]}. For a line bundle L on X, the sheaf L^{[n]} := q_{∗}p^{∗}L is locally free of rank n on X_{curv}^{[n]}, and the fiber over ξ ∈ X_{curv}^{[n]} is isomorphic to H^{0}(ξ, L ⊗ O_{ξ}).

There is a natural map

H^{0}(X, L) ⊗ O_{X_{curv}^{[n]}} → L^{[n]},

acting on the fiber over ξ ∈ X_{curv}^{[n]}, by s ↦ s|_{ξ}. In [V02] and [EGL] it is shown that, by taking wedge powers and global sections, this map induces an isomorphism:

∧^{n}H^{0}(X, L) ≅ H^{0}(X_{curv}^{[n]}, det L^{[n]}).

Voisin proves that there is an injective map

H^{0}(Ξ_{p+1}, det L^{[p+1]} ⊠ L^{q−1}) → ∧^{p}H^{0}(X, L) ⊗ H^{0}(X, L^{q})

whose image is isomorphic to the kernel of the Koszul differential. This eventually leads to the following result:

Theorem 2.10 (Voisin [V02]). For all integers p and q, the Koszul cohomology K_{p,q}(X, L) is isomorphic to the cokernel of the restriction map

H^{0}(X_{curv}^{[p+1]} × X, det L^{[p+1]} ⊠ L^{q−1}) → H^{0}(Ξ_{p+1}, det L^{[p+1]} ⊠ L^{q−1}|_{Ξ_{p+1}}).

In particular,

K_{p,1}(X, L) ≅ coker(H^{0}(X_{curv}^{[p+1]}, det L^{[p+1]}) →^{q^{∗}} H^{0}(Ξ_{p+1}, q^{∗}det L^{[p+1]}|_{Ξ_{p+1}})).

Remark 2.11. The group K_{p,q}(X, F, L) is obtained by replacing L^{q−1} by F ⊗ L^{q−1} in the statement of Theorem 2.10.

The main application of this approach is the proof of the generic Green Conjecture [V02]; see Subsection 4.1 for a more detailed discussion on the subject. The precise statement is the following.

Theorem 2.12 (Voisin [V02] and [V05]). Consider a smooth projective K3 surface S, such that Pic(S) is isomorphic to ℤ^{2}, and is freely generated by L and O_{S}(Δ), where Δ is a smooth rational curve such that deg(L_{|Δ}) = 2, and L is a very ample line bundle with L^{2} = 2g − 2, g = 2k + 1. Then K_{k+1,1}(S, L ⊗ O_{S}(Δ)) = 0 and

(4)  K_{k,1}(S, L) = 0.

Voisin’s result, apart from settling the Generic Green Conjecture, offers the possibility (via the cohomological calculations carried out in [Fa06b], see also Section 3), to give a much shorter proof of the Harris-Mumford Theorem [HM82] on the Kodaira dimension of \overline{M}_{g} in the case of odd genus. This proof does not use intersection theory on the stack of admissible coverings at all and is considerably shorter than the original proof. This approach has been described in full detail in [Fa08].

3. Geometric cycles on the moduli space

3.1. Brill-Noether cycles. We recall a few basic facts from Brill-Noether theory; see [ACGH85] for a general reference.

For a smooth curve C and integers r, d ≥ 0 one considers the Brill-Noether locus

W_{d}^{r}(C) := {L ∈ Pic^{d}(C) : h^{0}(C, L) ≥ r + 1}

as well as the variety of linear series of type 𝔤_{d}^{r} on C, that is,

G_{d}^{r}(C) := {(L, V) : L ∈ W_{d}^{r}(C), V ∈ G(r + 1, H^{0}(L))}.

The locus W_{d}^{r}(C) is a deteminantal subvariety of Pic^{d}(C) of expected dimension equal to the Brill-Noether number ρ(g, r, d) = g − (r + 1)(g − d + r). According to the Brill-Noether Theorem for a general curve [C] ∈ M_{g}, both W_{d}^{r}(C) and G_{d}^{r}(C) are irreducible varieties of dimension

dim W_{d}^{r}(C) = dim G_{d}^{r}(C) = ρ(g, r, d).

In particular, W_{d}^{r}(C) = ∅ when ρ(g, r, d) < 0. By imposing the condition that a curve carry a linear series 𝔤_{d}^{r} when ρ(g, r, d) < 0, one can define a whole range of geometric subvarieties of M_{g}.

We introduce the Deligne-Mumford stack σ : 𝔊_{d}^{r} → M_{g} classifying pairs [C, l] where [C] ∈ M_{g} and l = (L, V) ∈ G_{d}^{r}(C) is a linear series 𝔤_{d}^{r}, together with the projection σ[C, l] := [C]. The stack 𝔊_{d}^{r} has a determinantal structure inside a Grassmann bundle over the universal Picard stack 𝔓𝔦𝔠_{g}^{d} → M_{g} . In particular, each irreducible component of 𝔊_{d}^{r} has dimension at least 3g −3+ρ(g, r, d), cf. [AC81b]. We define the Brill-Noether cycle

M_{g,d}^{r} := σ_{∗}(𝔊_{d}^{r}) = {[C] ∈ M_{g} : W_{d}^{r}(C) ≠ ∅},

together with the substack structure induced from the determinantal structure of 𝔊_{d}^{r} via the morphism σ. A result of Steffen [St98] guarantees that each irreducible component of M_{g,d}^{r} has dimension at least 3g − 3 + ρ(g, r, d).

When ρ(g, r, d) = −1, Steffen’s result coupled with the Brill-Noether Theorem implies that the cycle M_{g,d}^{r} is pure of codimension 1 inside M_{g}. One has the following more precise statement due to Eisenbud and Harris [EH89]:

Theorem 3.1. For integers g, r and d such that ρ(g, r, d) = −1, the locus M_{g,d}^{r} is an irreducible divisor on M_{g}. The class of its compactification \overline{M}_{g,d}^{r} inside \overline{M}_{g} is given by the following formula:

\overline{M}_{g,d}^{r} ≡ c_{g,d,r} ((g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i=1}^{[g/2]} i(g − i)δ_{i}) ∈ Pic(\overline{M}_{g}).

The constant c_{g,d,r} has a clear intersection-theoretic interpretation using Schubert calculus. Note that remarkably, the slope of all the Brill-Noether divisors on \overline{M}_{g} is independent of d and r and

s(\overline{M}_{g,d}^{r}) = 6 + \frac{12}{g + 1}

for all r, d ≥ 1 satisfying ρ(g, r, d) = −1. For genera g such that g + 1 is composite, one has as many Brill-Noether divisors on \overline{M}_{g} as ways of non-trivially factoring g + 1. It is natural to raise the following:

Problem 3.2. Construct an explicit linear equivalence between various Brill-Noether divisors \overline{M}_{g,d}^{r} on \overline{M}_{g} for different integers r, d ≥ 1 with ρ(g, r, d) = −1.

The simplest case is g = 11 when there exist two (distinct) Brill-Noether divisors \overline{M}_{11,6}^{1} and \overline{M}_{11,9}^{2} and s(\overline{M}_{11,6}^{1}) = s(\overline{M}_{11,9}^{2}) = 7. These divisors can be understood in terms of Noether-Lefschetz divisors on the moduli space \overline{F}_{11} of polarized K3 surfaces of degree 2g − 2 = 20. We recall that there exists a rational ℙ^{11}-fibration

ϕ : \overline{M}_{11} −−→ \overline{F}_{11}, ϕ[C] := [S, O_{S}(C)],

where S is the unique K3 surface containing C, see [M94]. Noting that M_{11,6}^{1} = M_{11,14}^{5} it follows that

M_{11,6}^{1} = ϕ_{|M_{11}}^{∗}(NL_{1}),

where NL_{1} is the Noether-Lefschetz divisor on F_{11} of polarized K3 surfaces S with Picard lattice Pic(S) = ℤ · [O_{S}(1)] + ℤ · [C], where C^{2} = 20 and C · c_{1}(O_{S}(1)) = 14. Similarly, by Riemann-Roch, we have an equality of divisors M_{11,9}^{2} = M_{11,11}^{3}, and then

M_{11,9}^{2} = ϕ_{|M_{11}}^{∗}(NL_{2}),

with NL_{2} being the Noether-Lefschetz divisor whose general point corresponds to a quartic surface S ⊂ ℙ^{3} with Pic(S) = ℤ · [O_{S}(1)] + ℤ · [C], where C^{2} = 20 and C · c_{1}(O_{S}(1)) = 11. It is not clear whether NL_{1} and NL_{2} should be linearly equivalent on F_{11}.

