# The Rank of Syzygies of Canonical Curves

Michael Kemeny  University of Wisconsin-Madison, Department of Mathematics,
480 Lincoln Dr
WI 53706, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that the linear syzygy spaces of a general canonical curve are
spanned by syzygies of minimal rank.

##  0\. Introduction

The goal of this paper is to study the rank of the syzygies amongst the
equations defining a canonical curve.

Following Mumford’s influential work [M], one typically studies projective
varieties $X\subseteq{\mathbf{P}}_{\mathbb{C}}^{n}$ under the assumption that
the varieties are _defined by quadrics_ , or, in in algebraic terms, the
homogeneous ideal $I_{X}\subseteq\mathbb{C}[x_{0},\ldots,x_{n}]$ is generated
in degree two. This assumption is often satisfied in cases of interest. For
example, this assumption is satisfied for canonical curves of genus $g$ and
Clifford index at least two by the classical Petri Theorem [AS].

Consider a variety $X\subseteq{\mathbf{P}}_{\mathbb{C}}^{n}$ which is defined
by quadrics. It is natural to ask what is the minimal $r$ one can find such
that $X$ is defined by quadrics of rank $r$. For instance, Mumford proved in
[M, Thm. 1], that if $i:X\hookrightarrow{\mathbf{P}}^{n}$ is any projective
variety of degree $d_{0}$, and if we compose $i$ with the $d^{th}$ Veronese
embedding
$v_{d}:{\mathbf{P}}^{n}\hookrightarrow{\mathbf{P}}^{\binom{n+d}{d}-1}$, then
the new embedding

$X\hookrightarrow{\mathbf{P}}^{\binom{n+d}{d}-1}$

is defined by quadrics of rank at most four for $d\geq d_{0}$. This result has
been subsequently extended in interesting ways, see [SS] and [HLMP].

In the case of canonical curves of genus $g$ and Clifford index at least two,
Andreotti–Mayer [AM] and Arbarello–Harris [AH] proved that
$C\subseteq{\mathbf{P}}^{g-1}$ is defined by quadrics of rank at most four,
provided the curve is general. This was extended to arbitrary curves by Green
[Gr3].

In foundational work [Gr2], Green put the classical study of the projective
geometry of varieties into a much wider context. Instead of merely studying
the ideal $I_{X}\subseteq S:=\mathbb{C}[x_{0},\ldots,x_{n}]$, Green had the
insight that one should consider the entire _minimal free resolution_

$\ldots F_{2}\to F_{1}\to I_{X}\to 0,$

of the $S$-module $I$. Much of the _intrinsic_ geometry of the projective
variety reveals itself through the _extrinsic_ invariants encoded in the
resolution. The celebrated Green’s Conjecture postulates that one can read the
Clifford index of a curve $C$ off from the dimensions of the graded pieces of
the modules $F_{i}$ appearing in the resolution of the ideal of a canonical
curve $C\subseteq{\mathbf{P}}^{g-1}$.

Following Green’s philosophy of generalizing from an ideal $I_{X}$ to the
resolution $F_{\bullet}\to I_{X}$, Schreyer and von Bothmer defined a notion
of rank that applies to linear generators of the modules $F_{i}$. Let $X$ be a
projective variety, embedded via a very ample line bundle $L$, and assume that
$(X,L)$ is _normally generated_ , [M]. We decompose the free modules $F_{i}$
into their graded pieces by writing

$F_{i}=\bigoplus_{j\geq 1}\mathrm{K}_{i,j}(X,L)\otimes_{\mathbb{C}}S(-i-j),$

where the vector space $\mathrm{K}_{i,j}(X,L)$, called the $(i,j)^{th}$ Koszul
cohomology group. For any $\alpha\neq 0\in\mathrm{K}_{p,1}(X,L)$, we have a
well-defined notion of _rank_ , see Definition 1.5.

The following conjecture of Schreyer simultaneously unifies and generalises
both Voisin’s Theorem on generic Green’s Conjecture and the
Andreotti–Mayer–Arbarello–Harris Theorem on the ideal sheaf of a canonical
curve:

######  Conjecture 0.1 (The Geometric Syzygy Conjecture).

For a general curve of genus $g$, all linear syzygy spaces
$\mathrm{K}_{p,1}(C,\omega_{C})$ are spanned by syzygies of minimal rank
$p+1$.

Conjecture 0.1 was proven for curves of genus $g\leq 8$ in [vB4]. We
previously proved an important special case of Conjecture 0.1, namely the case
when $g=2k$ has even genus and $p=k-1$ is the _last_ nonzero linear syzygy
space, see [Ke3, Thm. 0.2].

The goal of this paper is to prove Conjecture 0.1 in full generality:

######  Theorem 0.2.

Conjecture 0.1 holds for a general curve of genus $g\geq 8$.

We always have the bound $\rm{rank}(\alpha)\geq p+1$ for
$\alpha\in\mathrm{K}_{p,1}(X,L)$. Thus Conjecture 0.1 amounts to the statement
that the spaces $K_{p,1}(C,\omega_{C})$ are generated by syzygies of the
lowest possible rank. Syzygies of lowest rank have a geometric interpretation,
[vB3], [AN1]. Indeed, if $\rm{rank}(\alpha)=p+1$, then $X$ lies on a rational
normal scroll

$X_{\alpha}\subseteq{\mathbf{P}}(H^{0}(L)^{\vee}),$

called the _syzygy scheme_ , which is constructed from $\alpha$ in an explicit
way, [Gr1]. The syzygy $\alpha$ then arises from a minimal rank syzygy at the
end of the resolution of $X_{\alpha}$, and as such has an explicit
description, see [Sch] and [FK2, §0.2]. For instance, when $p=1$, syzygies of
rank two provide linear determinantal equations for $C$ in the sense of [SS],
and in particular are quadrics of rank four.

Similarly, syzygies of rank $p+2$ arise from linear sections of Grassmannians.
Most general constructions of syzygies produce low rank syzygies. In
particular, an influential construction of Green–Lazarsfeld [Gr2] produces
syzygies $\alpha\in\mathrm{K}_{p,1}(X,L)$ of rank at most $p+2$ out of a
decomposition $L=L_{1}\otimes L_{2}$ with $h^{0}(L_{i})\geq 2$ for $i=1,2$,
[AN2, §3.4.2].

By applying the Green–Lazarsfeld construction to decompositions
$\omega_{C}\simeq L_{1}\otimes L_{2}$ of the canonical bundle of a curve,
Green obtained his famous conjecture predicting the vanishing of linear syzygy
spaces $K_{p,1}(C,\omega_{C})$ in terms of the intrinsic geometry of a curve.
Green’s Conjecture has been proven for general curves by Voisin in [V1], [V2];
see also [AFPRW] and [Ke3] for simpler proofs.

We end the introduction with a few words on the proof of Conjecture 0.1. Our
basic strategy is the technique of _projection of syzygies_ , originally
developed by Aprodu [AN2, §2.2] and further refined in the paper [Ke2]. This
method provides an inductive method to prove results on syzygies of a smooth
curves $C$. One first identifies pairs of points to create a nodal curve $D$,
for which the claim is covered by the base case of the induction. One then
projects from each node and uses formulae describing how syzygies change under
this operation. In our case the base case is provided by [Ke3].

In order to implement this strategy, several new tools are required. These
tools are likely to be useful in other contexts. In particular, we develop a
novel, _explicit_ formula for the projection map, see Proposition 3.1.
Moreover, we require an explicit formula for the construction of minimal rank
syzygies out of line bundles, see Proposition 1.9. Our formula has the feature
that it may be adapted to torsion free sheaves on singular curves which admit
embeddings into a surface, see Section 2. We then apply these formulae to
nodal curves which arise as deformations of (singular) curves on K3 surfaces,
see Section 4. A considerable amount of technical difficulty results from the
fact that we may not assume that the minimal pencils on such curves come from
line bundles, but are instead required to work with general torsion-free
sheaves.

Acknowledgements I thank Daniel Erman for comments on a draft.

##  1\. Rank of a Syzygy

###  1.1. Preliminaries

We begin by gathering a few preliminaries.

######  Lemma 1.1.

Let $X$ be a scheme which is locally of finite type over an algebraically
closed field $k$. Let $\phi:E\to F$ be a morphism of vector bundles. Assume
that, for any closed point $x\in X$, $\phi(x):E\otimes k(x)\to F\otimes k(x)$
is injective. Then $G:=\mathrm{Coker}(\phi)$ is locally free.

###### Proof.

Without loss of generality we may assume that $X=\mathrm{Spec}(A)$ is affine.
The dual map $\phi(x)^{\vee}$ is surjective for all $x\in X$. By Nakayama’s
Lemma, the morphism of sheaves $\phi^{\vee}$ is surjective. Consider
$\widetilde{G}:=\mathrm{Ker}(\phi^{\vee})$ which fits into the short exact
sequence

(1) $0\to\widetilde{G}\to F^{\vee}\to E^{\vee}\to 0.$

Then $\widetilde{G}$ is locally free, since $F^{\vee}$ and $E^{\vee}$ are,
[H1, Ex. III.6.5]. Dualizing (1) gives the claim. ∎

###  1.2. Notation and Background on Syzygies

If $V$ is a vector space and $M$ is a graded $S_{V}:=\mathrm{Sym}(V)$ module,
write $\mathrm{K}_{p,q}(M,V)$ for the middle cohomology of

$\bigwedge^{p+1}V\otimes M_{q-1}\to\bigwedge^{p}V\otimes
M_{q}\to\bigwedge^{p-1}V\otimes M_{q+1}.$

For any projective variety $X$, with line bundle $L$ and coherent sheaf $F$,
we define a graded $S_{L}:=\mathrm{Sym}(\mathrm{H}^{0}(X,L))$ module

$\Gamma_{X}(F,L):=\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}(X,L^{\otimes
q}\otimes F).$

Define
$\mathrm{K}_{p,q}(X,F,L):=\mathrm{K}_{p,q}(\Gamma_{X}(F,L),\mathrm{H}^{0}(X,L))$.
When $F=\mathcal{O}_{X}$ is trivial, we write $\Gamma_{X}(L)$ for
$\Gamma_{X}(\mathcal{O}_{X},L)$ and $\mathrm{K}_{p,q}(X,L)$ for
$\mathrm{K}_{p,q}(X,\mathcal{O}_{X},L)$. For any subspace
$W\subseteq\mathrm{H}^{0}(X,L)$, we may consider $\Gamma_{X}(F,L)$ as a
$S_{W}:=\mathrm{Sym}(W)$ module. Define
$\mathrm{K}_{p,q}(X,F,L;W):=\mathrm{K}_{p,q}(\Gamma_{X}(F,L),W)$. If
$F=\mathcal{O}_{X}$, we write $\mathrm{K}_{p,q}(X,L;W)$ for
$\mathrm{K}_{p,q}(X,\mathcal{O}_{X},L;W)$. We further define

$b_{p,q}(X,F,L;W):=\dim\mathrm{K}_{p,q}(X,F,L;W).$

The following proposition is a slight refinement of [AN2, Prop. 1.29, Cor.
1.31].

######  Proposition 1.2 (Semicontinuity of Koszul cohomology).

Suppose $\pi:\mathcal{X}\to S$ is a flat, projective morphism of finite-type
schemes over $\mathbb{C}$ and assume $S$ is integral. Let $\mathcal{L}$,
$\mathcal{F}$ be coherent sheaves on $\mathcal{X}$, flat over $S$, with
$\mathcal{L}$ a line bundle. Let $\mathcal{Z}\subseteq\mathcal{X}$ be a closed
subscheme, flat over $S$. Fix integers $p,q$. Assume

$h^{0}(\mathcal{X}_{S},\mathcal{L}_{s}\otimes
I_{\mathcal{Z}_{s}}),\;h^{0}(\mathcal{X}_{S},\mathcal{F}_{s}\otimes\mathcal{L}^{q-1}_{s}),\;h^{0}(\mathcal{X}_{S},\mathcal{F}_{s}\otimes\mathcal{L}^{q}_{s}),\;h^{0}(\mathcal{X}_{S},\mathcal{F}_{s}\otimes\mathcal{L}^{q+1}_{s})$

are all constant for all closed points $s\in S$. Then the function
$\Psi:S\to\mathbb{Z}$ with

$\displaystyle\Psi(s):=b_{p,q}\left(\mathcal{X}_{s},\mathcal{F}_{s},\mathcal{L}_{s};\mathrm{H}^{0}(\mathcal{X}_{S},\mathcal{L}_{s}\otimes
I_{\mathcal{Z}_{s}})\right)$

is upper semicontinuous.

###### Proof.

Without loss of generality, we may take $S=\mathrm{Spec}(R)$ to be affine. Set

$\displaystyle\mathcal{A}:=\pi_{*}(\mathcal{L}\otimes
I_{\mathcal{Z}}),\;\;\mathcal{B}_{i}:=\pi_{*}(\mathcal{F}\otimes\mathcal{L}^{\otimes
i}),\;i\in\\{q-1,q,q+1\\}.$

Then $\mathcal{A},\mathcal{B}_{q-1},\mathcal{B}_{q},\mathcal{B}_{q+1}$ are all
vector bundles on $S$ by Grauert’s Theorem, and we have a complex

$\bigwedge^{p+1}\mathcal{A}\otimes\mathcal{B}_{q-1}\xrightarrow{\delta_{1}}\bigwedge^{p}\mathcal{A}\otimes\mathcal{B}_{q}\xrightarrow{\delta_{2}}\bigwedge^{p-1}\mathcal{A}\otimes\mathcal{B}_{q+1}$

which specializes to the Koszul complex on $\mathcal{X}_{s}$ for any closed
point $s\in S$. We have

$\displaystyle\Psi(s)$ $\displaystyle:=\dim\mathrm{Ker}(\delta_{2}\otimes
k(s))-\dim\mathrm{Im}(\delta_{1}\otimes k(s))$
$\displaystyle=\mathrm{rk}(\wedge^{p}\mathcal{A}\otimes\mathcal{B}_{q})-\dim\mathrm{Im}(\delta_{2}\otimes
k(s))-\dim\mathrm{Im}(\delta_{1}\otimes k(s)).$

It suffices now to show that, for a morphism
$\Phi\;:\;\mathcal{O}_{S}^{a}\to\mathcal{O}_{S}^{b}$ of finitely-generated,
free modules, the function $s\mapsto\mathrm{rk}(\Phi\otimes k(s))$ is lower
semicontinuous. Indeed, the set $\\{s\in S\;:\;\mathrm{rk}(\Phi\otimes
k(s))\leq n\\}$ is closed with ideal defined by the entries of the matrix
$\wedge^{n}\Phi$, for any integer $n$. ∎

###  1.3. Rank of a Syzygy

We call a pair $(X,L)$ a polarized variety if $X$ is an integral, projective
variety and $L$ a very ample line bundle on $X$. We call $(X,L)$ _normally
generated_ if the map
$\mathrm{Sym}^{n}\mathrm{H}^{0}(X,L)\to\mathrm{H}^{0}(X,L^{\otimes n})$ is
surjective for all $n\geq 0$.