The next interesting case is g = 23, see [Fa00]: The three (distinct) Brill-Noether divisors \overline{M}_{23,12}^{1}, \overline{M}_{23,17}^{2} and \overline{M}_{23,20}^{1} are multicanonical in the sense that there exist explicitly known integers m, m_{1}, m_{2}, m_{3} ∈ ℤ_{>0} and an effective boundary divisor E ≡ ∑_{i=1}^{11} c_{i}δ_{i} ∈ Pic(\overline{M}_{23}) such that

m_{1} · \overline{M}_{23,12}^{1} + E ≡ m_{2} · \overline{M}_{23,17}^{2} + E = m_{3} · \overline{M}_{23,20}^{3} + E ∈ |mK_{\overline{M}_{23}}|.

Question 3.3. For a genus g such that g + 1 is composite, is there a good geometric description of the stable base locus

B(\overline{M}_{g}, |\overline{M}_{g,d}^{r}|) := ⋂_{n≥0} Bs(\overline{M}_{g}, |n\overline{M}_{g,d}^{r}|)

of the Brill-Noether linear system? It is clear that B(\overline{M}_{g}, |\overline{M}_{g,d}^{r}|) contains important subvarieties of \overline{M}_{g} like the hyperelliptic and trigonal locus, cf. [HaM90].

Of the higher codimension Brill-Noether cycles, the best understood are the d-gonal loci

M_{g,2}^{1} ⊂ M_{g,3}^{1} ⊂ ⋯ ⊂ M_{g,d}^{1} ⊂ ⋯ ⊂ M_{g}.

Each stratum M_{g,d}^{1} is an irreducible variety of dimension 2g + 2d − 5. The gonality stratification of M_{g}, apart from being essential in the statement of Green’s Conjecture, has often been used for cohomology calculations or for bounding the cohomological dimension and the affine covering number of M_{g}.

3.2. Koszul cycles. Koszul cohomology behaves like the usual cohomology in many regards. Notably, it can be computed in families, see [BG85], or the book [ApN08]:

Theorem 3.4. Let f : X → S be a flat family of projective varieties, parameterized by an integral scheme, L ∈ Pic(X/S) a line bundle and p, q ∈ ℤ. Then there exists a coherent sheaf K_{p,q}(X/S, L) on X and a nonempty Zariski open subset U ⊂ S such that for all s ∈ U one has that K_{p,q}(X/S, L) ⊗ k(s) ≅ K_{p,q}(X_{s}, L_{s}).

In the statement above, the open set U is precisely determined by the condition that all h^{i}(X_{s}, L_{s}) are minimal.

By Theorem 3.4, Koszul cohomology can be used to construct effective determinantal cycles on the moduli spaces of smooth curves. This works particularly well for Koszul cohomology of canonical curves, as h^{i} remain constant over the whole moduli space. More generally, Koszul cycles can be defined over the relative Picard stack over the moduli space. Under stronger assumptions, the canonically defined determinantal structure can be given a better description. To this end, one uses the description provided by Lazarsfeld kernel bundles.

In many cases, for example canonical curves, or 2-regular varieties, the property (N_{p}) reduces to the single condition K_{p,2}(X, L) = 0, see for instance [Ein87] Proposition 3. This phenomenon justifies the study of various loci given by the non-vanishing of K_{p,2}. Note however that extending this determinantal description over the boundary of the moduli stack (especially over the locus of reducible stable curves) poses considerable technical difficulties, see [Fa06a], [Fa06b]. We now describe a general set-up used to compute Koszul non-vanishing loci over a partial compactification M̃_{g} of the moduli space M_{g} inside \overline{M}_{g}. As usual, if M is a Deligne-Mumford stack, we denote by M its associated coarse moduli space.

We fix integers r, d ≥ 1, such that ρ(g, r, d) = 0 and denote by M_{g}^{0} ⊂ M_{g} the open substack classifying curves [C] ∈ M_{g} such that W_{d−1}^{r}(C) = ∅ and W_{d}^{r+1}(C) = ∅. Since ρ(g, r +1, d) ≤ −2 and ρ(g, r, d−1) = −r −1 ≤ −2, it follows from [EH89] that codim(M_{g} − M_{g}^{0}, M_{g}) ≥ 2. We further denote by Δ_{0}^{0} ⊂ Δ_{0} ⊂ \overline{M}_{g} the locus of nodal curves [C_{yq} := C/y ∼ q], where [C] ∈ M_{g−1} is a curve that satisfies the Brill-Noether Theorem and y, q ∈ C are arbitrary distinct points. Finally, Δ_{1}^{0} ⊂ Δ_{1} ⊂ \overline{M}_{g} denotes the open substack classifying curves [C ∪_{y} E], where [C] ∈ M_{g−1} is Brill-Noether general, y ∈ C is an arbitrary point and [E, y] ∈ \overline{M}_{1,1} is an arbitrary elliptic tail. Note that every Brill-Noether general curve [C] ∈ M_{g−1} satisfies

W_{d−1}^{r}(C) = ∅, W_{d}^{r+1}(C) = ∅ and dim W_{d}^{r}(C) = ρ(g − 1, r, d) = r.

We set M̃_{g} := M_{g}^{0} ∪ Δ_{0}^{0} ∪ Δ_{1}^{0} ⊂ \overline{M}_{g} and we regard it as a partial compactification of M_{g}. Then following [EH86] we consider the Deligne-Mumford stack

σ_{0} : 𝔊̃_{d}^{r} → M̃_{g}

classifying pairs [C, l] with [C] ∈ M̃_{g} and l a limit linear series of type 𝔤_{d}^{r} on C. We remark that for any curve [C] ∈ M_{g}^{0} ∪ Δ_{0}^{0} and L ∈ W_{d}^{r}(C), we have that h^{0}(C, L) = r + 1 and that L is globally generated. Indeed, for a smooth curve [C] ∈ M_{g}^{0} it follows that W_{d}^{r+1}(C) = ∅, so necessarily W_{d}^{r}(C) = G_{d}^{r}(C). For a point [C_{yq}] ∈ Δ_{0}^{0} we have the identification

σ_{0}^{−1}[C_{yq}] = {L ∈ W_{d}^{r}(C) : h^{0}(C, L ⊗ O_{C}(−y − q)) = r},

where we note that since the normalization [C] ∈ M_{g−1} is assumed to be Brill-Noether general, any sheaf L ∈ σ_{0}^{−1}[C_{yq}] satisfies

h^{0}(C, L ⊗ O_{C}(−y)) = h^{0}(C, L ⊗ O_{C}(−q)) = r

and h^{0}(C, L) = r + 1. Furthermore, \overline{W} _{d}^{r}(C_{yq}) = W_{d}^{r}(C_{yq}), where the left-hand-side denotes the closure of W_{d}^{r}(C_{yq}) inside the variety \overline{Pic}^{d}(C_{yq}) of torsion-free sheaves on C_{yq}. This follows because a non-locally free torsion-free sheaf in \overline{W} _{d}^{r}(C_{yq}) − W_{d}^{r}(C_{yq}) is of the form ν_{∗}(A), where A ∈ W_{d−1}^{r}(C) and ν : C → C_{yq} is the normalization map. But we know that W_{d−1}^{r}(C) = ∅, because [C] ∈ M_{g−1} satisfies the Brill-Noether Theorem. The conclusion of this discussion is that σ : 𝔊̃_{d}^{r} → M̃_{g} is proper. Since ρ(g, r, d) = 0, by general Brill-Noether theory, there exists a unique irreducible component of 𝔊_{d}^{r} which maps onto M_{g}^{0}.

In [Fa06b], a universal Koszul non-vanishing locus over a partial compactification of the moduli space of curves is introduced. Precisely, one constructs two locally free sheaves A and B over 𝔊̃_{d}^{r} such that for a point [C, l] corresponding to a smooth curve [C] ∈ M_{g}^{0} and a (necessarily complete and globally generated linear

series) l = (L, H^{0}(C, L)) ∈ G_{d}^{r}(C) inducing a map C ⟶^{|L|} ℙ^{r}, we have the following description of the fibres:

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

There is a natural vector bundle morphism ϕ : A → B given by restriction. From Grauert’s Theorem it follows that both A and B are vector bundles over 𝔊_{d}^{r} and from Bott’s Theorem (in the case of A) and Riemann-Roch (in the case of B) respectively, we compute their ranks

rank(A) = (p + 1)\binom{r + 2}{p + 2} and rank(B) = \binom{r}{p}(−\frac{pd}{r} + 2d + 1 − g).

Note that M_{L} is a stable vector bundle (again, one uses that [C] ∈ M_{g}^{0}), hence H^{1}(C, ∧^{p}M_{L} ⊗ L^{⊗2}) = 0 and then rank(B) = χ(C, ∧^{p}M_{L} ⊗ L^{⊗2}) can be computed from Riemann-Roch. We have the following result, cf. [Fa06b] Theorem 2.1:

Theorem 3.5. The cycle

U_{g,p} := {(C, L) ∈ 𝔊_{d}^{r} : K_{p,2}(C, L) ≠ 0},

is the degeneracy locus of the vector bundle map ϕ : A → B over 𝔊_{d}^{r}.

Under suitable numerical restrictions, when rank(A) = rank(B), the cycle constructed in Theorem 3.5 is a virtual divisor on 𝔊_{d}^{r}. This happens precisely when

r := 2s + sp + p, g := rs + s and d := rs + r.

for some p ≥ 0 and s ≥ 1. The first remarkable case occurs when s = 1. Set g = 2p + 3, r = g − 1 = 2p + 2, and d = 2g − 2 = 4p + 4. Note that, since the canonical bundle is the only 𝔤_{2g−2}^{g−1} on a curve of genus g, the Brill-Noether stack is isomorphic to M_{g}. The notable fact that the cycle in question is an actual divisor follows directly from Voisin’s Theorem 2.12 and from Green’s Hyperplane Section Theorem [V05].

Hence Z_{g,p} := σ_{∗}(U_{g,p}) is a virtual divisor on M_{g} whenever
g = s(2s + sp + p + 1).

In the next section, we explain how to extend the morphism ϕ : A → B to a morphism of locally free sheaves over the stack 𝔊̃_{d}^{r} of limit linear series and reproduce the class formula proved in [Fa06a] for the degeneracy locus of this morphism.

3.3. Divisors of small slope. In [Fa06a] it was shown that the determinantal structure of Z_{g,p} can be extended over \overline{M}_{g} in such a way that whenever s ≥ 2, the resulting virtual slope violates the Harris-Morrison Slope Conjecture. One has the following general statement:

Theorem 3.6. If σ : 𝔊̃_{d}^{r} → M̃_{g} denotes the compactification of 𝔊_{d}^{r} given by limit linear series, then there exists a natural extension of the vector bundle morphism ϕ : A → B over 𝔊̃_{d}^{r} such that \overline{Z}_{g,p} 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}(G_{p,2} − H_{p,2})) ≡ aλ − b_{0}δ_{0} − b_{1}δ_{1} − ⋯ − b_{[ \frac{g}{2} ]}δ_{[ \frac{g}{2} ]},

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

s(σ_{∗}(c_{1}(G_{p,2} − H_{p,2}))) = \frac{a}{b_{0}} = 6\frac{f(s, p)}{(p + 2) sh(s, p)}, with

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

50p + 41)s^{2} + (7p^{2} + 18p + 9)s + 2p + 2

and

h(s, p) = (p^{3} + 6p^{2} + 12p + 8)s^{6} + (p^{3} + 2p^{2} − 4p − 8)s^{5} − (p^{3} + 7p^{2} + 11p + 2)s^{4} − − (p^{3} − 5p)s^{3} + (4p^{2} + 5p + 1)s^{2} + (p^{2} + 7p + 11)s + 4p + 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,p} is a divisor on \overline{M}_{g} which gives a counterexample to the Slope Conjecture for g = s(2s + sp + p + 1).

A few remarks are necessary. In the case s = 1 and g = 2p + 3, the vector bundles A and B exist not only over a partial compactification of M̃_{g} but can be extended (at least) over the entire stack M_{g} ∪ Δ_{0} in such a way that B(C, ω_{C}) = H^{0}(C, ∧^{p}M_{ω_{C}} ⊗ ω_{C}^{2}) for any [C] ∈ M_{g} ∪ Δ_{0}. Theorem 3.6 reads in this case, see also [Fa08] Theorem 5.7:

(5) [\overline{Z}_{2p+3,p}]^{virt} = c_{1}(B−A) = \frac{1}{p + 2}\binom{2p}{p}(6(p+3)λ−(p+2)δ_{0}−6(p+1)δ_{1}−⋯),

in particular s([\overline{Z}_{2p+3,p}]^{virt}) = 6 + 12/(g + 1).

Particularly interesting is the case p = 0 when the condition K_{0,2}(C, L) = 0 for [C, L] ∈ 𝔊_{d}^{r}, is equivalent to the multiplication map

ν_{2}(L) : Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})

not being an isomorphism. Note that ν_{2}(L) is a linear map between vector spaces of the same dimension and Z_{g,0} is the failure locus of the Maximal Rank Conjecture:

Corollary 3.7. For g = s(2s+1), r = 2s, d = 2s(s+1) the slope of the virtual class of the locus of those [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 equal to

s(\overline{Z}_{s(2s+1),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)}.

4. Conjectures on Koszul cohomology of curves

4.1. Green’s Conjecture. In what follows, we consider (C, K_{C}) a smooth canonical curve of genus g ≥ 2. In this case, the Duality Theorem 2.7 applies, and the distribution of the numbers b_{p,q} := dim K_{p,q}(C, K_{C}) organized in a table (the Betti table) is the following:

[ \begin{matrix}b_{0,0} & 0 & 0 & … & 0 & 0 \\ 0 & b_{1,1} & b_{2,1} & … & b_{g−3,1} & b_{g−2,1} \\ b_{0,2} & b_{1,2} & b_{2,2} & … & b_{g−3,2} & 0 \\ 0 & 0 & 0 & … & 0 & b_{g−2,3} \end{matrix} ]

The Betti table is symmetric with respect to its center, that is, b_{i,j} = b_{g−2−i,3−j} and all the other entries not marked here are zero.

Trying to apply the Non-Vanishing Theorem 2.4 to the canonical bundle K_{C}, we obtain one condition and one quantity. The condition comes from the hypothesis that for a decomposition K_{C} = L_{1}⊗(K_{C} ⊗L_{1}^{∨}), Theorem 2.4 is applicable whenever

(6)  r_{1} + 1 := h^{0}(C, L_{1}) ≥ 2 and r_{2} + 1 := h^{1}(C, L_{1}) ≥ 2.

A line bundle L_{1} satisfying (6) is said to contribute to the Clifford index of C.

The quantity that appears in Theorem 2.4 is the Clifford index itself. More precisely

r_{1} + r_{2} − 1 = g − Cliff(L_{1}) − 2,

where

Cliff(L_{1}) := deg(L_{1}) − 2h^{0}(L_{1}) + 2.

Clifford’s Theorem [ACGH85] says that Cliff(L_{1}) ≥ 0, and Cliff(L_{1}) > 0 unless L_{1} is a 𝔤_{2}^{1}. Following [GL86] we define the Clifford index of C as the quantity

Cliff(C) := min{Cliff(L_{1}) : L_{1} contributes to the Clifford index of C}.

In general, the Clifford index will be computed by minimal pencils. Specifically, a general d-gonal curve [C] ∈ M_{g,d}^{1} (recall that the gonality strata are irreducible) will have Cliff(C) = d − 2. However, this equality is not valid for all curves, that is, there exist curves [C] ∈ M_{g} with Cliff(C) < gon(C)−2, basic examples being plane curves, or exceptional curves on K3 surfaces. Even in these exotic cases, Coppens and Martens [CM91] established the precise relation Cliff(C) = gon(C) − 3.