######  Proposition 1.3.

Let $(X,L)$ be normally generated. For any subspace
$W\subseteq\mathrm{H}^{0}(X,L)$ the natural map
$\mathrm{K}_{p,1}(X,L;W)\to\mathrm{K}_{p,1}(X,L)$ is injective for all $p\geq
0$.

###### Proof.

Set $V:=\mathrm{H}^{0}(L)$ and
${\mathbf{P}}:={\mathbf{P}}(\mathrm{H}^{0}(X,L))$. We have the exact Koszul
complex

$\ldots\to\bigwedge^{2}W\otimes S_{V}(-2)\to W\otimes S_{V}(-1)\to S_{V}\to
S_{V/W}\to 0$

of graded $S_{V}:=\mathrm{Sym}(V)$ modules, [P, §I.14]. Twisting this complex
appropriately, we see

$\wedge^{s+1}W\otimes\mathrm{Sym}^{t-1}(V)\to\wedge^{s}W\otimes\mathrm{Sym}^{t}(V)\to\wedge^{s-1}W\otimes\mathrm{Sym}^{t+1}(V)$

is exact for all $(s,t)\neq(0,0)$ and so $\mathrm{K}_{p,q}(S_{V},W)=0$ for
$(p,q)\neq(0,0)$.

By the assumption that $(X,L)$ is normally generated, we have an exact
sequence

$0\to\Gamma_{{\mathbf{P}}}(\mathcal{O}_{{\mathbf{P}}}(1),I_{X/{\mathbf{P}}})\to\Gamma_{{\mathbf{P}}}(\mathcal{O}_{{\mathbf{P}}}(1))\to\Gamma_{X}(L)\to
0$

of $S_{V}=\Gamma_{{\mathbf{P}}}(\mathcal{O}_{{\mathbf{P}}}(1))$ modules. We
obtain an exact sequence

$\to\mathrm{K}_{p,1}(S_{V},V)\to\mathrm{K}_{p,1}(X,L)\to\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1))\to\mathrm{K}_{p-1,2}(S_{V},V)\to.$

Thus
$\mathrm{K}_{p,1}(X,L)\simeq\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1))$.
Likewise, considering the short exact sequence as a sequence of $S_{W}$
modules,
$\mathrm{K}_{p,1}(X,L;W)\simeq\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1);W)$.

It remains to show that the natural map
$\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1);W)\to\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1))$
is injective. Since $\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(1)\otimes
I_{X/{\mathbf{P}}})=0$

$\displaystyle\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1);W)$
$\displaystyle=\mathrm{Ker}(\wedge^{p-1}W\otimes\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(2)\otimes
I_{X/{\mathbf{P}}})\to\wedge^{p-2}W\otimes\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(3)\otimes
I_{X/{\mathbf{P}}}),)$
$\displaystyle\mathrm{K}_{p-1,2}({\mathbf{P}},I_{X/{\mathbf{P}}},\mathcal{O}_{{\mathbf{P}}}(1))$
$\displaystyle=\mathrm{Ker}(\wedge^{p-1}V\otimes\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(2)\otimes
I_{X/{\mathbf{P}}})\to\wedge^{p-2}V\otimes\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(3)\otimes
I_{X/{\mathbf{P}}}))$

and the claim follows from the inclusion
$\wedge^{p-1}W\subseteq\wedge^{p-1}V$. ∎

The following proposition will be useful.

######  Proposition 1.4.

Let $(X,L)$ be a projective variety and let $W\subseteq\mathrm{H}^{0}(X,L)$ be
a space of sections generating $L$. Let $F$ be a coherent sheaf of $X$. Then
$\mathrm{K}_{p,q}(X,F,L;W)=0$ for $p\geq\dim W-1$ and any $q$.

###### Proof.

Let $M_{W}$ be the kernel bundle fitting into the exact sequence

$0\to M_{W}\to W\otimes\mathcal{O}_{X}\xrightarrow{ev}L\to 0,$

where $W\otimes\mathcal{O}_{X}\xrightarrow{ev}L$ is the evaluation map. We
have

$\mathrm{K}_{p,q}(X,F,L;W)=\mathrm{Ker}\left(\mathrm{H}^{1}(X,\bigwedge^{p+1}M_{W}\otimes
F\otimes L^{q-1})\to\bigwedge^{p+1}W\otimes\mathrm{H}^{1}(X,F\otimes
L^{q-1})\right),$

[AN2, Remark 2.6]. Since $\mathrm{rk}(M_{W})=\dim W-1$, the claim follows. ∎

We may now state the definition of rank of a syzygy.

######  Definition 1.5.

Let $(X,L)$ be normally generated and $\alpha\neq 0\in\mathrm{K}_{p,1}(X,L)$.
We define the rank of $\alpha$ as the dimension of the smallest subspace
$V\subseteq\mathrm{H}^{0}(X,L)$ such that
$\alpha\in\mathrm{K}_{p,1}(X,L;V)\subseteq\mathrm{K}_{p,1}(X,L)$.

Our definition of rank comes from [AN2, Chapter 3]. When $X$ is integral, for
any $\alpha\neq 0\in\mathrm{K}_{p,1}(X,L)$ we have the inequality
$\rm{rank}(\alpha)\geq p+1,$ [vB3]. The rank of a syzygy may alternatively be
described in terms of the maps in the linear free resolution of
$\Gamma_{X}(L)$. If

$\ldots\to
F_{2}\xrightarrow{\partial_{2}}F_{1}\xrightarrow{\partial_{1}}F_{0}\to\Gamma_{X}(L)\to
0$

is the minimal free resolution, then $F_{i}\simeq\oplus_{j\geq
1}K_{i,j}(X,L)\otimes_{\mathbb{C}}S(-i-j)$ for $i\geq 1$. Restricting the
$p^{th}$ map

$\partial_{p}\;:\;F_{p}\to F_{p-1}$

in the resolution to $K_{p,1}(X,L)\otimes S(-p-1)$ and composing with the
projection $F_{p-1}\to K_{p-1,1}(X,L)\otimes S(-p)$, we obtain a map
$\mathrm{K}_{p,1}(X,L)\otimes S(-p-1)\to\mathrm{K}_{p-1,1}(X,L)\otimes S(-p).$
Taking the strand of degree $p+1$ we obtain a linear map

$\partial^{\ell}_{p}\;:\;\mathrm{K}_{p,1}(X,L)\to\mathrm{K}_{p-1,1}(X,L)\otimes_{\mathbb{C}}\mathrm{H}^{0}(X,L),$

of vector spaces.

We may describe the rank of a syzygy in terms of the complex valued matrix
$\partial^{\ell}_{p}$. Indeed this was von Bothmer’s original definition of
rank of a syzygy [vB4, Definition 3.9]. Recall that we have a natural
identification

$\mathrm{K}_{p,1}(X,L)\simeq\mathrm{Ker}(\bigwedge^{p-1}\mathrm{H}^{0}(X,L)\otimes
I_{2}\to\bigwedge^{p-2}\mathrm{H}^{0}(X,L)\otimes I_{3}),$

from the proof of Proposition 1.3, where $I:=\mathrm{Ker}(S\to\Gamma_{X}(L))$
is the homogeneous ideal of $X$.

######  Theorem 1.6 ([EG], Appendix to Section 1).

Let $\alpha=\sum_{i}\sigma_{i}\otimes
Q_{i}\in\mathrm{Ker}(\bigwedge^{p-1}\mathrm{H}^{0}(X,L)\otimes
I_{2}\to\bigwedge^{p-2}\mathrm{H}^{0}(X,L)\otimes I_{3})$. Then

$\partial^{\ell}_{p}(\alpha)=\sum_{i}d\sigma_{i}\otimes
Q_{i}\in\wedge^{p-1}\mathrm{H}^{0}(X,L)\otimes\mathrm{H}^{0}(X,L)\otimes
I_{2},$

where
$d:\wedge^{p-1}\mathrm{H}^{0}(X,L)\to\wedge^{p-1}\mathrm{H}^{0}(X,L)\otimes\mathrm{H}^{0}(X,L)$
is the differential.

We first need a simple lemma.

######  Lemma 1.7.

Let $V$ be a finite-dimensional, complex vector space with subspace
$W\subseteq V$. Let $d:\wedge^{p}V\to\wedge^{p-1}V\otimes V$ be the
differential and let $\sigma\in\wedge^{p}V$. Then
$d\sigma\in\wedge^{p-1}V\otimes W$ if and only if $\sigma\in\wedge^{p}W$.

###### Proof.

Let $\\{e_{1},\ldots e_{m}\\}$, where $m=\dim W$, be a basis for $W$ and
extend it to a basis $\\{e_{1},\ldots,e_{n}\\}$, $n=\dim V$ of $V$. For all
integers $\ell$ let $I_{\ell}\subseteq\\{1,\ldots,m\\}^{\times\ell}$ denote
the set of tuples $\bar{v}=(v_{1},\ldots,v_{\ell})$ with
$v_{1}<v_{2}\ldots<v_{\ell}$. For $\bar{v}\in\\{1,\ldots,m\\}^{\times\ell}$,
we use the notation $e_{\bar{v}}$ for the element $e_{v_{1}}\wedge\ldots\wedge
e_{v_{\ell}}$, where $v_{i}$ denotes the $i^{th}$ component of $\bar{v}$. We
denote by $\\{\bar{v}\\}\subseteq\\{1,\ldots,m\\}$ the set
$\\{v_{1},\ldots,v_{\ell}\\}$ and define

$\bar{v}_{j}:=(v_{1},\ldots,v_{j-1},v_{j+1},\ldots v_{\ell})\in I_{\ell-1}.$

The set $\\{e_{\bar{v}}\;|\;\bar{v}\in I_{p}\\}$ forms a basis for
$\wedge^{p}V$. Let $\sigma=\sum_{\bar{v}\in
I_{p}}\alpha_{\bar{v}}\,e_{\bar{v}}\in\wedge^{p}V,$ with
$\alpha_{\bar{v}}\in\mathbb{C}$ for all $\bar{v}\in I_{p}$. Then

$\displaystyle d\sigma=\sum_{\bar{v}\in
I_{p}}\sum_{j=1}^{p}(-1)^{j}\alpha_{\bar{v}}\,e_{\bar{v}_{j}}\otimes
e_{v_{j}}=\sum_{\bar{u}\in
I_{p-1}}e_{\bar{u}}\otimes\sum_{i\in\\{1,\ldots,m\\}\setminus\\{\bar{u}\\}}\pm\alpha_{\bar{u}\cup\\{i\\}}e_{i},$

where, for $i\in\\{1,\ldots,m\\}\setminus\\{\bar{u}\\}$, we denote by
$\bar{u}\cup\\{i\\}\in I_{p}$ the unique element in $I_{p}$ with underlying
set $\\{\bar{u}\cup\\{i\\}\\}=\\{\bar{u}\\}\cup\\{i\\}$. Thus
$d\sigma\in\wedge^{p-1}V\otimes W$ if and only if $\alpha_{\bar{v}}=0$ for all
$\bar{v}\in I_{p}$ with a component $v_{i}>m$ which occurs if and only if
$\sigma\in\wedge^{p}W$. ∎

We may now characterize the rank of a syzygy in terms of the map
$\partial^{\ell}_{p}:\mathrm{K}_{p,1}(X,L)\to\mathrm{K}_{p-1,1}(X,L)\otimes_{\mathbb{C}}\mathrm{H}^{0}(X,L)$.

######  Proposition 1.8.

Let $(X,L)$ be a normally generated, polarized variety. Let $\alpha\neq
0\in\mathrm{K}_{p,1}(X,L)$. Then $\alpha$ has rank $\leq r$ if and only if
there exists a vector space $V\subseteq\mathrm{H}^{0}(X,L)$ of dimension at
most $r$ and
$\partial^{\ell}_{p}(\alpha)\in\mathrm{K}_{p-1,1}(X,L)\otimes_{\mathbb{C}}V$.

###### Proof.

By definition $\alpha$ has rank $\leq r$ if and only if there exists a
subspace $V\subseteq\mathrm{H}^{0}(X,L)$ of dimension at most $r$ with
$\alpha\in\mathrm{K}_{p,1}(X,L;V)$. As in the proof of Proposition 1.3, we
have a natural identification
$\mathrm{K}_{p,1}(X,L)\simeq\mathrm{Ker}(\wedge^{p-1}\mathrm{H}^{0}(X,L)\otimes
I_{2}\to\wedge^{p-2}\mathrm{H}^{0}(X,L)\otimes I_{3}),$ and $\alpha$ has rank
$\leq r$ if and only if there exists a subspace
$V\subseteq\mathrm{H}^{0}(X,L)$ of dimension at most $r$ with
$\alpha\in\wedge^{p-1}V\otimes I_{2}$. By Lemma 1.7 and Theorem 1.6, this is
equivalent to
$\partial^{\ell}_{p}(\alpha)\in\wedge^{p-1}\mathrm{H}^{0}(X,L)\otimes V\otimes
I_{2}$. ∎

The rank of a syzygy $\alpha$ can be thought of as a measure of complexity.
Syzygies of minimal rank $p+1$ are the simplest, and all such syzygies can be
constructed explicitly [vB3], [AN2, CH. 3]. Precisely, all such syzygies are
the restriction of syzygies of a rational normal scroll. Alternatively, at
least if $X$ is smooth, all minimal rank syzygies arise from the
Green–Lazarsfeld construction of syzygies, [Gr2, Appendix A]. Indeed, if
$\alpha\neq 0\in\mathrm{K}_{p,1}(X,L;W)$ with $\dim W=p+1$, then by
Proposition 1.4, $W$ cannot be base-point free. The base-locus contains a
divisor $D\subseteq X$ moving in a pencil and $\alpha$ arises from the
Green–Lazarsfeld construction applied to $D,L(-D)$.