Theorem 2.4 implies the following non-vanishing

K_{g−Cliff(C)−2,1}(C, K_{C}) ≠ 0,

and Green’s Conjecture predicts optimality of Theorem 2.4 for canonical curves:

Conjecture 4.1. For any curve [C] ∈ M_{g} we have the vanishing K_{p,1}(C, K_{C}) = 0 for all p ≥ g − Cliff(C) − 1.

In the statement of Green’s Conjecture, it suffices to prove the vanishing of K_{g−Cliff(C)−1,1}(C, K_{C}) or, by duality, that K_{Cliff(C)−1,2}(C, K_{C}) = 0.

We shall analyze some basic cases:

Example 4.2. Looking at the group K_{0,2}(C, K_{C}), Green’s Conjecture predicts that it is zero for all non-hyperelliptic curves. Or, the vanishing of K_{0,2}(C, K_{C}) is equivalent to the projective normality of the canonical curve. This is precisely the content of the classical Max Noether Theorem [ACGH85], p. 117.

Example 4.3. For a non-hyperelliptic curve, we know that K_{1,2}(C, K_{C}) = 0 if and only if the canonical curve C ⊂ ℙ^{g−1} is cut out by quadrics. Green’s Conjecture predicts that K_{1,2}(C, K_{C}) = 0 unless the curve is hyperelliptic, trigonal or a smooth plane quintic. This is precisely the Enriques-Babbage-Petri Theorem, see [ACGH85], p. 124.

Thus Conjecture 4.1 appears as a sweeping generalization of two famous classical theorems. Apart from these classical results, strong evidence has been found for Green’s Conjecture (and one should immediately add, that not a shred of evidence has been found suggesting that the conjecture might fail even for a single curve [C] ∈ M_{g}). For instance, the conjecture is true for general curves in any gonality stratum M_{g,d}^{1}, see [Ap05], [Tei02] and [V02]. The proof of this fact relies on semicontinuity. Since M_{g,d}^{1} is irreducible, it suffices to find one example of a d-gonal curve that satisfies the conjecture, for any 2 ≤ d ≤ (g + 2)/2; here we also need the fact mentioned above, that the Clifford index of a general d-gonal curve is d − 2. The most important and challenging case, solved by Voisin [V05], was the case of curves of odd genus g = 2d − 1 and maximal gonality d + 1. Following Hirschowitz and Ramanan [HR98] one can compare the Brill-Noether divisor M_{g,d}^{1} of curves with a 𝔤_{d}^{1} and the virtual divisor of curves [C] ∈ M_{g} with K_{d−1,1}(C, K_{C}) ≠ 0. The non-vanishing Theorem 2.4 gives a set-theoretic inclusion M_{g,d}^{1} ⊂ Z_{g,d−2}. Now, we compare the class [Z_{g,d−2}]^{virt} ∈ Pic(M_{g}) of the virtual divisor Z_{g,d−2} to the class [M_{g,d}^{1}] computed in [HM82]. One finds the following relation

[Z_{g,d−2}]^{virt} = (d − 1)[M_{g,d}^{1}] ∈ Pic(M_{g})

cf. [HR98]; Theorem 3.6 in the particular case s = 1 provides an extension of this equality to a partial compactification of M_{g}. Green’s Conjecture for general curves of odd genus [V05] implies that Z_{g,d−2} is a genuine divisor on M_{g}. Since a general curve [C] ∈ M_{g,d}^{1} satisfies

dim K_{d−1,1}(C, K_{C}) ≥ d − 1,

cf. [HR98], one finds the set-theoretic equality M_{g,d}^{1} = Z_{g,d−2}. In particular we obtain the following strong characterization of curves of odd genus and maximal gonality:

Theorem 4.4 (Hirschowitz-Ramanan, Voisin). If C is a smooth curve of genus g = 2d − 1 ≥ 7, then K_{d−1,1}(C, K_{C}) ≠ 0 if and only if C carries a 𝔤_{d}^{1}.

Voisin proved Theorem 2.12, using Hilbert scheme techniques, then she applied Green’s Hyperplane Section Theorem [Gr84a] to obtain the desired example of a curve [C] ∈ M_{g} satisfying Green’s Conjecture.

Starting from Theorem 4.4, all the other generic d-gonal cases are obtained in the following refined form, see [Ap05]:

Theorem 4.5. We fix integers g and d ≥ 2 such that 2 ≤ d ≤ [g/2] + 1. For any smooth curve [C] ∈ M_{g} satisfying the condition

(7)  dim W_{g−d+2}^{1}(C) ≤ g − 2d + 2 = ρ(g, 1, g − d + 2),

we have that K_{g−d+1,1}(C, K_{C}) = 0. In particular, C satisfies Green’s Conjecture.

Note that the condition d ≤ [g/2]+1 excludes the case already covered by Theorem 4.4. The proof of Theorem 4.5 relies on constructing a singular stable curve [C′] ∈ \overline{M}_{2g+3−2d} of maximal gonality g + 3 − d (that is, [C′] ∉ \overline{M}_{2g+3−d,g+2−d}^{1}), starting from any smooth curve [C] ∈ M_{g} satisfying (7). The curve C′ is obtained from C by gluing together g +3−2d pairs of general points of C, and then applying an analogue of Theorem 4.4 for singular stable curves, [Ap05], see Section 4.2. The version in question is the following, cf. [Ap05] Proposition 7. The proof we give here is however slightly different:

Theorem 4.6. For any nodal curve [C′] ∈ M_{g′} ∪ Δ_{0}, with g′ = 2d′ − 1 ≥ 7 such that K_{d′−1,1}(C′, ω_{C′}) ≠ 0, it follows that [C′] ∈ \overline{M}_{g′,d′}^{1}.

Proof. By duality, we obtain the following equality of cycles on M̃_{g′}:

\{[C′] : K_{d′−1,1}(C′, ω_{C′}) ≠ 0\} = \{[C′] : K_{d′−2,2}(C′, ω_{C′}) ≠ 0\} =: \overline{Z}_{g′,d′−2}.

Theorem 3.6 shows that this locus is a virtual divisor on M̃_{g′} whose class is given by formula (5) and Theorem 2.12 implies that \overline{Z}_{g′,d′−2} is actually a divisor. Comparing its class against the class of the Hurwitz divisor \overline{M}_{g′,d′}^{1} [HM82], we find that

\overline{Z}_{g′,d′−2} ≡ (d′ − 1)\overline{M}_{g′,d′}^{1} ∈ Pic(M̃_{g′}).

Note that this is a stronger statement than [HR98] Proposition 3.1, being an equality of codimension 1 cycles on the compactified moduli space M̃_{g′}, rather than on M_{g′}. The desired statement follows immediately since for any curve [C′] ∈ M_{g′,d′}^{1} one has dim K_{d′−1,1}(C′, ω_{C′}) ≥ d′ − 1, hence the degeneracy locus \overline{Z}_{g′,d′−2}

contains \overline{M}_{g′,d′}^{1} with multiplicity at least d′ − 1. □

We return to the discussion on Theorem 4.5 (the proof will be resumed in the next subsection). By duality, the vanishing in the statement above can be rephrased as

K_{d−3,2}(C, K_{C}) = 0.

The condition (7) is equivalent to a string of inequalities

dim W_{d+n}^{1}(C) ≤ n

for all 0 ≤ n ≤ g − 2d + 2, in particular gon(C) ≥ d. This condition is satisfied for a general d-gonal curve, cf. [Ap05]. More generally, if [C] ∈ M_{g,d}^{1} is a general d-gonal curve then any irreducible component

Z ≠ W_{d}^{1}(C) + W_{n−d}(C)

of W_{n}^{1}(C) has dimension ρ(g, 1, n). In particular, for ρ(g, 1, n) < 0 it follows that W_{n}^{1}(C) = W_{d}^{1}(C) + W_{n−d}(C) which of course implies (7). For g = 2d − 2, the inequality (7) becomes necessarily an equality and it reads: the curve C carries finitely many 𝔤_{d}^{1}’s of minimal degree.

We make some comments regarding condition (7). Let us suppose that C is non-hyperelliptic and d ≥ 3. From Martens’ Theorem [ACGH85] p.191, it follows that dim W_{g−d+2}^{1}(C) ≤ g − d − 1. Condition (7) requires the better bound g − 2d + 2 ≤ g − d − 1. However, for d = 3, the two bounds are the same, and Theorem 4.5 shows that K_{0,0}(C, K_{C}) = 0, for any non-hyperelliptic curve, which is Max Noether’s Theorem, see also Example 4.2. Applying Mumford’s Theorem [ACGH85] p.193, we obtain the better bound dim W_{g−d+2}^{1}(C) ≤ g − d − 2 for d ≥ 4, unless the curve is trigonal, a smooth plane quintic or a double covering of an elliptic curve. Therefore, if C is not one of the three types listed above, then K_{1,2}(C, K_{C}) = 0, and we recover the of Enriques-Babbage-Petri Theorem, see also Example 4.3 (note, however, the exception made to bielliptic curves).

Keem has improved the dimension bounds for W_{g−d+2}^{1}(C). For d ≥ 5 and C a curve that has neither a 𝔤_{4}^{1} nor is a smooth plane sextic, one has the inequality dim W_{g−d+2}^{1}(C) ≤ g − d − 3, cf. [Ke90] Theorems 2.1 and 2.3. Consequently, Theorem 4.5 implies the following result which is a complete solution to Green’s Conjecture for 5-gonal curves:

Theorem 4.7 (Voisin [V88], Schreyer [Sch91]). If K_{2,2}(C, K_{C}) ≠ 0, then C is hyperelliptic, trigonal, tetragonal or a smooth plane sextic, that is, Cliff(C) ≤ 2.

Geometrically, the vanishing of K_{2,2}(C, K_{C}) is equivalent to the ideal of the canonical curve being generated by quadrics, and the minimal relations among the generators being linear.

Theorem 3.1 from [Ke90] gives the next bound dim W_{g−d+2}^{1}(C) ≤ g −d−4, for d ≥ 6 and C with gon(C) ≥ 6 which does not admit a covering of degree two or three on another curve, and which is not a plane curve. The following improvement of Theorem 4.7 is then obtained directly from Theorem 4.5 and [Ap05] Theorem 3.1:

Theorem 4.8. If g ≥ 12 and K_{3,2}(C, K_{C}) ≠ 0, then C is one of the following: hyperelliptic, trigonal, tetragonal, pentagonal, double cover of an genus 3 curve, triple cover of an elliptic curve, smooth plane septic. In other words, if Cliff(C) ≥ 4 then K_{3,2}(C, K_{C}) = 0.

Theorem 4.8 represents the solution to Green’s Conjecture for hexagonal curves. Likewise, Theorem 4.5 can be used together with the Brill-Noether theory to prove Green’s Conjecture for any gonality d and large genus. The idea is to apply Coppens’ results [Co83].

Theorem 4.9. If g ≥ 10 and d ≥ 5 are two integers such that g > (d−2)(2d−7), and C is any d-gonal curve of genus g which does not admit any morphism of degree less than d onto another smooth curve, then Cliff(C) = d−2 and Green’s Conjecture is verified for C, i.e. K_{g−d+1,1}(C, K_{C}) = 0.

The statement of Conjecture 4.1 (meant as a vanishing result), is empty for hyperelliptic curves is, hence the interesting cases begin with d ≥ 3.

It remains to verify Green’s Conjecture for curves which do not verify (7). One result in this direction was proved in [ApP06].

Theorem 4.10. Let S be a K3 surface with Pic(S) = ℤ · H ⊕ Z · ℓ, with H very ample, H^{2} = 2r − 2 ≥ 4, and H · ℓ = 1. Then any smooth curve in the linear system |2H + ℓ| verifies Green’s conjecture.

Smooth curves in the linear system |2H + ℓ| count among the few known examples of curves whose Clifford index is not computed by pencils, i.e. Cliff(C) = gon(C)−3, [ELMS89] (other obvious examples are plane curves, for which Green’s Conjecture was verified before, cf. [Lo89]). Such curves are the most special ones in the moduli space of curves from the point of view of the Clifford dimension. Hence, this case may be considered as opposite to that of a general curve of fixed gonality. Note that these curves carry a one-parameter family of pencils of minimal degree, hence the condition (7) is not satisfied.

4.2. The Gonality Conjecture. The Green-Lazarsfeld Gonality Conjecture [GL86] predicts that the gonality of a curve can be read off the Koszul cohomology with values in any sufficiently positive line bundle.

Conjecture 4.11 (Green-Lazarsfeld). Let C be a smooth curve of gonality d, and L a sufficiently positive line bundle on C. Then

K_{h^{0}(L)−d,1}(C, L) = 0.

Theorem 2.8 applied to L written as a sum of a minimal pencil and the residual bundle yields

K_{h^{0}(L)−d−1,1}(C, L) ≠ 0.

Note that if L is sufficiently positive, then the Green-Lazarsfeld Nonvanishing Theorem is optimal when applied for a decomposition where one factor is a pencil. Indeed, consider any decomposition L = L_{1} ⊗ L_{2} with r_{1} = h^{0}(C, L_{1}) − 1 ≥ 2, and r_{2} = h^{0}(C, L_{2}) − 1 ≥ 2. Since L is sufficiently positive, the linear system |K_{C}^{⊗2} ⊗ L^{∨}| is empty, and finiteness of the addition map of divisors shows that at least one of the two linear systems |K_{C} ⊗ L_{i}^{∨}| is empty. Suppose |K_{C} ⊗ L_{2}^{∨}| = ∅, choose a point x ∈ C−Bs(|L_{1}|) and consider a new decomposition L = L′_{1}⊗L′_{2}, with L′_{1} = L_{1} ⊗O_{C}(−x), and L′_{2} = L_{2} ⊗O_{C}(x). Denoting as usual r′_{i} = h^{0}(C, L′_{i})−1, we find that r′_{1} + r′_{2} − 1 = r_{1} + r_{2} − 1, whereas r′_{1} = r_{1} − 1, and L′_{2} is again non-special. We can apply an inductive argument until r_{1} becomes 1. Hence the Gonality Conjecture predicts the optimality of the Green-Lazarsfeld Nonvanishing Theorem. However, one major disadvantage of this statement is that “sufficiently positive” is not a precise condition. It was proved in [Ap02] that by adding effective divisors to bundles that verify the Gonality Conjecture we obtain again bundles that verify the conjecture. Hence, in order to find a precise statement for Conjecture 4.11 one has to study the edge cases.

In most generic cases (general curves in gonality strata, to be more precise), the Gonality Conjecture can be verified for line bundles of degree 2g, see [ApV03] and [Ap05]. The test bundles are obtained by adding two generic points to the canonical bundle.

Theorem 4.12 ([Ap05]). For any d-gonal curve [C] ∈ M_{g} with d ≤ [g/2] + 2 which satisfies the condition (7), and for general points x, y ∈ C, we have that K_{g−d+1,1}(C, K_{C} ⊗ O_{C}(x + y)) = 0.

The case not covered by Theorem 4.12 is slightly different. A general curve C of odd genus carries infinitely many minimal pencils, hence a bundle of type K_{C}⊗O_{C}(x+y) can never verify the vanishing predicted by the Gonality Conjecture. Indeed, for any two points x and y there exists a minimal pencil L_{1} such that H^{0}(C, L_{1}(−x − y)) ≠ 0, and we apply Theorem 2.4. However, adding three points to the canonical bundle solves the problem, cf. [Ap04], [Ap05].

Theorem 4.13. For any curve [C] ∈ M_{2d−1} of maximal gonality gon(C) = d+ 1 and for general points x, y, z ∈ C, we have that K_{d,1}(C, K_{C} ⊗O_{C}(x+y +z)) = 0.