######  Proposition 1.9.

Let $(X,L)$ be a projective variety. Let $M$ be a line bundle on $X$ with
$h^{0}(X,M)=2$ and $h^{0}(X,L\otimes M^{\vee})=p+1$. Pick $s\neq
0\in\mathrm{H}^{0}(M)$. Assume $L\otimes M^{\vee}$ is base-point free. Then we
have a natural isomorphism

$\mathrm{K}_{p,1}(X,L;\mathrm{H}^{0}(L\otimes
M^{\vee}))\simeq\bigwedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee}))\otimes\frac{\mathrm{H}^{0}(M)}{\mathbb{C}\langle s\rangle}.$

In particular, $\dim\mathrm{K}_{p,1}(X,L;\mathrm{H}^{0}(L\otimes
M^{\vee}))=1.$

###### Proof.

The syzygy space $\mathrm{K}_{p,1}(X,L;\mathrm{H}^{0}(L\otimes M^{\vee}))$ is
the middle cohomology of

$\bigwedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})\xrightarrow{f}\bigwedge^{p}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(L)\xrightarrow{g}\bigwedge^{p-1}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(L^{2}).$

The map $\wedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})\xrightarrow{f}\wedge^{p}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(L)\subseteq\wedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)$
factors through
$d:\wedge^{p+1}\mathrm{H}^{0}(L)\to\wedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)$
via the inclusion $\mathrm{H}^{0}(L\otimes
M^{\vee})\subseteq\mathrm{H}^{0}(X,L)$ induced by $s$. The differential $d$ is
injective since if
$\wedge:\wedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)\to\wedge^{p+1}\mathrm{H}^{0}(L)$
is the wedge map then $\wedge\circ d=(p+1)Id$. Thus $f$ is injective and we
have an isomorphism

$\mathrm{K}_{p,1}(X,L;\mathrm{H}^{0}(L\otimes
M^{\vee}))\simeq\frac{\mathrm{Ker}(g)}{\wedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})}.$

We have $\mathrm{K}_{p,1}(X,M,L\otimes M^{\vee})=0$ by Proposition 1.4. Thus
the sequence

$\bigwedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(M)\xrightarrow{f^{\prime}}\bigwedge^{p}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(L)\xrightarrow{g^{\prime}}\bigwedge^{p-1}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(L^{2}\otimes M^{\vee})$

is exact. Further $\mathrm{K}_{p+1,0}(X,M,L\otimes
M^{\vee})=\mathrm{Ker}(f^{\prime})$ since $\wedge^{p+2}\mathrm{H}^{0}(L\otimes
M^{\vee})=0$. Thus $f^{\prime}$ is injective by Proposition 1.4. Thus
$\mathrm{Ker}(g)=\mathrm{Ker}(g^{\prime})\simeq\wedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(M).$ This gives

$\mathrm{K}_{p,1}(X,L;\mathrm{H}^{0}(L\otimes
M^{\vee}))\simeq\frac{\wedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})\otimes\mathrm{H}^{0}(M)}{\wedge^{p+1}\mathrm{H}^{0}(L\otimes
M^{\vee})},$

as required. ∎

We now recall some facts about rational normal scrolls, [Sch]. Consider the
locally free sheaf

$\mathcal{E}=\mathcal{O}_{{\mathbf{P}}^{1}}(e_{1})\oplus\ldots\oplus\mathcal{O}_{{\mathbf{P}}^{1}}(e_{d})$

of rank $d$ on ${\mathbf{P}}^{1}$ with the assumptions

$e_{1}\geq e_{2}\geq\ldots\geq e_{d}\geq
0\;\;\text{and}\;\;f:=e_{1}+\ldots+e_{d}\geq 2.$

Set $X:={\mathbf{P}}(\mathcal{E})$ and let $\pi:X\to{\mathbf{P}}^{1}$ denote
the projection. Then $\mathcal{O}_{X}(1)$ is base point free and the
associated morphism $j:X\to{\mathbf{P}}^{r}:={\mathbf{P}}$, $r=f+d-1$, has
image $j(X)\subseteq{\mathbf{P}}^{r}$ a rational normal scroll of minimal
degree $f$ in ${\mathbf{P}}^{r}$, [EH]. We let $H:=\mathcal{O}_{X}(1)$ and
$R:=\pi^{*}\mathcal{O}_{{\mathbf{P}}^{1}}(1)$. Then $H$ and $R$ generate the
integral Picard group of $X$.

The syzygy spaces $\mathrm{K}_{p,q}(X,H)$ may be described explicitly, [Sch].
In particular, $b_{p,q}(X,H)=0$ for $q\geq 2$, whereas
$b_{p,1}(X,H)=p\binom{f}{p+1}$. It follows readily from the fact that $H$ and
$R$ generate $\mathrm{Pic}(X)$ that $H-R$ is base-point free for $f\geq 2$. We
have $h^{0}(X,H-R)=h^{0}(X,H)-d=f$ and $h^{0}(X,R)=2$. Proposition 1.9 shows
that

$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\frac{\mathrm{H}^{0}(X,R)}{<s>}\simeq\mathrm{K}_{f-1,1}(X,H;\mathrm{H}^{0}(X,H-R)).$

Note further that it makes sense to talk about the rank of syzygies in
$\mathrm{K}_{p,1}(X,H)$, even if $H$ is not very ample, since we have a
natural isomorphism
$\mathrm{K}_{p,1}(X,H)\simeq\mathrm{K}_{p,1}(j(X),\mathcal{O}_{j(X)}(1))$ and
the scroll $j(X)\subseteq{\mathbf{P}}^{r}$ is normally generated.

######  Lemma 1.10.

Let $X:={\mathbf{P}}(\mathcal{E})$, for
$\mathcal{E}=\mathcal{O}_{{\mathbf{P}}^{1}}(e_{1})\oplus\ldots\oplus\mathcal{O}_{{\mathbf{P}}^{1}}(e_{d})$
with

$e_{1}\geq e_{2}\geq\ldots\geq e_{d}\geq
0\;\;\text{and}\;\;f:=e_{1}+\ldots+e_{d}\geq 2.$

We have a well-defined morphism
$\phi:{\mathbf{P}}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right)\to{\mathbf{P}}(\mathrm{K}_{f-1,1}(X,H))$
with

$\displaystyle\phi([s]):=[\,\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\frac{\mathrm{H}^{0}(X,R)}{<s>}\,].$

Further,
$\phi^{*}\mathcal{O}_{{\mathbf{P}}(\mathrm{K}_{f-1,1}(X,H))}(1)\simeq\mathcal{O}_{{\mathbf{P}}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right)}(f-2).$

###### Proof.

Set
${\mathbf{P}}:={\mathbf{P}}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right)$.
The dual of the evaluation morphism
$\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(1))\otimes\mathcal{O}_{{\mathbf{P}}}\twoheadrightarrow\mathcal{O}_{{\mathbf{P}}}(1)$
gives an inclusion
$i:\mathcal{O}_{{\mathbf{P}}}(-1)\hookrightarrow\mathrm{H}^{0}(\mathcal{O}_{X}(R))\otimes\mathcal{O}_{{\mathbf{P}}}$.
At the fibre $i\otimes k([s])$ at $[s]\in{\mathbf{P}}$ is the inclusion
$<s>\subseteq\mathrm{H}^{0}(X,R)$. Let
$F:=\mathrm{Coker}(i)\simeq\mathcal{O}_{{\mathbf{P}}}(1)$ and consider the
line bunde

$L:=\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes F(1-f),$

which has degree $2-f$. We have the differential

$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\mathrm{H}^{0}(X,R)\to\wedge^{f-1}\mathrm{H}^{0}(X,H-R)\otimes\mathrm{H}^{0}(X,H).$

Twisting by $\mathcal{O}_{{\mathbf{P}}}(1-f)$ we obtain a morphism

$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\mathrm{H}^{0}(X,R)\otimes\mathcal{O}_{{\mathbf{P}}}(1-f)\to\wedge^{f-1}(\mathrm{H}^{0}(X,H-R)\otimes\mathcal{O}_{{\mathbf{P}}}(-1))\otimes\mathrm{H}^{0}(X,H).$

Using the composition
$\mathrm{H}^{0}(H-R)\otimes\mathcal{O}_{{\mathbf{P}}}(-1)\to\mathrm{H}^{0}(H-R)\otimes\mathrm{H}^{0}(R)\otimes\mathcal{O}_{{\mathbf{P}}}\to\mathrm{H}^{0}(H)\otimes\mathcal{O}_{{\mathbf{P}}},$
we get a morphism

$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\mathrm{H}^{0}(X,R)\otimes\mathcal{O}_{{\mathbf{P}}}(1-f)\to\wedge^{f-1}\mathrm{H}^{0}(X,H)\otimes\mathrm{H}^{0}(X,H)\otimes\mathcal{O}_{{\mathbf{P}}},$

which induces a morphism

$\delta\;:\;\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\mathrm{H}^{0}(X,R)\otimes\mathcal{O}_{{\mathbf{P}}}(1-f)\to\frac{\wedge^{f-1}\mathrm{H}^{0}(X,H)\otimes\mathrm{H}^{0}(X,H)}{\wedge^{f}\mathrm{H}^{0}(X,H)}\otimes\mathcal{O}_{{\mathbf{P}}}.$

By the proof of Proposition 1.9, $\delta$ vanishes on the subbundle
$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\mathcal{O}_{{\mathbf{P}}}(-f)$ and
hence induces a morphism

$\delta^{\prime}\;:\;L\to\frac{\wedge^{f-1}\mathrm{H}^{0}(X,H)\otimes\mathrm{H}^{0}(X,H)}{\wedge^{f}\mathrm{H}^{0}(X,H)}\otimes\mathcal{O}_{{\mathbf{P}}},$

which lands in $\mathrm{K}_{f-1,1}(X,H)\otimes\mathcal{O}_{{\mathbf{P}}}$. The
resulting morphism

$\widetilde{\delta}\;:\;L\to\mathrm{K}_{f-1,1}(X,H)\otimes\mathcal{O}_{{\mathbf{P}}}$

has fibres $\widetilde{\delta}\otimes k([s])$ which are naturally identified
with the natural map

$\wedge^{f}\mathrm{H}^{0}(X,H-R)\otimes\frac{\mathrm{H}^{0}(X,R)}{<s>}\to\mathrm{K}_{f-1,1}(X,H),$

which is injective by Proposition 1.9 and Proposition 1.3, noting that we may
naturally identify $\mathrm{K}_{p,1}(X,H)$ with
$\mathrm{K}_{p,1}(j(X),\mathcal{O}_{j(X)}(1))$, since $j(X)$ has rational
singularities and is normally generated [Sch]. The dual $\delta^{\vee}$ is a
surjective morphism of vector bundles and induces the desired morphism
$\phi:{\mathbf{P}}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right)\to{\mathbf{P}}(\mathrm{K}_{f-1,1}(X,H))$,
[H1, II.7.12]. ∎

We end this section with a description of the last syzygy space of a scroll.

######  Proposition 1.11 (cf. [vB4], Prop. 4.3.).

The morphism $\phi$ from Lemma 1.10 induces an isomorphism

$\phi^{*}\;:\;\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{K}_{f-1,1}(X,H))}(1))\xrightarrow{\sim}\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{H}^{0}(\mathcal{O}_{X}(R)))}(f-2)).$

###### Proof.

By the Eagon–Northcott complex resolving $j(X)\subseteq{\mathbf{P}}^{r}$, we
have an isomorphism

$\mathrm{K}_{f-1,1}(X,H)\simeq\mathrm{Sym}^{f-2}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right),$

[Sch]. Thus
$h^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{K}_{f-1,1}(X,H))}(1))=h^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{H}^{0}(\mathcal{O}_{X}(R)))}(f-2))$
and it suffices to show that $\phi^{*}$ is injective, i.e. that the image of
$\phi$ is not contained in any hyperplane. By Proposition 1.9 and the
definition of $\phi$, it suffices to show that $\mathrm{K}_{f-1,1}(X,H)$ is
spanned by the one-dimensional subspaces
$\mathrm{K}_{f-1,1}(X,H;\mathrm{H}^{0}(X,H\otimes I_{Z(\beta)}))$,
$\beta\in\mathrm{H}^{0}(\mathcal{O}_{X}(R))$. But von Bothmer shows that the
element
$\beta^{f-2}\in\mathrm{Sym}^{f-2}\left(\mathrm{H}^{0}(\mathcal{O}_{X}(R))\right)\simeq\mathrm{K}_{f-1,1}(X,H)$
lies in $\mathrm{K}_{f-1,1}(X,H;\mathrm{H}^{0}(X,H\otimes I_{Z(\beta)}))$ for
$\beta\in\mathrm{H}^{0}(\mathcal{O}_{X}(R))$. Since such elements span
$\mathrm{K}_{f-1,1}(X,H)$, this completes the proof. ∎

By Proposition 1.11, $\mathrm{K}_{f-1,1}(X,H)$ is spanned by the rank $f$
syzygies

$\alpha\in\mathrm{K}_{f-1,1}(X,H;\mathrm{H}^{0}(X,H\otimes
I_{Z(s)})),\;\;\text{for}\;\;s\in\mathrm{H}^{0}(\mathcal{O}_{X}(R)).$

##  2\. Brill–Noether Theory for Integral Curves Lying on a Surface

Let $C$ be an integral, projective, curve which admits an embedding into a
smooth surface, or equivalently, each singularity of $C$ has embedding
dimension at most two, [AK1, Corollary 9]. Consider the compactified Jacobian
$\bar{J}^{d}(C)$ of rank one, torsion free sheaves of degree $d$ on $C$, [R],
[AK2]. Then $\bar{J}^{d}(C)$ is a projective, integral, local complete
intersection scheme of dimension $g$. The Picard scheme $\mathrm{Pic}^{d}(C)$
of degree $d$ line bundles is a dense, open subset of $\bar{J}^{d}(C)$. Let
$\mathrm{Hilb}^{d}(C)$ denote the Hilbert scheme parametrizing zero-
dimensional closed subschemes $Z\subseteq C$ of degree $d$. From [AK2], we
have the _Abel map_

$\rho\;:\;\mathrm{Hilb}^{d}(C)\to\bar{J}^{d}(C)$

taking the closed subscheme $Z\subseteq C$ to the torsion-free sheaf
$I^{\vee}_{Z}:=\mathcal{H}om_{\mathcal{O}_{C}}(I_{Z},\mathcal{O}_{C})$, where
$I_{Z}$ is the ideal sheaf of $Z$.

Note that $C$ is Gorenstein. All torsion-free sheaves on an integral,
Gorenstein curve are reflexive, [H2, Lemma 1.1]. In particular, for each
$[Z]\in\mathrm{Hilb}^{d}(C)$, the ideal sheaf $I_{Z}$ is reflexive, as is
$A:=\rho([Z])=I_{Z}^{\vee}.$ We may thus identify $\rho^{-1}([A])$ with the
projective space ${\mathbf{P}}(\mathrm{Hom}_{C}(A^{\vee},\mathcal{O}_{C}))$,
since any nonzero homomorphism between rank one, torsion-free sheaves on $C$
is injective. Dualizing each morphism in
$\mathrm{Hom}_{C}(A^{\vee},\mathcal{O}_{C})$ we identify $\rho^{-1}([A])$ with
${\mathbf{P}}(\mathrm{H}^{0}(C,A))$.

We have the Brill–Noether loci

$W^{r}_{d}(C):=\\{[A]\in\bar{J}^{d}(C)\;|\;h^{0}(A)\geq r+1\\},$

which are determinantal subschemes of $\bar{J}^{d}(C)$, see the construction
in [BP] (which is stated for nodal curves, but goes through unchanged for
integral curves which embed into a surface). Let $\mathcal{A}$ be the
universal, torsion-free sheaf of rank one on $C\times W^{r}_{d}(C)$, i.e.
$\mathcal{A}_{|_{C\times[A]}}\simeq A.$ Further let $p:C\times W^{r}_{d}(C)\to
C$, $q:C\times W^{r}_{d}(C)\to W^{r}_{d}(C)$, denote the projection morphisms.
We have the locus of ${g^{r}_{d}}^{\prime}s$

$G^{r}_{d}(C):=\\{(A,V)\;|A\in
W^{r}_{d}(C),\;V\in\mathrm{Gr}(r+1,\mathrm{H}^{0}(A))\\}$

[ACG, IV.3] (note that the construction goes through when $C$ embeds into a
surface).

We next generalize the base-point free pencil trick, [ACG, §III.3] to torsion-
free sheaves.

######  Lemma 2.1.

Let $C$ be an integral curve lying on a smooth surface and let $[A]\in
W^{1}_{d}(C)\setminus W^{2}_{d}(C)$ be globally generated. Then we have an
exact sequence

$0\to A^{\vee}\xrightarrow{i}\mathrm{H}^{0}(A)\otimes
O_{C}\xrightarrow{ev}A\to 0,$

where $\mathrm{ev}:\mathrm{H}^{0}(A)\otimes O_{C}\twoheadrightarrow A$ is the
evaluation morphism, and where $i$ is given by the formula

$i(s)=y\otimes\rho(xs)-x\otimes\rho(ys),$

for $x,y$ a basis of $\mathrm{H}^{0}(A)$ and $\rho:\mathrm{H}^{0}(A)\otimes
A^{\vee}\to\mathcal{O}_{C}$ the natural map.

###### Proof.

Let $K:=\mathrm{Ker}(\mathrm{ev})$. Then $K$ is a rank-one, torsion free
sheaf. We need to show $K\simeq A^{\vee}$. Let $Z:=Z(x)$ be the zero scheme of
$x\in\mathrm{H}^{0}(A)$. The claim now follows by observing the commutative
diagram

${0}$${0}$${\mathcal{O}_{C}}$${\mathcal{O}_{C}}$${0}$${K}$${\mathrm{H}^{0}(A)\otimes\mathcal{O}_{C}}$${A}$${0}$${0}$${I_{Z}\simeq
A^{\vee}}$${\mathcal{O}_{C}}$${\mathcal{O}_{Z}}$${0}$${0}$${0}$$\scriptstyle{\sim}$$\scriptstyle{\alpha}$$\scriptstyle{\simeq}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$$\scriptstyle{j}$

with exact rows and columns, where $\alpha$ is the inclusion of the coordinate
$x\in\mathrm{H}^{0}(A)$, $\phi$ is the projection to the coordinate
$y\in\mathrm{H}^{0}(A)$, $j:A^{\vee}\to\mathcal{O}_{C}$ is multiplication by
$x$ and $\psi=\phi_{|_{K}}$. Clearly the sheaf morphism
$i:A^{\vee}\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{C}$ lands in the subsheaf
$K$, and we have $\phi\circ i=j$, i.e. $\psi\circ i=id$. Thus $i$ is the
inverse to the isomorphism $\psi$, as required. ∎

We now need a version of Proposition 1.9 with $M$ assumed merely to be
torsion-free.

######  Lemma 2.2.

Let $C$ be an integral curve which lies on a smooth surface and let $[A]\in
W^{1}_{d}(C)\setminus W^{2}_{d}(C)$. Choose $s\neq 0\in\mathrm{H}^{0}(A)$. For
any $p$, we have a natural map

$\delta_{s}\;:\;\bigwedge^{p+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}\to\mathrm{K}_{p,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})).$

###### Proof.

We have a morphism $A^{\vee}\otimes A\to\mathcal{O}_{C}$ of sheaves, as well
as a natural composition

$\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(A)\to\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee}\otimes A)\to\mathrm{H}^{0}(\omega_{C}).$

Contracting by $s$ induces an map $m_{s}:\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\to\mathrm{H}^{0}(\omega_{C}),$ which is injective since $m_{s}$ is
induced from the inclusion $A^{\vee}\hookrightarrow\mathcal{O}_{C}$ dual to
$s:\mathcal{O}_{C}\to A$. We have a natural inclusion
$\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\subseteq\mathrm{H}^{0}(\omega_{C})$ and
$\mathrm{K}_{p,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes A^{\vee}))$ is
well defined for $A$ torsion free.

We have the natural complex

$\bigwedge^{p+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(A)\xrightarrow{d_{1}}\bigwedge^{p}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(\omega_{C})\xrightarrow{d_{2}}\bigwedge^{p-1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(\omega_{C}^{\otimes^{2}}\otimes A^{\vee}).$

So we have a map

$d_{1}\;:\;\bigwedge^{p+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(A)\to\mathrm{Ker}(d_{2})\subseteq\bigwedge^{p}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathrm{H}^{0}(\omega_{C}).$

Since $\mathrm{K}_{p,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})):=\frac{\mathrm{Ker}(d_{2})}{d_{1}(\bigwedge^{p+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\mathbb{C}\langle s\rangle)}$, we have a natural map

$\bigwedge^{p+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}\to\mathrm{K}_{p,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})).$

∎

######  Remark 2.3.

If $A$ is locally free and $p=g-d$ then $h^{0}(\omega_{C}\otimes
A^{\vee})=p+1$ by Riemann–Roch and Serre duality. If in addition
$(C,\omega_{C})$ is normally generated, then $\delta_{s}$ coincides with the
isomorphism from Proposition 1.9.

######  Proposition 2.4.

Let $C$ be an integral curve lying on a surface and fix $d,r\geq 0$. Suppose
that we have the equality $h^{0}(A)=r+1$ for all points $y=[A]\in
W^{r}_{d}(C)$. Then $q_{*}\mathcal{A}$ and
$q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})$ are locally free sheaves
with natural isomorphisms

$q_{*}\mathcal{A}\otimes
k(y)\simeq\mathrm{H}^{0}(C,A),\;\;q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes
k(y)\simeq\mathrm{H}^{0}(C,\omega_{C}\otimes A^{\vee}).$

######  Remark 2.5.

Note that one cannot conclude the above proposition using Grauert’s Theorem,
as we are not assuming that $W^{r}_{d}(C)$ is reduced.

###### Proof.

Under the assumption $h^{0}(A)=r+1$ for all $y=[A]\in W^{r}_{d}(C)$, the
natural morphism $c:G^{r}_{d}(C)\to\bar{J}^{d}(C)$ is a closed immersion, see
[ACG, IV.3, IV.4]. The image of $c$ is $W^{r}_{d}(C)$, so we have a natural
identification $G^{r}_{d}(C)\simeq W^{r}_{d}(C)$ under our assumptions. By
construction of $G^{r}_{d}(C)\simeq W^{r}_{d}(C)$, there is a locally free
subsheaf $F\subseteq q_{*}\mathcal{A}$ such that, for any $y=[A]\in
W^{r}_{d}(C)$, the natural composition

$F\otimes k(y)\to q_{*}\mathcal{A}\otimes k(y)\to\mathrm{H}^{0}(C,A)$

is injective. By our assumptions, this composition must be an isomorphism, and
in particular the natural composition $q_{*}\mathcal{A}\otimes
k(y)\to\mathrm{H}^{0}(C,A)$ is surjective. By the Base Change Theorem, [H1,
Ch. II, Thm. 12.11], $q_{*}\mathcal{A}$ is locally free and we have an
isomorphism $q_{*}\mathcal{A}\otimes k(y)\simeq\mathrm{H}^{0}(C,A)$. Further,
since $q$ has one dimensional fibres, and further $R^{2}q_{*}\mathcal{A}=0$ by
[H1, Ch. II, Cor. 11.2], the Base Change Theorem implies that the natural map
$R^{1}q_{*}\mathcal{A}\otimes k(y)\to\mathrm{H}^{1}(C,A)$ is an isomorphism.
Combining this with the fact that $q_{*}\mathcal{A}\otimes
k(y)\simeq\mathrm{H}^{0}(C,A)$, another application of the Base Change Theorem
gives us that $R^{1}q_{*}\mathcal{A}$ is locally free. By the Relative Duality
Theorem [Bi, Ch. 2], we have natural isomorphisms
$q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\simeq(R^{1}q_{*}\mathcal{A})^{\vee}$,
which gives the claim (using usual Serre duality
$\mathrm{H}^{1}(C,A)^{\vee}\simeq\mathrm{H}^{0}(C,\omega_{C}\otimes
A^{\vee})$). ∎

Assume that $r,d\geq 0$ are such that $W^{r+1}_{d}(C)=\emptyset$. By
Proposition 2.4, we have a vector bundle $q_{*}\mathcal{A}$ on $W^{r}_{d}(C)$.
In this situation, we define the projective bundle

$\pi:X^{r}_{d}(C)\to W^{r}_{d}(C)$

by $X^{r}_{d}(C):={\mathbf{P}}(q_{*}\mathcal{A}),$ where we are use the
convention
${\mathbf{P}}(q_{*}\mathcal{A}):=\text{Proj}\left((q_{*}\mathcal{A})^{\vee}\right)$,
where the Proj functor is defined in [H1, Ch. II.7]. We let

$p^{\prime}\;:\;C\times X^{r}_{d}(C)\to C,\;\;q^{\prime}\;:\;C\times
X^{r}_{d}(C)\to X^{r}_{d}(C),$

denote the projections. Also let $\pi^{\prime}:C\times X^{r}_{d}(C)\to C\times
W^{r}_{d}(C)$ denote the morphism $id\times\pi$. We denote by $\mathcal{O}(1)$
the line bundle $\mathcal{O}_{{\mathbf{P}}(q_{*}\mathcal{A})}(1)$ on
$X^{r}_{d}(C)$. With our conventions,
$\pi_{*}\mathcal{O}(1)\simeq(q_{*}\mathcal{A})^{\vee}$.

######  Lemma 2.6.

We have a natural inclusion
${\pi^{\prime}}^{*}\mathcal{A}^{\vee}\otimes{q^{\prime}}^{*}\mathcal{O}(-1)\hookrightarrow\mathcal{O}_{C\times
X^{r}_{d}(C)}.$

###### Proof.

Dualizing, it is enough to give a canonical, nonzero, section $s$ of
${\pi^{\prime}}^{*}\mathcal{A}\otimes{q^{\prime}}^{*}\mathcal{O}(1)$. By the
projection formula and flat base change we have natural identifications

$\displaystyle\mathrm{H}^{0}(C\times
X^{r}_{d}(C),{\pi^{\prime}}^{*}\mathcal{A}\otimes{q^{\prime}}^{*}\mathcal{O}(1))$
$\displaystyle\simeq\mathrm{H}^{0}(X^{r}_{d}(C),{q^{\prime}}_{*}{\pi^{\prime}}^{*}\mathcal{A}\otimes\mathcal{O}(1))$
$\displaystyle\simeq\mathrm{H}^{0}(X^{r}_{d}(C),\pi^{*}q_{*}\mathcal{A}\otimes\mathcal{O}(1))$
$\displaystyle\simeq\mathrm{H}^{0}(W^{r}_{d}(C),q_{*}\mathcal{A}\otimes
q_{*}\mathcal{A}^{\vee})$
$\displaystyle\simeq\text{Hom}_{\mathcal{O}_{W^{r}_{d}(C)}}(q_{*}\mathcal{A},q_{*}\mathcal{A}).$

We then take the section $s$ corresponding to the identity
$id\in\text{Hom}_{\mathcal{O}_{W^{r}_{d}(C)}}(q_{*}\mathcal{A},q_{*}\mathcal{A})$.
∎

The _gonality_ of an integral curve $C$ lying on a surface is defined to be
the minimal $d$ such that $W^{1}_{d}(C)\neq\emptyset$.

######  Proposition 2.7.

Let $C$ be an integral curve which may be embedded into a surface, and fix
integers $p,d\geq 0$. Assume that $\omega_{C}$ is very ample and
$(C,\omega_{C})$ is normally generated. Assume further
$W^{2}_{d}(C)=\emptyset$ and that, for any $[A]\in W^{1}_{d}(C)$ and $s\neq
0\in\mathrm{H}^{0}(C,A)$, the map

$\delta_{s}\;:\;\bigwedge^{g-d+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}\to\mathrm{K}_{g-d,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee}))$

from Lemma 2.2 is injective. We have a well-defined morphism
$S:X^{1}_{d}(C)\to{\mathbf{P}}(\mathrm{K}_{g-d,1}(C,\omega_{C}))$

$\displaystyle S([(A,s)]):=[\bigwedge^{g-d+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle s\rangle}].$

###### Proof.

The proof is similar to that of Lemma 1.10. Define
$W_{s}:=\wedge^{g-d+1}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle s\rangle}.$ By
assumption, $h^{0}(A)=2$ and, by Riemann–Roch and Serre duality,
$h^{0}(\omega_{C}\otimes A^{\vee})=g-d+1$, so $\dim W_{s}=1$. We have an
inclusion $\mathrm{K}_{g-d,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee}))\hookrightarrow\mathrm{K}_{g-d,1}(C,\omega_{C})$ by Proposition 1.3.
Under the assumption that $\delta_{s}$ is injective, we may consider $W_{s}$
as a subspace of $\mathrm{K}_{g-d,1}(C,\omega_{C})$. The map $S$ is thus well-
defined on closed points of $X^{1}_{d}(C)$.

To show that $S$ is defined as a morphism of schemes, we show that there is a
line subbundle
$i:\mathcal{L}\hookrightarrow\mathrm{K}_{g-d,1}(C,\omega_{C})\otimes\mathcal{O}_{X^{1}_{d}(C)}$
such that, for any closed point $x=[(A,s)]\in X^{1}_{d}(C)$,

$\mathcal{L}\otimes k(x)\simeq W_{s}$

and further $i\otimes k(x)$ is identified with the inclusion
$W_{s}\hookrightarrow\mathrm{K}_{p,1}(C,\omega_{C})$ from Lemma 2.2 and
Proposition 1.3. The claim then follows from [H1, II.7.12]. We have a natural
evaluation morphism
$\pi^{*}\pi_{*}\mathcal{O}(1)\twoheadrightarrow\mathcal{O}(1)$ on
$X^{1}_{d}(C)={\mathbf{P}}(q_{*}\mathcal{A})$. Since
$\pi_{*}\mathcal{O}(1)\simeq(q_{*}\mathcal{A})^{\vee}$, we may dualize the
evaluation morphism to get an inclusion

$j\;:\;\mathcal{O}(-1)\hookrightarrow\pi^{*}q_{*}\mathcal{A},$

such that $j\otimes k(x)$ corresponds to $\mathbb{C}\langle
s\rangle\hookrightarrow\mathrm{H}^{0}(A)$. Let
$\mathcal{F}:=\mathrm{Coker}(j)$. We set

$\mathcal{L}:=\bigwedge^{g-d+1}{\pi}^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\mathcal{F}(d-g).$

Then $\mathcal{L}$ is a line bundle with $\mathcal{L}\otimes k(x)\simeq
W_{s}$.