The proofs of Theorems 4.5, 4.12 and 4.13 are all based on the same idea. We start with a smooth curve C and construct a stable curve of higher genus out of it, in such a way that the Koszul cohomology does not change. Then we apply a version of Theorem 4.4 for singular curves. Proof of Theorems 4.5 and 4.12. We start with [C] ∈ M_{g} satisfying the condition (7). We claim that if we choose δ := g + 3 − 2d pairs of general points x_{i}, y_{i} ∈ C for 1 ≤ i ≤ δ, then the resulting stable curve

[C′ := \frac{C}{x_{1} ∼ y_{1}, …, x_{δ} ∼ y_{δ}} ] ∈ \overline{M}_{2g+3−2d}

is a curve of maximal gonality, that is, g + 3 − d. Indeed, otherwise [C′] ∈ \overline{M}_{2g+3−2d,g+2−d}^{1} and this implies that there exists a degree g + 2 − d admissible covering f : C̃ → R from a nodal curve C̃ that is semi-stably equivalent to C′, onto a genus 0 curve R. The curve C is a subcurve of C̃ and if deg(f_{|C}) = n ≤ g + 2 − d, then it follows that f_{|C} induces a pencil 𝔤_{n}^{1} such that f_{|C}(x_{i}) = f_{|C}(y_{i}) for 1 ≤ i ≤ δ. Since the points x_{i}, y_{i} ∈ C are general, this implies that dim W_{n}^{1}(C)+δ ≥ 2δ, which contradicts (7).

To conclude, apply Theorem 4.6 and use the following inclusions, [V02], [ApV03]:

K_{g−d+1,1}(C, K_{C}) ⊂ K_{g−d+1,1}(C, K_{C}(x + y)) ⊂ K_{g−d+1,1}(C′, ω_{C′}).
□

Remark 4.14. The proofs of Theorems 4.5, 4.6 and 4.12 indicate an interesting phenomenon, completely independent of Voisin’s proof of the generic Green Conjecture. They show that Green’s Conjecture for general curves of genus g = 2d − 1 and maximal gonality d + 1 is equivalent to the Gonality Conjecture for bundles of type K_{C} ⊗ O_{C}(x + y) for general pointed curves [C, x, y] ∈ M_{2d−2,2}. We refer to [Ap02] and [ApV03] for further implications between the two conjectures, in both directions.

Proof of Theorem 4.13. For C as in the hypothesis, and for general points x, y, z ∈ C, we construct a stable curve [C′] ∈ M_{2d+1} by adding a smooth rational component passing through the points x, y and z. Using admissible covers one can show, as in the proofs of Theorems 4.5 and 4.12, that C′ is of maximal gonality, that is d + 2. From Theorem 4.6, we obtain K_{d,1}(C′, ω_{C′}) = 0. The conclusion follows from the observation that K_{d,1}(C, K_{C} ⊗ O_{C}(x + y + z)) ≅ K_{d,1}(C′, ω_{C′}). □

It is natural to ask the following:

Question 4.15. For a curve C and points x, y ∈ C, can one give explicit conditions on Koszul cohomology ensuring that x + y is contained in a fiber of a minimal pencil?

We prove here the following result, which can be considered as a precise version of the Gonality Conjecture for generic curves.

Theorem 4.16. Let [C] ∈ M_{2d−2}, and x, y ∈ C arbitrarily chosen distinct points. Then K_{d−1,1}(C, K_{C} ⊗O_{C}(x+y)) ≠ 0 if and only if there exists A ∈ W_{d}^{1}(C) such that h^{0}(C, A ⊗ O_{C}(−x − y)) ≠ 0.

Proof. Suppose there exists A ∈ W_{d}^{1}(C) such that h^{0}(C, A(−x − y)) ≠ 0. Theorem 2.8 applied to the decomposition K_{C} ⊗ O_{C}(x + y) = A ⊗ B, with B = K_{C}(x+y)⊗A^{∨} produces nontrivial classes in the group K_{d−1,1}(C, K_{C} ⊗O_{C}(x+y)).

For the converse, we consider C′ the stable curve obtained from C by gluing together the points x and y and denote by ν : C → C′ the normalization morphism. Clearly [C′] ∈ \overline{M}_{2d−1}. We observe that

K_{d−1,1}(C, K_{C} ⊗ O_{C}(x + y)) ≅ K_{d−1,1}(C′, ω_{C′}).

From Theorem 4.6, it follows that [C′] ∈ \overline{M}_{2d−1,d}^{1}, hence there exists a map

f : C̃ ⟶^{d:1} R

from a curve C̃ semistably equivalent to C′ onto a rational nodal curve R. The curve C is a subcurve of C̃ and f_{|C} provides the desired pencil. □

As mentioned above, the lower possible bound for explicit examples of line bundles that verify the Gonality Conjecture found so far was 2g. One can raise the question whether this bound is optimal or not and the sharpest statement one can make is Conjecture 1.4 discussed in the introduction of this paper.

5. The Strong Maximal Rank Conjecture

Based mainly on work carried out in [Fa06a] and [Fa06b] we propose a conjecture predicting the resolution of an embedded curve with general moduli. This statement unifies two apparently unrelated deep results in the theory of algebraic curves: The Maximal Rank Conjecture which predicts the number of hypersurfaces of each degree containing a general embedded curve C ⊂ ℙ^{r} and Green’s Conjecture on syzygies of canonical curves.

We begin by recalling the statement of the classical Maximal Rank Conjecture. The modern formulation of this conjecture is due to Harris [H82] p. 79, even though it appears that traces of a similar statement can be found in the work of Max Noether: We fix integers g, r and d such that ρ(g, r, d) ≥ 0 and denote by ℑ_{d,g,r} the unique component of the Hilbert scheme Hilb_{d,g,r} of curves C ⊂ ℙ^{r} with Hilbert polynomial h_{C}(t) = dt + 1 − g, containing curves with general moduli. In other words, the variety ℑ_{d,g,r} is characterized by the following properties:

(1) The general point [C ↪ ℙ^{r}] ∈ ℑ_{d,g,r} corresponds to a smooth curve C ⊂ ℙ^{r} with deg(C) = d and g(C) = g.

(2) The moduli map m : ℑ_{d,g,r} −−→ M_{g}, m([C ↪ ℙ^{r}]) := [C] is dominant.

Conjecture 5.1. (Maximal Rank Conjecture) A general embedded smooth curve [C ↪ P^{r}] ∈ ℑ_{d,g,r} is of maximal rank, that is, for all integers n ≥ 1 the restriction maps

ν_{n}(C) : H^{0}(ℙ^{r}, O_{ℙ^{r}}(n)) → H^{0}(C, O_{C}(n))

are of maximal rank, that is, either injective or surjective.

Thus if a curve C ⊂ ℙ^{r} lies on a hypersurface of degree d, then either hypersurfaces of degree d cut out the complete linear series |O_{C}(d)| on the curve, or else C is special in its Hilbert scheme. Since C can be assumed to be a Petri general curve, it follows that H^{1}(C, O_{C}(n)) = 0 for n ≥ 2, so h^{0}(C, O_{C}(n)) = nd + 1 − g and Conjecture 5.1 amounts to knowing the Hilbert function of C ⊂ ℙ^{r}, that is, the value of h^{0}(ℙ^{r}, I_{C/ℙ^{r}}(n)) for all n.

Example 5.2. We consider the locus of curves C ⊂ ℙ^{3} with deg(C) = 6 and g(C) = 3 that lie on a quadric surface, that is, ν_{2}(C) fails to be an isomorphism. Such curves must be of type (2, 4) on the quadric, in particular, they are hyperelliptic. This is a divisorial condition on ℑ_{6,3,3}, that is, for a general [C ↪ ℙ^{3}] ∈ ℑ_{6,3,3} the map ν_{2}(C) is an isomorphism.

Conjecture 5.1 makes sense of course for any component of ℑ_{d,g,r} but is known to fail outside the Brill-Noether range, see [H82]. The Maximal Rank Conjecture is known to hold in the non-special range, that is when d ≥ g + r, due to work of Ballico and Ellia relying on the méthode d’Horace of Hirschowitz, see [BE87]. Vosin has also proved cases of the conjecture when h^{1}(C, O_{C}(1)) = 2, cf. [V92]. Finally, Conjecture 5.1 is also known in the case ρ(g, r, d) = 0 when it has serious implications for the birational geometry of \overline{M}_{g}. This case can be reduced to the case when dim Sym^{n}H^{0}(C, O_{C}(1)) = dim H^{0}(C, O_{C}(n)), that is,

\binom{n + r}{n} = nd + 1 − g,

when Conjecture 5.1 amounts to constructing one smooth curve [C ↪ ℙ^{r}] ∈ ℑ_{d,g,r} such that H^{0}(ℙ^{r}, I_{C/ℙ^{r}}(n)) = 0. In this situation, the failure locus of Conjecture 5.1 is precisely the virtual divisor Z_{g,0} on M_{g} whose geometry has been discussed in Section 3, Corollary 3.7. The most interesting case (at least from the point of view of slope calculations) is that of n = 2. One has the following result [Fa06b] Theorem 1.5:

Theorem 5.3. For each s ≥ 1 we fix integers

g = s(2s + 1), r = 2s and d = 2s(s + 1),

hence ρ(g, r, d) = 0. The locus

Z_{g,0} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such that ν_{2}(L) : Sym^{2}H^{0}(C, L) →^{≇} H^{0}(C, L^{⊗2})}

is an effective divisor on M_{g}. In particular, a general curve [C] ∈ M_{g} satisfies the Maximal Rank Conjecture with respect to all linear series L ∈ W_{d}^{r}(C).