We have the differential

$\wedge^{g-d+1}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\to\wedge^{g-d}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee}).$

Tensoring the above morphism by $\pi^{*}q_{*}\mathcal{A}(d-g)$ and using the
natural composition

$q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes q_{*}\mathcal{A}\to
q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee}\otimes\mathcal{A})\to
q_{*}(p^{*}\omega_{C}),$

we obtain a morphism

$\wedge^{g-d+1}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\pi^{*}q_{*}\mathcal{A}(d-g)\to\wedge^{g-d}\left(\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})(-1)\right)\otimes\pi^{*}q_{*}(p^{*}\omega_{C}).$

Using the natural composition

$\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})(-1)\hookrightarrow\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\pi^{*}q_{*}\mathcal{A}\to\pi^{*}q_{*}(p^{*}\omega_{C})\simeq\mathrm{H}^{0}(\omega_{C})\otimes\mathcal{O}_{X^{1}_{d}(C)},$

we arrive at a natural morphism

$\displaystyle\wedge^{g-d+1}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\pi^{*}q_{*}\mathcal{A}(d-g)\to\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})\otimes\mathcal{O}_{X^{1}_{d}(C)},$

and composing with the quotient

$\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})\otimes\mathcal{O}_{X^{1}_{d}(C)}\to\frac{\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})}{\wedge^{g-d+1}\mathrm{H}^{0}(\omega_{C})}\otimes\mathcal{O}_{X^{1}_{d}(C)}$

we arrive at

$\delta\;:\;\wedge^{g-d+1}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})\otimes\pi^{*}q_{*}\mathcal{A}(d-g)\to\frac{\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})}{\wedge^{g-d+1}\mathrm{H}^{0}(\omega_{C})}\otimes\mathcal{O}_{X^{1}_{d}(C)}.$

By the proof of Lemma 2.2, $\delta$ vanishes on
$\wedge^{g-d+1}\pi^{*}q_{*}(p^{*}\omega_{C}\otimes\mathcal{A}^{\vee})(-1+d-g)$
and induces a morphism

$\delta^{\prime}\;:\;\mathcal{L}\to\frac{\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})}{\wedge^{g-d+1}\mathrm{H}^{0}(\omega_{C})}\otimes\mathcal{O}_{X^{1}_{d}(C)}.$

Again by Lemma 2.2, $\delta^{\prime}$ lands in the subbundle
$\frac{\mathrm{Ker}(d)}{\wedge^{g-d+1}\mathrm{H}^{0}(\omega_{C})}\otimes\mathcal{O}_{X^{1}_{d}(C)}$,
where $d$ is the differential

$d\;:\;\wedge^{g-d}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C})\to\wedge^{g-d-1}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C}^{\otimes
2}).$

Hence $\delta^{\prime}$ induces a morphism
$i:\mathcal{L}\to\mathrm{K}_{g-d,1}(C,\omega_{C})\otimes\mathcal{O}_{X^{1}_{d}(C)}$
with the required properties. ∎

######  Remark 2.8.

Note that, in the situation where $W^{1}_{d}(C)$ is zero-dimensional and
reduced, then any coherent sheaf of the form
$\pi^{*}\mathcal{A}\in\mathrm{Coh}(X^{1}_{d}(C))$ is free, for
$\mathcal{A}\in\mathrm{Coh}(W^{1}_{d}(C))$. We see from this that
$\mathcal{L}\simeq\mathrm{det}(\mathcal{L})\simeq\mathcal{O}(d+1-g)$, where
$\mathcal{L}$ is the line bundle from the above proof. Thus

$S^{*}\mathcal{O}_{{\mathbf{P}}}(1)\simeq\mathcal{O}_{X^{1}_{d}(C)}(g-d-1)$

for ${\mathbf{P}}:={\mathbf{P}}(\mathrm{K}_{g-d,1}(C,\omega_{C})),$ under the
assumption that $W^{1}_{d}(C)$ is zero-dimensional and reduced.

We end this section with a result on the Brill–Noether variety of minimal
pencils on nodal curves of even genus.

######  Proposition 2.9.

Let $D$ be an general integral curve of arithmetic genus $g=2k$ with precisely
$m$ nodes for $m\leq k-1$. Then $D$ has gonality $k+1$. Further,
$W^{2}_{k+1}(D)=\emptyset$, any closed point $[A]\in W^{1}_{k+1}(D)$
corresponds to a locally free sheaf $A$ on $D$, and $W^{1}_{k+1}(D)$ is zero-
dimensional and reduced.

###### Proof.

We first prove $D$ has gonality $k+1$, i.e. there is no $d\leq k$ with
$W^{1}_{d}(D)\neq\emptyset$. By degenerating $D$ to a rational curve, it
suffices to prove there exists a rational, nodal curve $D^{\prime}$ of genus
$g=2k$ and gonality $k+1$. By Giesker’s Theorem, there exists such a
$D^{\prime}$ with the property that the Petri map is injective for all _line
bundles_ on $D^{\prime}$, [Gi, Prop. 1.2]. Furthermore, the same property
holds for all partial normalizations of $D^{\prime}$. We will show that
$D^{\prime}$ has gonality $k+1$. Let $A\in W^{1}_{d}(D^{\prime})$ for $d\leq
k$. By Gieseker’s Theorem, $A$ cannot be locally free. Thus there exists some
partial normalization $\mu:\widetilde{D}\to D^{\prime}$ at $n>0$ nodes of
$D^{\prime}$ such that $A=\mu_{*}\widetilde{A}$ for some $\widetilde{A}\in
W^{1}_{d-n}(\widetilde{D})$, with $\widetilde{A}$ locally free. As Petri’s
Condition holds for the line bundle $\widetilde{A}$ on the desingularization
$\widetilde{D}$, we have

$d-n\geq\lfloor\frac{g(\widetilde{D})+3}{2}\rfloor\geq\frac{2k-n+2}{2}.$

Thus $d\geq\frac{2k+n+2}{2}\geq k+1$, which is a contradiction.

To complete the proof, by degeneration it suffices to show that
$W^{1}_{k+1}(D^{\prime})$ is zero-dimensional and reduced, that
$W^{2}_{k+1}(D^{\prime})=\emptyset$ and that each closed point of
$W^{1}_{k+1}(D^{\prime})$ corresponds to a locally free sheaf, for the
rational curve $D^{\prime}$ above. The fact that each closed point $[A]$
corresponds to a locally free sheaf follows from the previous paragraph.
Indeed, otherwise $A=\mu_{*}\widetilde{A}$ for some $\widetilde{A}\in
W^{1}_{k+1-n}(\widetilde{D})$, with $\widetilde{A}$ locally free and
$\mu:\widetilde{D}\to D^{\prime}$ a partial normalization at $n>0$ nodes. As
above, this implies $k+1-n\geq\frac{2k-n+2}{2}$ which is impossible for $n>0$.
Since Petri’s Condition holds for all line bundles on $D^{\prime}$, it follows
that $W^{1}_{k+1}(D^{\prime})$ is zero-dimensional and reduced outside of
$W^{2}_{k+1}(D^{\prime})$. It remains to prove
$W^{2}_{k+1}(D^{\prime})=\emptyset$. If $[A]\in W^{2}_{k+1}(D^{\prime})$,
write $A=\mu_{*}\widetilde{A}$, for $\widetilde{A}$ locally free and
$\mu:\widetilde{D}\to D^{\prime}$ a partial normalization at $n\geq 0$ nodes
(notice that we are allowing $n=0$). Since Petri’s Condition holds for
$\widetilde{A}$ on $\widetilde{D}$, we have

$(r+1)(g-n-(k+1-n)+r)=(r+1)(g-k+r)\leq g-n=g(\widetilde{D}),$

where $r+1=h^{0}(\widetilde{D},\widetilde{A})=h^{0}(D,A)\geq 3$. But

$(r+1)(g-k+r)\geq 3(g-k+2)=3(k+2)>g=2k,$

which is a contradiction. ∎

##  3\. The Projection Map

We now recall an important construction from [Ap1, §2], see also [AN2,
§2.2.1]. Let $M$ be a graded $S_{L}$ module and let
$f:\mathrm{H}^{0}(X,L)\twoheadrightarrow\mathbb{C}$ be a surjection. Set
$W:=\mathrm{Ker}(f)$. We have the _Aprodu projection map_

$pr_{f}\;:\;\mathrm{K}_{p,1}(M,\mathrm{H}^{0}(L))\to\mathrm{K}_{p-1,1}(M,W),$

which is induced on Koszul cohomology from the map

$\displaystyle\iota_{f}\;:\;\bigwedge^{p}\mathrm{H}^{0}(L)\otimes M_{1}$
$\displaystyle\to\bigwedge^{p-1}W\otimes M_{1}$ $\displaystyle
v_{1}\wedge\ldots\wedge v_{p}\otimes m$
$\displaystyle\mapsto\sum_{i}(-1)^{i}v_{1}\wedge\ldots\wedge\hat{v_{i}}\wedge\ldots\wedge
v_{p}\otimes f(v_{i})m.$

Let $L$ be an effective line bundle on an integral curve $C$ and let $C_{sm}$
denote the smooth locus. If $x\in C_{sm}$ is not a base point of $L$, we let

$pr_{x}\,:\,\mathrm{K}_{p,1}(C,L)\to\mathrm{K}_{p-1,1}\left(\Gamma_{C}(L),\mathrm{H}^{0}(L(-x))\right)$

denote the projection map $pr_{x}:=pr_{ev_{x}}$ associated to the evaluation
map
$ev_{x}:\mathrm{H}^{0}(C,L)\twoheadrightarrow\mathbb{C}\simeq\mathrm{H}^{0}(\mathcal{O}_{x})$.

We first develop a more explicit description of the projection map $pr_{x}$.
Let $\overline{\alpha}\in\mathrm{K}_{p,1}(C,L)$,

$\alpha\in\mathrm{Ker}\left(\bigwedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)\xrightarrow{d}\bigwedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2})\right).$

We have $\mathrm{H}^{0}(L)=\mathrm{H}^{0}(L(-x))\oplus\mathbb{C}\langle
s\rangle$ where $s\in\mathrm{H}^{0}(L)$ is a section with $ev_{x}(s)=1$, so
that

$\bigwedge^{p}\mathrm{H}^{0}(L)=\bigwedge^{p}\mathrm{H}^{0}(L(-x))\bigoplus\left(\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\right)\wedge
s.$

Set

$\displaystyle V_{1}$
$\displaystyle:=\bigwedge^{p}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x)),$
$\displaystyle V_{2}$
$\displaystyle:=\bigwedge^{p}\mathrm{H}^{0}(L(-x))\otimes s,$ $\displaystyle
V_{3}$ $\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\right)\wedge
s\otimes\mathrm{H}^{0}(L(-x)),$ $\displaystyle V_{4}$
$\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\right)\wedge
s\otimes s.$

Then $\bigwedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)=V_{1}\oplus
V_{2}\oplus V_{3}\oplus V_{4}$. For
$\alpha\in\bigwedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)$, write

$\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\;\text{ for
$\alpha_{i}\in V_{i}$, $1\leq i\leq 4$}.$

We further write $\alpha_{2}=\alpha_{2}^{\prime}\otimes
s,\alpha_{3}=\alpha_{3}^{\prime}\wedge s$ for
$\alpha_{2}^{\prime}\in\wedge^{p}\mathrm{H}^{0}(L(-x))$,
$\alpha_{3}^{\prime}\in\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x))$.

######  Proposition 3.1.

Let $L$ be an effective line bundle on an integral curve $C$. Suppose $L$ is
globally generated at $x\in C_{sm}$.

  1. (1)

Aprodu’s projection map $pr_{x}$ is the composite of the map

$\displaystyle q_{x}\;:\;\mathrm{K}_{p,1}(C,L)$
$\displaystyle\to\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x))$
$\displaystyle\bar{\alpha}$ $\displaystyle\mapsto
d\alpha_{2}^{\prime}+(-1)^{p}\alpha_{3}^{\prime}$

with the natural quotient map

$\mathrm{Ker}(d)\subseteq\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x))\to\mathrm{K}_{p-1,1}(\Gamma_{C}(L),\mathrm{H}^{0}(L(-x))).$

In particular, the image of $pr_{x}$ is contained in

$\mathrm{K}_{p-1,1}(C,L(-x))\subseteq\mathrm{K}_{p-1,1}(\Gamma_{C}(L),\mathrm{H}^{0}(L(-x))).$

  2. (2)

If $x$ is not a base point of $L(-x)$, $q_{x}$ lands in
$\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-2x)).$ In
particular, $pr_{x}$ lands in the image of the multiplication map

$\gamma_{x}\,:\,\mathrm{K}_{p-1,1}(C,-x,L(-x))\to\mathrm{K}_{p-1,1}(C,L(-x))$

induced from the inclusion $\Gamma_{C}(-x,L)\hookrightarrow\Gamma_{C}(L).$

######  Remark.

When $L$ is very ample, a different proof of the second statement in part
$(1)$ of the above proposition is given in [Ap1, Lemma 3.1].

###### Proof.

  1. (1)

Let
$\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}\in\wedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)$
for $\alpha_{i}\in V_{i}$, $1\leq i\leq 4$ with $d\alpha=0$. Write
$\alpha_{2}=\alpha_{2}^{\prime}\otimes s$ and
$\alpha_{3}=\alpha_{3}^{\prime}\wedge s$ for

$\alpha_{2}^{\prime}\in\wedge^{p}\mathrm{H}^{0}(L(-x)),\;\;\alpha_{3}^{\prime}\in\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x)).$

Let $\beta\in\wedge^{p+1}\mathrm{H}^{0}(L)$. Write

$\beta=\beta_{1}+\beta_{2}\wedge s,\;\;\text{with
$\beta_{1}\in\wedge^{p+1}\mathrm{H}^{0}(L(-x))$,
$\beta_{2}\in\wedge^{p}\mathrm{H}^{0}(L(-x))$}.$

Then $d\beta_{1}\in V_{1}$ and $d(\beta_{2}\wedge s)=d\beta_{2}\wedge
s+(-1)^{p+1}\beta_{2}\otimes s\in V_{3}\oplus V_{2}.$ We have the
decomposition

$\alpha+d\beta=(\alpha_{1}+d\beta_{1})+(\alpha_{2}^{\prime}+(-1)^{p+1}\beta_{2})\otimes
s+(\alpha_{3}^{\prime}+d\beta_{2})\wedge s+\alpha_{4},$

from which the well-definedness of $q_{x}$ follows.

Next, observe that

$\iota_{x}(\alpha_{1})=\iota_{x}(\alpha_{2})=0,$

where
$\iota_{x}\;:\wedge^{p}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)\to\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L)$
is defined as $\iota_{x}:=\iota_{ev_{x}}$. We have
$\mathrm{H}^{0}(L^{2})=\mathrm{H}^{0}(L^{2}(-x))\oplus\mathbb{C}\langle
s^{2}\rangle$ so that

$\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2})=\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2}(-x))\oplus\wedge^{p-1}\mathrm{H}^{0}(L)\otimes
s^{2}.$

Let
$\pi_{1}:\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2})\to\wedge^{p-1}\mathrm{H}^{0}(L)\otimes
s^{2}$ denote the projection. Observe

$d(V_{i})\subseteq\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2}(-x)),\;\text{for
$i=1,2,3$,}$

and hence $\pi_{1}\circ(d\alpha_{i})=0$ for $i=1,2,3$. The composition
$\pi_{1}\circ d$ is, up to sign, the natural inclusion

$\wedge^{p-1}\mathrm{H}^{0}(L(-x))\wedge s\otimes
s\xrightarrow{\sim}\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes
s^{2}\subseteq\wedge^{p-1}\mathrm{H}^{0}(L)\otimes s^{2}.$

Since $\pi_{1}\circ d(\alpha)=0$, we conclude $\alpha_{4}=0.$ Since
$\iota_{x}(\alpha_{1})=\iota_{x}(\alpha_{2})=0$, we have

$\iota_{x}(\alpha)=\iota_{x}(\alpha_{3})=\iota_{x}(\alpha_{3}^{\prime}\wedge
s)=(-1)^{p}\alpha^{\prime}_{3},$

since
$\alpha_{3}^{\prime}\in\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-x))$
and $ev_{x}(s)=1$. Hence

$q_{x}(\alpha)=\iota_{x}(\alpha)+d\alpha_{2}^{\prime}.$

Since $d(\alpha)=0$ we have $d(\iota_{x}(\alpha))=0$, [Ap1, §2]. It follows,
firstly, that $q_{x}$ lands in $\mathrm{Ker}(d)$ and, secondly,

$\overline{q_{x}(\alpha)}=\overline{\iota_{x}(\alpha)}=pr_{x}(\bar{\alpha})$

as required. This gives $(1)$. Note that, since the inclusion
$\Gamma_{C}(L(-x))\hookrightarrow\Gamma_{C}(L)$ of $S_{L(-x)}$ modules is an
isomorphism in degree zero, we have an inclusion

$\mathrm{K}_{q,1}(C,L(-x))\hookrightarrow\mathrm{K}_{q,1}(\Gamma_{C}(L),\mathrm{H}^{0}(L(-x)))$

for any $q$.

  2. (2)

We need to show
$d\alpha_{2}^{\prime}+(-1)^{p}\alpha_{3}^{\prime}\in\wedge^{p-1}\mathrm{H}^{0}(L(-x))\otimes\mathrm{H}^{0}(L(-2x)).$
Assuming that $L(-x)$ is globally generated at $x$, let
$t\in\mathrm{H}^{0}(L(-x))$ be a section with $ev_{x}(t)=1$. Then
$\mathrm{H}^{0}(L(-x))=\mathrm{H}^{0}(L(-2x))\oplus\mathbb{C}\langle
t\rangle$. We set

$\displaystyle V_{2,1}$
$\displaystyle:=\bigwedge^{p}\mathrm{H}^{0}(L(-2x))\otimes s,$ $\displaystyle
V_{2,2}$ $\displaystyle:=\bigwedge^{p-1}\mathrm{H}^{0}(L(-2x))\wedge t\otimes
s,$ $\displaystyle V_{3,1}$
$\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\right)\wedge
s\otimes\mathrm{H}^{0}(L(-2x)),$ $\displaystyle V_{3,2}$
$\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(L(-x))\right)\wedge
s\otimes t$

We have decompositions $V_{i}=V_{i,1}\oplus V_{i,2}$ for $i=2,3$. Write
$\alpha_{i}=\alpha_{i,1}+\alpha_{i,2}$ for $\alpha_{i,1}\in V_{i,1}$,
$\alpha_{i,2}\in V_{i,2}$ and $i=2,3$. Let

$\alpha_{3,2}=\omega_{1}\wedge s\otimes t,\;\;\alpha_{2,2}=\omega_{2}\wedge
t\otimes s$

for
$\omega_{1}\in\wedge^{p-1}\mathrm{H}^{0}(L(-x)),\omega_{2}\in\wedge^{p-1}\mathrm{H}^{0}(L(-2x))$.
Since

$d(\omega_{2}\wedge t)=d\omega_{2}\wedge t+(-1)^{p}\omega_{2}\otimes t,$

and
$d\omega_{2}\in\wedge^{p-2}\mathrm{H}^{0}(L(-2x))\otimes\mathrm{H}^{0}(L(-2x))$,
it suffices to prove $\omega_{1}=-\omega_{2}.$

We have
$\alpha=\alpha_{1}+\alpha_{2,1}+\alpha_{2,2}+\alpha_{3,1}+\alpha_{3,2},$
recalling that $\alpha_{4}=0$. Observe

$d(V_{1}),d(V_{2,1}),d(V_{3,1})\subseteq\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2}(-2x)).$

We have

$\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2})=\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2}(-2x))\oplus\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathbb{C}\langle
s^{2},st\rangle.$

Let
$\pi_{2}:\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathrm{H}^{0}(L^{2})\to\wedge^{p-1}\mathrm{H}^{0}(L)\otimes\mathbb{C}\langle
s^{2},st\rangle$ be the projection. Then

$0=\pi_{2}(d\alpha)=\pi_{2}(d\alpha_{2,2})+\pi_{2}(d\alpha_{3,2}).$

We have

$\displaystyle\pi_{2}(d(\alpha_{2,2}))$
$\displaystyle=\pi_{2}(d(\omega_{2}\wedge t\otimes
s))=(-1)^{p}\omega_{2}\otimes st,$ $\displaystyle\pi_{2}(d(\alpha_{3,2}))$
$\displaystyle=\pi_{2}(d(\omega_{1}\wedge s\otimes
t))=(-1)^{p}\omega_{1}\otimes st$

Thus the equation $\pi_{2}(d\alpha_{2,2})+\pi_{2}(d\alpha_{3,2})=0$ gives
$\omega_{1}+\omega_{2}=0$ as required.

∎

Our first goal is to offer an improvement of Proposition 3.3 from [Ke2]. We
recall the set-up from §3 of [Ke2]. Let $C$ be an integral, nodal curve of
arithmetic genus $g\geq 3$, and assume that the canonical linear system
$\omega_{C}$ is very ample and $C\subseteq{\mathbf{P}}^{g-1}$ is normally
generated. Assume $C$ has precisely $m$ nodes and no other singularities.
Choose general points $x,y\in C_{sm}$ in the smooth locus of $C$ and let $D$
be the nodal curve of genus $g+1$ obtained by identifying $x,y$. Then
$\omega_{D}$ is very ample, [CFHR, Thm. 3.6]. Let $p\in D$ be the node over
$x,y$ and let $\mu:C\to D$ be the partial normalization map.

We embed $D$ in ${\mathbf{P}}^{g}$ via the canonical linear system. Note that
$D$ is normally generated. Indeed, $D$ being normally generated is equivalent
to $K_{0,2}(D,\omega_{D})=0$ which is in turn equivalent to
$\mathrm{K}_{g+1-3,1}(D,\omega_{D})=0$ by duality. This is equivalent to
$\mathrm{K}_{g-2,1}(C,\omega_{C}(x+y))=0$ and is implied by
$\mathrm{K}_{g-3,1}(C,\omega_{C})=0$ by [Ap1, Thm. 3]. This last vanishing is
equivalent to $(C,\omega_{C})$ being normally generated by duality.

We now consider the projection

$\pi_{p}:{\mathbf{P}}^{g}\dashrightarrow{\mathbf{P}}^{g-1}$

from the node $p$. Then $C\subseteq{\mathbf{P}}^{g-1}$ is the projection
$\pi_{p}(D)$. We let $Z\subseteq{\mathbf{P}}^{g}$ denote the cone over
$C=\pi_{p}(D)$ with vertex at $p$. Then $D\subseteq Z$. We denote by
$\nu:\widetilde{Z}\to Z$ the desingularization of $Z$. The strict transform
$D^{\prime}\subseteq\widetilde{Z}$ of $D$ is isomorphic to $C$ with
$\mu\sim\nu_{|_{D^{\prime}}}$. From [Ke2, Lemma 3.2] we have
$\widetilde{Z}\simeq{\mathbf{P}}(\mathcal{O}_{C}\oplus\omega_{C})$ and

$\mathcal{O}_{\widetilde{Z}}(D^{\prime})\simeq\mathcal{H}\otimes\iota^{*}\mathcal{O}_{C}(x+y),$

where $\mathcal{H}$ is the pullback of $\mathcal{O}_{{\mathbf{P}}^{g}}(1)$ and
$\iota:{\mathbf{P}}(\mathcal{O}_{C}\oplus\omega_{C})\to C$ is the projection.

Denote by $S$ the graded rings
$\mathrm{Sym}\,\mathrm{H}^{0}(\widetilde{Z},\mathcal{H})$. We have a morphism

$f\;:\;\bigoplus_{q}\mathrm{H}^{0}(\widetilde{Z},\mathcal{H}^{\otimes
q})\to\bigoplus_{q}\mathrm{H}^{0}(C,\omega_{C}^{\otimes q}(qx+qy))$

of graded $S$ modules, given by restriction to $D^{\prime}\simeq C$. We let
$\mathbb{M}\subseteq\bigoplus_{q}\mathrm{H}^{0}(C,\omega_{C}^{\otimes
q}(qx+qy))$ denote the image of $f$. Pullback induces an identification
$\mathrm{H}^{0}(\widetilde{Z},\mathcal{H})\simeq\mathrm{H}^{0}(\mathcal{O}_{Z}(1))$,
and so we may consider the canonical ring
$\displaystyle{\Gamma_{D}(\omega_{D}):=\bigoplus_{q}\mathrm{H}^{0}(\omega^{\otimes
q}_{D})}$ as an $S$ module. The inclusion
$\mathcal{O}_{D}\hookrightarrow\mu_{*}\mathcal{O}_{C}$ induces a natural
inclusion

$\Gamma_{D}(\omega_{D})\subseteq\bigoplus_{q}\mathrm{H}^{0}(C,\omega_{C}^{\otimes
q}(qx+qy)),$

since $\mu^{*}\omega_{D}\simeq\omega_{C}(x+y)$. Under this inclusion,
$\Gamma_{D}(\omega_{D})$ is the image of $\alpha$, giving the isomorphism

$\mathbb{M}\simeq\Gamma_{D}(\omega_{D}),$

since $D\subseteq{\mathbf{P}}^{g}$ is normally generated. As in [Ke2, §3], we
consider the short exact sequence

$0\to\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}\left(\mathcal{H}^{\otimes
q-1}(-\iota^{*}(x+y))\right)\to\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}(\mathcal{H}^{\otimes
q})\xrightarrow{f}\Gamma_{D}(\omega_{D})\to 0,$

of $S$ modules, where we have used the isomorphism
$\mathbb{M}\simeq\Gamma_{D}(\omega_{D})$. Taking the long exact sequence of
Koszul cohomology gives an exact sequence

$0\to\mathrm{K}_{p,1}(C,\omega_{C})\to\mathrm{K}_{p,1}(D,\omega_{D})\to\mathrm{K}_{p-1,1}(\widetilde{Z},-\iota^{*}(x+y),\mathcal{H})\to\mathrm{K}_{p-1,2}(C,\omega_{C})\to\mathrm{K}_{p-1,2}(D,\omega_{D})\to\ldots$

cf. [Ke2, Prop. 3.3].

######  Lemma 3.2.

Embed $C$ in
$\widetilde{Z}\simeq{\mathbf{P}}(\mathcal{O}_{C}\oplus\omega_{C})$ as a
hyperplane section, (equivalently, as the section induced by the trivial
quotient $\mathcal{O}_{C}\oplus\omega_{C}\twoheadrightarrow\omega_{C}$). Then,
for any $q$ the restriction maps

$\mathrm{H}^{0}(\widetilde{Z},\mathcal{H}^{\otimes
q}\otimes\iota^{*}\mathcal{O}_{C}(-x-y))\to\mathrm{H}^{0}(C,\omega_{C}^{\otimes
q}(-x-y))$

are surjective.

###### Proof.

We have $\displaystyle{\mathrm{H}^{0}(\widetilde{Z},\mathcal{H}^{\otimes
q}\otimes\iota^{*}\mathcal{O}_{C}(-x-y))\simeq\mathrm{H}^{0}(C,\mathrm{Sym}^{q}(\mathcal{O}_{C}\oplus\omega_{C})(-x-y)).}$
The trivial projection map
$\mathcal{O}_{C}\oplus\omega_{C}\twoheadrightarrow\omega_{C}$ induces the
restriction map

$\mathrm{H}^{0}(C,\mathrm{Sym}^{q}(\mathcal{O}_{C}\oplus\omega_{C})(-x-y))\to\mathrm{H}^{0}(C,\omega_{C}^{\otimes
q}(-x-y)).$

Since the projection
$\mathcal{O}_{C}\oplus\omega_{C}\twoheadrightarrow\omega_{C}$ splits, we
deduce that $\omega_{C}^{\otimes q}$ is a direct summand of
$\mathrm{Sym}^{q}(\mathcal{O}_{C}\oplus\omega_{C})$ and hence the restriction
map above is surjective. ∎

As a consequence of the above lemma we deduce:

######  Lemma 3.3.

We have
$\mathrm{K}_{p,q}(\widetilde{Z},-\iota^{*}(x+y),\mathcal{H})\simeq\mathrm{K}_{p,q}(C,-x-y,\omega_{C})$,
for integers $p,q$.