For s = 1 we have the equality Z_{3,0} = M_{3,2}^{1}, and we recover the hyperelliptic locus on M_{3}. The next case, s = 2 and g = 10, has been treated in detail in [FaPo05]. One has a scheme-theoretic equality Z_{10,0} = 42 · K_{10} on M_{10}, where 42 = #(W_{12}^{4}(C)) is the number of minimal pencils 𝔤_{6}^{1} = K_{C}(−𝔤_{12}^{4}) on a general curve [C] ∈ M_{10}. Thus a curve [C] ∈ M_{10} fails the Maximal Rank Conjecture for a linear series L ∈ W_{12}^{4}(C) if and only if it fails it for all the 42 linear series 𝔤_{12}^{4}! This incarnation of the K3 divisor K_{10} is instrumental in being able to compute the class of \overline{K}_{10} on \overline{M}_{10}, cf. [FaPo05].

In view of Theorem 5.3 it makes sense to propose a much stronger form of Conjecture 5.1, replacing the generality assumption of [C ↪ ℙ^{r}] ∈ ℑ_{d,g,r} by a generality assumption of [C] ∈ M_{g} with respect to moduli and asking for the maximal rank of the curve with respect to all linear series 𝔤_{d}^{r}.

We fix positive integers g, r, d such that g − d + r ≥ 0 and satisfying

0 ≤ ρ(g, r, d) < r − 2.

We also fix a general curve [C] ∈ M_{g}. The numerical assumptions imply that all the linear series l ∈ G_{d}^{r}(C) are complete (the inequality ρ(g, r + 1, d) < 0 is satisfied), as well as very ample. For each (necessarily complete) linear series l = (L, H^{0}(C, L)) ∈ G_{d}^{r}(C) and integer n ≥ 2, we denote by

ν_{n}(L) : Sym^{n}H^{0}(C, L) → H^{0}(C, L^{⊗n})

the multiplication map of global sections. We then choose a Poincaré line bundle on C × Pic^{d}(C) and construct two vector bundles E_{n} and F_{n} over G_{d}^{r}(C) with rank(E_{n}) = \binom{r+n}{n} and rank(F_{n}) = h^{0}(C, L^{⊗n}) = nd + 1 − g, together with a bundle morphism ϕ_{n} : E_{n} → F_{n}, such that for L ∈ G_{d}^{r}(C) we have that

E_{n}(L) = Sym^{n}H^{0}(C, L) and F_{n}(L) = H^{0}(C, L^{⊗n})

and ν_{n}(L) is the map given by multiplication of global sections.

Conjecture 5.4. (Strong Maximal Rank Conjecture) We fix integers g, r, d ≥ 1 and n ≥ 2 as above. For a general curve [C] ∈ M_{g}, the determinantal variety

Σ_{n,g,d}^{r}(C) := {L ∈ G_{d}^{r}(C) : ν_{n}(L) is not of maximal rank}

has expected dimension, that is,

dim Σ_{n,g,d}^{r}(C) = ρ(g, r, d) − 1 − |rank(E_{n}) − rank(F_{n})|,

where by convention, negative dimension means that Σ_{n,g,d}^{r}(C) is empty.

For instance, in the case ρ(g, r, d) < nd + 2 − g − \binom{r+n}{n}, the conjecture predicts that for a general [C] ∈ M_{g} we have that Σ_{n,g,d}^{r}(C) = ∅, that is,

H^{0}(ℙ^{r}, I_{C/ℙ^{r}}(n)) = 0

for every embedding C ↪^{|L|} ℙ^{r} given by L ∈ G_{d}^{r}(C).

When ρ(g, r, d) = 0 (and in particular whenever r ≤ 3), using a standard monodromy argument showing the uniqueness the component ℑ_{d,g,r}, the Strong Maximal Rank Conjecture is equivalent to Conjecture 5.1, and it states that ν_{n}(L)

is of maximal rank for a general [C ↪^{L} ℙ^{r}] ∈ ℑ_{d,g,r}.

For ρ(g, r, d) ≥ 1 however, Conjecture 5.4 seems to be a more difficult question than Conjecture 5.1 because one requires a way of seeing all linear series L ∈ G_{d}^{r}(C) at once.

Remark 5.5. The bound ρ(g, r, d) < r − 2 in the statement of Conjecture 5.4 implies that all linear series L ∈ G_{d}^{r}(C) on a general curve C, are very ample.

Remark 5.6. We discuss Conjecture 5.4 when r = 4 and ρ(g, r, d) = 1. The conjecture is trivially true for g = 1. The first interesting case is g = 6 and d = 9. For a general curve [C] ∈ M_{6} we observe that there is an isomorphism C ≅ W_{9}^{4}(C) given by C ∋ x ↦ K_{C} ⊗ O_{C}(−x). Since rank(E_{2}) = 15 and rank(F_{2}) = 13, Conjecture 5.4 predicts that,

ν_{2}(K_{C}(−x)) : Sym^{2}H^{0}(C, K_{C} ⊗ O_{C}(−x)) ↠ H^{0}(C, K_{C}^{⊗2} ⊗ O_{C}(−2x)),

for all x ∈ C, which is true (use the Base-Point-Free Pencil Trick).

The next case is g = 11, d = 13, when the conjecture predicts that the map

ν_{2}(L) : Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2})

is injective for all L ∈ W_{13}^{4}(C). This follows (non-trivially) from [M94]. Another case that we checked is r = 5, g = 14 and d = 17, when ρ(14, 5, 17) = 2.

Proposition 5.7. The Strong Maximal Rank Conjecture holds for general nonspecial curves, that is, when r = d − g.

Proof. This is an immediate application of a theorem of Mumford’s stating that for any line bundle L ∈ Pic^{d}(C) with d ≥ 2g + 1, the map ν_{2}(L) is surjective, see e.g. [GL86]. The condition ρ(g, r, d) < r − 2 forces in the case r = d − g the inequality d ≥ 2g + 3. Since the expected dimension of Σ_{n,g,d}^{d−g}(C) is negative, the conjecture predicts that Σ_{n,g,d}^{d−g}(C) = ∅. This is confirmed by Mumford’s result. □

5.1. The Minimal Syzygy Conjecture. Interpolating between Green’s Conjecture for generic curves (viewed as a vanishing statement) and the Maximal Rank Conjecture, it is natural to expect that the Koszul cohomology groups of line bundles on a general curve [C] ∈ M_{g} should be subject to the vanishing suggested by the determinantal description provided by Theorem 3.5. For simplicity we restrict ourselves to the case ρ(g, r, d) = 0:

Conjecture 5.8. We fix integers r, s ≥ 1 and set d := rs + r and g := rs + s, hence ρ(g, r, d) = 0. For a general curve [C] ∈ M_{g} and for every integer

0 ≤ p ≤ \frac{r − 2s}{s + 1}

we have the vanishing K_{p,2}(C, L) = 0, for every linear series L ∈ W_{d}^{r}(C).

As pointed out in Theorem 3.5, in the limiting case p = \frac{r−2s}{s+1} ∈ ℤ, Conjecture 5.8 would imply that the failure locus

Z_{g,p} := {[C] ∈ M_{g} : ∃L ∈ W_{d}^{r}(C) such that K_{p,2}(C, L) ≠ ∅}

is an effective divisor on M_{g} whose closure \overline{Z}_{g,p} violates the Slope Conjecture.