###### Proof.

By Lemma 3.2, we have a short exact sequence

$0\to\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}(\mathcal{H}^{\otimes
q-1}\otimes\iota^{*}\mathcal{O}_{C}(-x-y))\to\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}(\mathcal{H}^{\otimes
q}\otimes\iota^{*}\mathcal{O}_{C}(-x-y))\to\bigoplus_{q\in\mathbb{Z}}\mathrm{H}^{0}(C,\omega_{C}^{\otimes
q}(-x-y))\to 0.$

The claim now follows from the proof of the Green–Lefschetz Theorem, see [Gr2,
Thm. 3.b.7] and [FK1, Lemma 2.2]. ∎

By Lemma 3.3, we have the long exact sequence

(2)
$0\to\mathrm{K}_{p,1}(C,\omega_{C})\to\mathrm{K}_{p,1}(D,\omega_{D})\xrightarrow{\delta}\mathrm{K}_{p-1,1}(C,-x-y,\omega_{C})\to\mathrm{K}_{p-1,2}(C,\omega_{C})\to\ldots$

Let $D$ be the nodal curve of arithmetic genus $g+1$ obtained by identifying
general points $x,y\in C_{sm}$ on an integral, nodal curve $C$ with
$\omega_{C}$ very ample and $(C,\omega_{C})$ normally generated. Let $\mu:C\to
D$ be the partial normalization morphism. Let

$\alpha\neq 0\in\mathrm{K}_{p,1}(D,\omega_{D})$

be a syzygy of rank $p+1$. By [vB3, Corollary 5.2], [AN2, Lemma 3.21], the
syzygy scheme

$X_{\alpha}:=\mathrm{Syz}(\alpha)\subseteq{\mathbf{P}}^{g}$

of $\alpha$ is a rational normal scroll of degree $p+1$ and codimension $p$
containing $D\subseteq{\mathbf{P}}^{g}$. Letting $\widetilde{X}_{\alpha}$
denote the desingularization of $X_{\alpha}$, the Picard group of the
projective bundle $\widetilde{X}_{\alpha}$ is generated by
$\mathcal{O}_{X_{\alpha}}(1)$, together with the class of a ruling $R$. Assume
that $D$ lies in the smooth locus of $X_{\alpha}$. As $D$ lies in the smooth
locus, we may treat $D$ as a subscheme of $\widetilde{X}_{\alpha}$, and we
denote the restriction of the ruling to $D$ as the line bundle

$L_{\alpha}:=\mathcal{O}_{D}(R).$

Let
$\displaystyle{\gamma_{x,y}\;:\;\mathrm{K}_{p,1}(C,-x-y,\omega_{C})\to\mathrm{K}_{p,1}(C,\omega_{C})}$
be the natural morphism induced by the inclusions
$\mathrm{H}^{0}(\omega_{C}^{\otimes
q}(-x-y))\subseteq\mathrm{H}^{0}(\omega_{C}^{\otimes q})$ for all
$q\in\mathbb{Z}$.

######  Proposition 3.4.

Let $D$ be the nodal curve of arithmetic genus $g(D):=g+1$ obtained by
identifying general points $x,y\in C_{sm}$ on an integral, nodal curve $C$ of
genus $g$ with $\omega_{C}$ very ample and $(C,\omega_{C})$ normally
generated. Let $\displaystyle{\alpha\neq 0\in\mathrm{K}_{p,1}(D,\omega_{D})}$
be a syzygy of least possible rank $p+1$ as above. Assume $D$ lies in the
smooth locus of the syzygy scheme $X_{\alpha}$ and
$h^{0}(C,\mu^{*}L_{\alpha})=h^{0}(D,L_{\alpha})=2$. Assume further that
$\deg(L_{\alpha})=g(D)-p$.

Then $\gamma_{x,y}(\delta(\alpha))$ is a linear combination of syzygies
$\sigma$ of minimal rank $p$. Moreover, $C$ lies in the smooth locus of
$X_{\sigma}$ and has associated line bundles
$L_{\sigma}\simeq\mu^{*}L_{\alpha}$.

###### Proof.

Let $\pi_{p}\;:\;{\mathbf{P}}^{g}\dashrightarrow{\mathbf{P}}^{g-1}$ be the
projection from $p$. The scroll $X_{\alpha}$ coincides with the scroll induced
by the pencil $|L_{\alpha}|$, by the assumptions $\deg(L_{\alpha})=g(D)-p$ and
$h^{0}(D,L_{\alpha})=2$, [EH, Thm. 2], [Sch, §4]. Since $L_{\alpha}$ is base-
point free from the definition, the same is true for the line bundle
$\mu^{*}L_{\alpha}$ on $C$. The projection $\pi_{p}(X_{\alpha})$ is the scroll
induced by $\mu^{*}L_{\alpha}$. By the assumption
$h^{0}(C,\mu^{*}L_{\alpha})=h^{0}(D,L_{\alpha})=2$, $\pi_{p}(X_{\alpha})$ is a
rational normal scroll of degree $p$, [Sch, §4]. The curve $C$ lies in the
smooth locus of $\pi_{p}(X_{\alpha})$ as $L_{\alpha}$ is base-point free.

We now follow [Ke2, §3]. Let $Y_{\alpha}\supseteq X_{\alpha}$ be the cone over
$\pi_{p}(X_{\alpha})$. We may resolve $Y_{\alpha}$ by a scroll
$\widetilde{Y}_{\alpha}$, [Sch]. Let
$\mu_{Y_{\alpha}}:\widetilde{Y}_{\alpha}\to Y_{\alpha}$ be the resolution of
singularities and let $X^{\prime}_{\alpha}$ be the strict transform of
$X_{\alpha}$. Let $\mathcal{O}_{\widetilde{Y}_{\alpha}}(1)$ be the class of a
hyperplane, and let $R$ be the class of the ruling on the scroll
$\widetilde{Y}_{\alpha}$. As in the proof of [Ke2, Thm. 3.6], pull-back gives
isomorphisms
$\mathrm{H}^{0}(\mathcal{O}_{X_{\alpha}}(q))\simeq\mathrm{H}^{0}(\mathcal{O}_{X^{\prime}_{\alpha}}(q))$
for all $q$, so we have natural isomorphisms
$\displaystyle{\mathrm{K}_{p,1}(X_{\alpha},\mathcal{O}_{X_{\alpha}}(1))\simeq\mathrm{K}_{p,1}(X^{\prime}_{\alpha},\mathcal{O}_{X^{\prime}_{\alpha}}(1))}$.
We further have a map
$\displaystyle{\mathrm{K}_{p,1}(X^{\prime}_{\alpha},\mathcal{O}_{X^{\prime}_{\alpha}}(1))\xrightarrow{\Delta}\mathrm{K}_{p-1,1}(\widetilde{Y}_{\alpha},-R;\mathcal{O}_{\widetilde{Y}_{\alpha}}(1)).}$
Let $s\in\mathrm{H}^{0}(\mathcal{O}_{\widetilde{Y}_{\alpha}}(R))$ be a section
such that the ruling $Z(s)$ defined by $s$ has the property
$p\in\mu_{Y_{\alpha}}(Z(s))$. Then multiplication by $s$ induces a morphism

$\mathrm{K}_{p,1}(\widetilde{Y}_{\alpha},-R;\mathcal{O}_{\widetilde{Y}_{\alpha}}(1))\xrightarrow{\gamma_{s}}\mathrm{K}_{p,1}(\widetilde{Y}_{\alpha},\mathcal{O}_{\widetilde{Y}_{\alpha}}(1)).$

Letting $Z$ denote the cone over $C=\pi_{p}(D)$ as above, note that the
resolution of singularities $\widetilde{Z}=Bl_{p}(Z)$ naturally lies in the
blow-up $Bl_{p}(Y_{\alpha})$ of $Y_{\alpha}$ at $p$. We have a natural,
birational, morphism $g:Bl_{p}(Y_{\alpha})\to\widetilde{Y}_{\alpha}$. Further,
as above, let $D^{\prime}\subseteq\widetilde{Z}$ be the strict transform of
$D$, so $C\simeq D^{\prime}$. Then

${g^{*}_{s}}_{|_{D^{\prime}}}\in\mathrm{H}^{0}(C,\mu^{*}L_{\alpha}),$

is a section which vanishes at $x$ and $y$. As in [Ke2, §3], we have a
commutative diagram

${\mathrm{K}_{p,1}(X^{\prime}_{\alpha},\mathcal{O}_{X^{\prime}_{\alpha}}(1))}$${\mathrm{K}_{p-1,1}(\widetilde{Y}_{\alpha},\mathcal{O}_{\widetilde{Y}_{\alpha}}(1))}$${\mathrm{K}_{p,1}(D,\omega_{D})}$${\mathrm{K}_{p-1,1}(C,\omega_{C}).}$$\scriptstyle{\gamma_{s}\circ\Delta}$$\scriptstyle{r_{D^{\prime}}}$$\scriptstyle{r_{\widetilde{Z}}}$$\scriptstyle{\gamma_{x,y}\circ\delta}$

where $r_{D^{\prime}}$ is obtained by restricting to $D^{\prime}$ and using
the natural identifications
$\mathrm{K}_{p,1}(D,\omega_{D})\simeq\mathrm{K}_{p,1}(D^{\prime},\mathcal{O}_{D^{\prime}}(1))$,
whereas $r_{\widetilde{Z}}$ is obtained by pulling back to $\widetilde{Z}$ and
using the natural identifications
$\mathrm{K}_{p-1,1}(C,\omega_{C})\simeq\mathrm{K}_{p-1,1}(\widetilde{Z},\mathcal{H})$.

By definition of the syzygy scheme $\mathrm{Syz}(\alpha)=X_{\alpha}$, there
exists some
$\beta\in\mathrm{K}_{p,1}(X^{\prime}_{\alpha},\mathcal{O}_{X^{\prime}_{\alpha}}(1))\simeq\mathrm{K}_{p,1}(X_{\alpha},\mathcal{O}_{X_{\alpha}}(1))$
with $r_{D^{\prime}}(\beta)=\alpha$. Then
$\gamma_{x,y}(\delta(\alpha))\in\mathrm{Im}(r_{\widetilde{Z}})$. By
Proposition 1.11,
$\mathrm{K}_{p-1,1}(\widetilde{Y}_{\alpha},\mathcal{O}_{\widetilde{Y}_{\alpha}}(1))$
is generated by syzygies $\sigma^{\prime}$ of rank $p$. The pull-back map
$r_{\widetilde{Z}}$ does not change the rank of a syzygy. Thus
$\gamma_{x,y}(\delta(\alpha))$ is a linear combination of syzygies
$\sigma:=r_{\widetilde{Z}}(\sigma^{\prime})$ of rank $p$. We have isomorphisms

$\displaystyle\mathrm{K}_{p-1,1}(\widetilde{Y}_{\alpha},\mathcal{O}_{\widetilde{Y}_{\alpha}}(1))\simeq\mathrm{K}_{p-1,1}(Y_{\alpha},\mathcal{O}_{{{\mathbf{P}}}^{g}_{|_{Y_{\alpha}}}}(1))\simeq\mathrm{K}_{p-1,1}(\pi_{p}(X_{\alpha}),\mathcal{O}_{{{\mathbf{P}}}^{g-1}_{|_{\pi_{p}(X_{\alpha})}}}(1)).$

Under this identification, $r_{\widetilde{Z}}$ becomes the natural inclusion

$\mathrm{K}_{p-1,1}(\pi_{p}(X_{\alpha}),\mathcal{O}_{\pi_{p}(X_{\alpha})}(1))\hookrightarrow\mathrm{K}_{p-1,1}(C,\omega_{C}).$

Thus $\pi_{p}(X_{\alpha})\subseteq\mathrm{Syz}(\sigma)$ by definition of the
syzygy scheme, [AN2, Remark 3.6]. By [vB3, Corollary 5.2], the syzygy scheme
$\mathrm{Syz}(\sigma)$ is a rational normal scroll of degree $p$ and
codimension $p-1$, and therefore $\mathrm{Syz}(\sigma)=\pi_{p}(X_{\alpha})$,
as required. ∎

######  Lemma 3.5.

Let $C$ be a general, integral $m$-nodal curve of arithmetic genus $g\geq 2$
for any $0\leq m\leq g$. Let $x,y\in C$ be general points in the smooth locus
of $C$. For $n\leq\lfloor\frac{g}{2}\rfloor-2$, we have

$\mathrm{K}_{n,2}(C,-x-y,\omega_{C})=0.$

Further, if $n\neq g-2$ then $\mathrm{K}_{n,3}(C,-x-y,\omega_{C})=0$.

###### Proof.

By Equation (2) we have the exact sequence

$\to\mathrm{K}_{n+1,2}(D,\omega_{D})\to\mathrm{K}_{n,2}(C,-x-y,\omega_{C})\to\mathrm{K}_{n,3}(D,\omega_{D})\to$

But $\mathrm{K}_{p,3}(D,\omega_{D})=0$ unless $p=g-1$ for the canonical curve
$(D,\omega_{D})$, whereas $\mathrm{K}_{n+1,2}(D,\omega_{D})=0$ for
$n\leq\lfloor\frac{g}{2}\rfloor-2$ by Voisin’s Theorem [V2], which holds for a
general $m$-nodal curve as there exist $j$-nodal curves in the primitive
linear system on a general K3 surface of genus $\ell$ for all $0\leq
j\leq\ell$, [C]. From the exact sequence

$\to\mathrm{K}_{n+1,3}(D,\omega_{D})\to\mathrm{K}_{n,3}(C,-x-y,\omega_{C})\to\mathrm{K}_{n,4}(D,\omega_{D})\to$

and the vanishings $\mathrm{K}_{n,4}(D,\omega_{D})$, valid for all $n$,
$\mathrm{K}_{n+1,3}(D,\omega_{D})=0$, valid for $n+1\neq(g+1)-2$, we see
$\mathrm{K}_{n,3}(C,-x-y,\omega_{C})=0$.