Conjecture 5.8 generalizes Green’s Conjecture for generic curves: When s = 1, it reads like K_{p,2}(C, K_{C}) = 0 for g ≥ 2p+3, which is precisely the main result from [V05]. Next, in the case p = 0, Conjecture 5.8 specializes to Theorem 5.3. The conjecture is also known to hold when s = 2 and g ≤ 22 (cf. [Fa06a] Theorems 2.7 and 2.10).

References

[Ap02] Aprodu, M.: On the vanishing of higher syzygies of curves. Math. Zeit., 241, 1–15 (2002)

[Ap04] Aprodu, M.: Green-Lazarsfeld gonality Conjecture for a generic curve of odd genus. Int. Math. Res. Notices, 63, 3409–3414 (2004)

[Ap05] Aprodu, M.: Remarks on syzygies of d-gonal curves. Math. Res. Lett., 12, 387–400 (2005)

[ApN08] Aprodu, M., Nagel, J.: Koszul cohomology and algebraic geometry, Max Planck Institut preprint 08-52, (2008).

[ApP06] Aprodu, M., Pacienza, G.: The Green Conjecture for Exceptional Curves on a K3 Surface. Int. Math. Res. Notices (2008) 25 pages.

[ApV03] Aprodu, M., Voisin, C.: Green-Lazarsfeld’s Conjecture for generic curves of large gonality. C.R.A.S., 36, 335–339 (2003)

[ACGH85] Arbarello, E., Cornalba, M., Griffiths, P. A., Harris, J.: Geometry of algebraic curves, Volume I. Grundlehren der mathematischen Wissenschaften 267, Springer-Verlag (1985)

[AC81a] Arbarello, E., Cornalba, M.: Footnotes to a paper of Beniamino Segre. Math. Ann., 256, 341–362 (1981)

[AC81b] Arbarello E ., Cornalba, M.: Su una congettura di Petri. Comment. Math. Helv. 56, no. 1, 1–38 (1981)

[BE87] Ballico, E., Ellia, P.: The maximal rank Conjecture for nonspecial curves in ℙ^{n}. Math. Zeit. 196, no.3, 355–367 (1987)

[BG85] Boratyńsky M., Greco, S.: Hilbert functions and Betti numbers in a flat family. Ann. Mat. Pura Appl. (4), 142, 277–292 (1985)

[Co83] Coppens, M.: Some sufficient conditions for the gonality of a smooth curve. J. Pure Appl. Algebra 30, no. 1, 5–21 (1983).

[CM91] Coppens, M., Martens, G.: Secant spaces and Clifford’s Theorem, Compositio Math., 78, 193–212 (1991)

[Ein87] Ein, L.: A remark on the syzygies of the generic canonical curves. J. Differential Geom. 26, 361–365 (1987)

[Ei92] Eisenbud, D.: Green’s Conjecture: An orientation for algebraists. In Free Resolutions in Commutative Algebra and Algebraic Geometry, Boston 1992.

[Ei06] Eisenbud, D.: Geometry of Syzygies. Graduate Texts in Mathematics, 229, Springer Verlag (2006)

[EH86] Eisenbud, D., Harris, J., Limit linear series: basic theory, Invent. Math. 85 (1986), 337-371

[EH87] Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus ≥ 23. Invent. Math. 90, no. 2, 359–387 (1987)

[EH89] Eisenbud, D., Harris, J: Irreducibility of some families of linear series with Brill-Noether number −1, Ann. Sci. École Normale Superieure 22 (1989), 33–53.

[ELMS89] Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compositio Math. 72, 173–204 (1989)

[EGL] Ellingsrud, G., Göttsche, L., Lehn, M.: On the cobordism class of the Hilbert scheme of a surface. J. Algebraic Geom. 10, 81-100 (2001).

[FaL08] Farkas, G., Ludwig, K.: The Kodaira dimension of the moduli space of Prym varieties. J. European Math. Soc 12, 755-795 (2010).

[FaPo05] Farkas, G., Popa, M.: Effective divisors on \overline{M}_{g}, curves on K3 surfaces, and the Slope Conjecture. J. Algebraic Geom. 14, 241–267 (2005)

[Fa00] Farkas, G.: The geometry of the moduli space of curves of genus 23, Math. Ann. 318, 43-65 (2000).

[Fa06a] Farkas, G.: Syzygies of curves and the effective cone of \overline{M}_{g}. Duke Math. J., 135, No. 1, 53–98 (2006)

[Fa06b] Farkas, G.: Koszul divisors on moduli space of curves. American J. Math. 131 (2009), 819-867.

[Fa08] Farkas, G.: Aspects of the birational geometry of \overline{M}_{g}. Surveys in Differential Geometry Vol. 14 (2010), 57-111.

[Gr84a] Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differential Geom., 19, 125-171 (1984)

[Gr84b] Green, M.: Koszul cohomology and the geometry of projective varieties. II. J. Differential Geom., 20, 279–289 (1984)

[Gr89] Green, M.: Koszul cohomology and geometry. In: Cornalba, M. (ed.) et al., Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 177–200 (1989)

[GL84] Green, M., Lazarsfeld, R.: The nonvanishing of certain Koszul cohomology groups. J. Differential Geom., 19, 168–170 (1984)

[GL86] Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math., 83, 73–90 (1986)

[H82] Harris, J: Curves in projective space, Presses de l’Université de Montréal 1982.

[HM82] Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. Math., 67, 23–86 (1982)

[HaM90] Harris, J., Morrison, I.: Slopes of effective divisors on the moduli space of stable curves. Invent. Math., 99, 321–355 (1990)

[HR98] Hirschowitz, A., Ramanan, S.: New evidence for Green’s Conjecture on syzygies of canonical curves. Ann. Sci. École Norm. Sup. (4), 31, 145–152 (1998)

[Ke90] Keem, C.: On the variety of special linear systems on an algebraic curve. Math. Ann. 288, 309–322 (1990)

[La89] Lazarsfeld, R.: A sampling of vector bundle techniques in the study of linear series. In: Cornalba, M. (ed.) et al., Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 500–559 (1989)

[Lo89] Loose, F.: On the graded Betti numbers of plane algebraic curves. Manuscripta Math., 64, 503–514 (1989)

[Ma82] Martens, G.: Über den Clifford-Index algebraischer Kurven. J. Reine Angew. Math., 336, 83–90 (1982)

[M94] Mukai, S.: Curves and K3 surfaces of genus eleven. Moduli of vector bundles (Sanda 1994; Kyoto 1994), 189-197, Lecture Notes in Pure and Appl. Math. 1996.

[PR88] Paranjape, K., Ramanan, S.: On the canonical ring of a curve. Algebraic geometry and commutative algebra, Vol. II, 503–516, Kinokuniya, Tokyo, 1988.

[Sch86] Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann., 275, 105–137 (1986)

[Sch89] Schreyer, F.-O.: Green’s Conjecture for general p-gonal curves of large genus. Algebraic curves and projective geometry, Trento, 1988, Lecture Notes in Math., 1389, Springer, Berlin-New York 254–260 (1989)

[Sch91] Schreyer, F.-O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math., 421, 83–123 (1991)

[St98] Steffen, F.: A generalized principal ideal Theorem with applications to Brill-Noether theory. Invent. Math. 132 73-89 (1998)

[Tei02] Teixidor i Bigas, M.: Green’s Conjecture for the generic r-gonal curve of genus g ≥ 3r − 7. Duke Math. J., 111, 195–222 (2002)

[V88] Voisin, C.: Courbes tétragonales et cohomologie de Koszul. J. Reine Angew. Math., 387, 111–121 (1988)

[V92] Voisin, C.: Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri. Acta Mathematica 168, 249-272 (1992).

[V93] Voisin, C.: Déformation des syzygies et théorie de Brill-Noether. Proc. London Math. Soc. (3) 67 no. 3, 493–515 (1993).

[V02] Voisin, C.: Green’s generic syzygy Conjecture for curves of even genus lying on a K3 surface. J. European Math. Soc., 4, 363–404 (2002).

[V05] Voisin, C.: Green’s canonical syzygy Conjecture for generic curves of odd genus. Compositio Math., 141 (5), 1163–1190 (2005).

Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, RO-014700 Bucharest, and Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, RO-010702 Bucharest, Romania

E-mail address: aprodu@imar.ro

Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin, Germany

E-mail address: farkas@math.hu-berlin.de