∎

Since $\mu_{*}\omega_{C}\simeq\omega_{D}\otimes I_{p}$ we have an exact
sequence
$\displaystyle{0\to\mathrm{H}^{0}(C,\omega_{C})\to\mathrm{H}^{0}(D,\omega_{D})\xrightarrow{ev_{p}}\mathbb{C}\to
0.}$ We have the Aprodu projection map $pr:=pr_{ev_{p}}$ acting on
$\mathrm{K}_{p,1}(D,\omega_{D})$. By [Ap1], Lemma 3.1, the image of $pr$ lands
in $\mathrm{K}_{p-1,1}(C,\omega_{C})$ since
$\pi_{p}(D)=C\subseteq{\mathbf{P}}^{g-1}$, where
$\pi_{p}:{\mathbf{P}}^{g}\dashrightarrow{\mathbf{P}}^{g-1}$ is the projection
away from $p\in D$, i.e. we have the map

$pr\,:\,\mathrm{K}_{p,1}(D,\omega_{D})\to\mathrm{K}_{p-1,1}(C,\omega_{C}).$

######  Lemma 3.6.

Let $C$ be an integral $m$-nodal curve of arithmetic genus $g$ for any $0\leq
m\leq g$ such that $(C,\omega_{C})$ is normally generated. Let $D$ be the
$m+1$ nodal curve obtained by identifying general points $x,y$ in the smooth
locus of $C$. For any $p\geq 0$, the image of
$pr:\mathrm{K}_{p,1}(D,\omega_{D})\to\mathrm{K}_{p-1,1}(C,\omega_{C})$ is
contained in

$\gamma_{x,y}(\mathrm{K}_{p-1,1}(C,-x-y,\omega_{C}))\subseteq\mathrm{K}_{p-1,1}(C,\omega_{C}).$

###### Proof.

We have isomorphisms
$\mathrm{H}^{0}(D,\omega_{D})\simeq\mathrm{H}^{0}(C,\omega_{C}(x+y)),\;\mathrm{H}^{0}(C,\omega_{C})\simeq\mathrm{H}^{0}(C,\omega_{C}(x)),$
inducing natural isomorphisms

$\mathrm{K}_{p,1}(D,\omega_{D})\simeq\mathrm{K}_{p,1}(C,\omega_{C}(x+y)),\;\;\mathrm{K}_{p-1,1}(C,\omega_{C})\simeq\mathrm{K}_{p-1,1}(C,\omega_{C}(x)).$

We may then identify
$pr:\mathrm{K}_{p,1}(D,\omega_{D})\to\mathrm{K}_{p-1,1}(C,\omega_{C})$ with
$pr_{y}:\mathrm{K}_{p,1}(C,\omega_{C}(x+y))\to\mathrm{K}_{p-1,1}(C,\omega_{C}(x))$.
Precisely, let $s\in\mathrm{H}^{0}(\omega_{D})$ be a section with
$ev_{p}(s)=1$, which corresponds to a section
$s^{\prime}\in\mathrm{H}^{0}(\omega_{C}(x+y))$ with
$ev_{x}(s^{\prime})=ev_{y}(s^{\prime})=1$, identifying $\mathcal{O}_{x}$ and
$\mathcal{O}_{y}$ with $\mathcal{O}_{p}$ via $\mu$. The decomposition

$\displaystyle\wedge^{p}\mathrm{H}^{0}(\omega_{C}(x+y))\otimes\mathrm{H}^{0}(\omega_{C}(x+y))$
$\displaystyle\simeq\wedge^{p}\mathrm{H}^{0}(\omega_{C}(x))\otimes\mathrm{H}^{0}(\omega_{C}(x))\oplus\wedge^{p}\mathrm{H}^{0}(\omega_{C}(x))\otimes
s^{\prime}$
$\displaystyle\oplus\wedge^{p-1}\mathrm{H}^{0}(\omega_{C}(x))\wedge
s^{\prime}\otimes\mathrm{H}^{0}(\omega_{C}(x))\oplus\wedge^{p-1}\mathrm{H}^{0}(\omega_{C}(x))\wedge
s^{\prime}\otimes s^{\prime}$

corresponds to the decomposition

$\wedge^{p}\mathrm{H}^{0}(\omega_{D})\otimes\mathrm{H}^{0}(\omega_{D})\simeq
V_{1}\oplus V_{2}\oplus V_{3}\oplus V_{4}$

with

$\displaystyle V_{1}$
$\displaystyle:=\bigwedge^{p}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C}),$
$\displaystyle V_{2}$
$\displaystyle:=\bigwedge^{p}\mathrm{H}^{0}(\omega_{C})\otimes s,$
$\displaystyle V_{3}$
$\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(\omega_{C})\right)\wedge
s\otimes\mathrm{H}^{0}(\omega_{C}),$ $\displaystyle V_{4}$
$\displaystyle:=\left(\bigwedge^{p-1}\mathrm{H}^{0}(\omega_{C})\right)\wedge
s\otimes s.$

Applying Proposition 3.1 to $pr_{y}$ and using the above identifications we
see

$pr(\bar{\alpha})=\overline{(-1)^{p}\alpha^{\prime}_{3}+d\alpha^{\prime}_{2}}$

for $\alpha=\alpha_{1}+\alpha^{\prime}_{2}\otimes s+\alpha^{\prime}_{3}\wedge
s+\alpha_{4}$, with $\alpha_{1}\in V_{1}$, $\alpha^{\prime}_{2}\otimes s\in
V_{2}$, $\alpha^{\prime}_{3}\wedge s\in V_{3}$, $\alpha_{4}\in V_{4}$.
Further, by the second part of Proposition 3.1, applied to $pr_{y}$ again,

$(-1)^{p}\alpha^{\prime}_{3}+d\alpha^{\prime}_{2}\in\wedge^{p-1}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C}(-y))\simeq\wedge^{p-1}\mathrm{H}^{0}(\omega_{C}(x))\otimes\mathrm{H}^{0}(\omega_{C}(x-y)).$

Swapping the roles of $x$ and $y$, we see

$(-1)^{p}\alpha^{\prime}_{3}+d\alpha^{\prime}_{2}\in\wedge^{p-1}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C}(-x))$

so that
$(-1)^{p}\alpha^{\prime}_{3}+d\alpha^{\prime}_{2}\in\wedge^{p-1}\mathrm{H}^{0}(\omega_{C})\otimes\mathrm{H}^{0}(\omega_{C}(-x-y)),$
which gives the claim.

∎

We have a short exact sequence

(3)
$0\to\Gamma_{C}(-x-y,\omega_{C})\to\Gamma_{C}(\omega_{C})\to\mathbb{T}_{x,y}(C)\to
0$

for some graded $S$-module $\mathbb{T}_{x,y}(C)$.

######  Proposition 3.7.

Let $C$ be a general, integral $m$-nodal curve of arithmetic genus $g\geq 2$
for any $0\leq m\leq g$. Let $x,y\in C$ be general points in the smooth locus
of $C$. Fix $p\leq\lfloor\frac{g-3}{2}\rfloor$. As the points $x,y\in C$ vary
over $C$, the subspaces
$\mathrm{Im}(\gamma_{x,y})\subseteq\mathrm{K}_{p,1}(C,\omega_{C})$ span
$\mathrm{K}_{p,1}(C,\omega_{C})$.

###### Proof.

In fact, we will prove that if $x,y,s,t\in C$ are general then
$\mathrm{Im}(\gamma_{x,y})\cup\mathrm{Im}(\gamma_{s,t})$ spans
$\mathrm{K}_{p,1}(C,\omega_{C})$. From the long exact sequence of Koszul
cohomology associated to the short exact sequence 3, we have an exact sequence

$\mathrm{K}_{p,1}(C,-x-y,\omega_{C})\xrightarrow{\gamma_{x,y}}\mathrm{K}_{p,1}(C,\omega_{C})\xrightarrow{\alpha}\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))\to
0,$

since $\mathrm{K}_{p-1,2}(C,-x-y,\omega_{C})=0$ by Lemma 3.5. Thus, to prove
the claim it suffices to show

$\alpha_{|_{\mathrm{Im}(\gamma_{s,t})}}\;:\;\mathrm{Im}(\gamma_{s,t})\to\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))$

is surjective.

Let $D^{\prime}$ be the nodal curve of genus $g+1$ obtained by identifying $s$
and $t$, with partial normalization map $\mu^{\prime}:C\to D^{\prime}$. We
have a short exact sequence

$0\to\Gamma_{D^{\prime}}(-x-y,\omega_{D^{\prime}})\to\Gamma_{D^{\prime}}(\omega_{D^{\prime}})\to\mathbb{T}_{x,y}(D^{\prime})\to
0$

of
$\overline{S}:=\mathrm{Sym}\left(\mathrm{H}^{0}(D^{\prime},\omega_{D^{\prime}})\right)$-modules.
By Lemma 3.5 we have a surjection

$\alpha^{\prime}\;:\;\mathrm{K}_{p+1,1}(D^{\prime},\omega_{D^{\prime}})\twoheadrightarrow\mathrm{K}_{p+1,1}(\mathbb{T}_{x,y}(D^{\prime}),\mathrm{H}^{0}(\omega_{D^{\prime}}))$

since $p\leq\lfloor\frac{g+1}{2}\rfloor-2=\lfloor\frac{g-3}{2}\rfloor$. We
have a short exact sequence

$0\to\mathrm{H}^{0}(C,\omega_{C})\xrightarrow{\mu^{\prime}_{*}}\mathrm{H}^{0}(D,\omega_{D})\to\mathbb{C}\to
0,$

inducing an inclusion $S\hookrightarrow\overline{S}$ of graded ring. By
restriction of scalars, we may consider $\mathbb{T}_{x,y}(D^{\prime})$ as an
$S$-module, denoted $(\mathbb{T}_{x,y}(D^{\prime}))_{S}$. Then we may identify
$(\mathbb{T}_{x,y}(D^{\prime}))_{S}$ with $\mathbb{T}_{x,y}(C)$. We have a
projection map

$\mathrm{pr}\,:\,\mathrm{K}_{p+1,1}(\mathbb{T}_{x,y}(D^{\prime}),\mathrm{H}^{0}(\omega_{D^{\prime}}))\to\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C})),$

by [Ap1, §2], which fits into an exact sequence

$\to\mathrm{K}_{p+1,1}(\mathbb{T}_{x,y}(D^{\prime}),\mathrm{H}^{0}(\omega_{D^{\prime}}))\xrightarrow{pr}\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))\to\mathrm{K}_{p,2}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))\to\ldots.$

We have the exact sequence

$\to\mathrm{K}_{p,2}(C,\omega_{C})\to\mathrm{K}_{p,2}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))\to\mathrm{K}_{p-1,3}(C,-x-y,\omega_{C})\to\ldots$

We have $\mathrm{K}_{p,2}(C,\omega_{C})=0$ by Voisin’s Theorem, whereas
$\mathrm{K}_{p-1,3}(C,-x-y,\omega_{C})=0$ by Lemma 3.5. Thus
$\mathrm{K}_{p,2}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C}))=0$ and we
have a _surjective_ map

$\mathrm{pr}\,:\,\mathrm{K}_{p+1,1}(\mathbb{T}_{x,y}(D^{\prime}),\mathrm{H}^{0}(\omega_{D^{\prime}}))\twoheadrightarrow\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C})).$

We have a commutative diagram

${\mathrm{K}_{p+1,1}(D^{\prime},\omega_{D^{\prime}})}$${\mathrm{K}_{p+1,1}(\mathbb{T}_{x,y}(D^{\prime}),\mathrm{H}^{0}(\omega_{D^{\prime}}))}$${\mathrm{K}_{p,1}(C,\omega_{C})}$${\mathrm{K}_{p,1}(\mathbb{T}_{x,y}(C),\mathrm{H}^{0}(\omega_{C})),}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{pr}$$\scriptstyle{pr}$$\scriptstyle{\alpha}$

by functoriality of projection maps, [Ap1, §2]. Since

$pr(\mathrm{K}_{p+1,1}(D^{\prime},\omega_{D^{\prime}}))\subseteq\mathrm{Im}(\gamma_{s,t})\subseteq\mathrm{K}_{p,1}(C,\omega_{C})$

by Lemma 3.6, we see that $\alpha_{|_{\mathrm{Im}(\gamma_{s,t})}}$ is
surjective, as required. ∎

##  4\. The Geometric Syzygy Conjecture

We will prove the Geometric Syzygy Conjecture by induction on genus. The next
lemma provides the induction step.

######  Proposition 4.1.

Let $C$ be a general, integral $m$-nodal curve of arithmetic genus $g\geq 2$
for any $0\leq m\leq g$. Let $x,y\in C_{sm}$ be general and let $D$ be the
nodal curve obtained by identifying $x$ and $y$. Let
$p\leq\lfloor\frac{g-1}{2}\rfloor$. Suppose $\mathrm{K}_{p,1}(D,\omega_{D})$
is spanned by syzygies $\alpha$ satisfying the following properties:

  1. (i)

$\alpha$ has minimal rank $p+1$.

  2. (ii)

The curve $D$ lies in the smooth locus of the syzygy scheme
$X_{\alpha}:=\mathrm{Syz}(\alpha)$.

  3. (iii)

The associated line bundle $L_{\alpha}:=\mathcal{O}_{D}(R)$ satisfies
$\deg(L_{\alpha})=g+1-p$ and $h^{0}(C,\mu^{*}L_{\alpha})=2$, where $\mu:C\to
D$ is the partial normalization map.

Then $\mathrm{K}_{p-1,1}(C,\omega_{C})$ is spanned by syzygies $\sigma$ of
rank $p$ such that $X_{\sigma}:=\mathrm{Syz}(\sigma)$ is a scroll with $C$
lying in the smooth locus of $X_{\sigma}$ and with associated line bundle of
the form $L_{\sigma}=\mu^{*}L_{\alpha}$, where $L_{\alpha}$ is associated to a
syzygy $\alpha\in\mathrm{K}_{p,1}(D,\omega_{D})$ of rank $p+1$.

###### Proof.

By Proposition 3.7, $\mathrm{K}_{p-1,1}(C,\omega_{C})$ is spanned by the
subspaces $\mathrm{Im}(\gamma_{x,y})\subseteq\mathrm{K}_{p-1,1}(C,\omega_{C})$
as $x,y\in C_{sm}$, vary. Thus it suffices to show $\mathrm{Im}(\gamma_{x,y})$
is spanned by syzygies $\sigma$ of rank $p$ and with the required properties.
We have $\mathrm{K}_{p-1,2}(C,\omega_{C})=0$ by generic Green’s Conjecture
[V2], and thus a surjection

$\delta\;:\;\mathrm{K}_{p,1}(D,\omega_{D})\twoheadrightarrow\mathrm{K}_{p-1,1}(C,-x-y,\omega_{C}).$

Since $\mathrm{K}_{p,1}(D,\omega_{D})$ is spanned by syzygies $\alpha$
satisfying the properties $(i),(ii),(iii)$, it suffices to show that, for such
a syzygy $\alpha$, the syzygy $\gamma_{x,y}(\delta(\alpha))$ is a linear
combination of syzygies of rank $p$ with the required properties. This follows
from Proposition 3.4. ∎

Let $(X,L)$ be a primitively polarized K3 surface of genus $g$, i.e.
$(L)^{2}=2g-2$.

######  Lemma 4.2.

Let $(X,L)$ be a very general, primitively polarized, K3 surface of even genus
$g=2k$ for $g\geq 2$. Then any curve $C\in|L|$ is integral, with $\omega_{C}$
very ample, and further $(C,\omega_{C})$ is normally generated. We have
$W^{1}_{d}(C)=\emptyset$ for $d\leq k$ and, further, each $[A]\in
W^{1}_{k+1}(C)$ is globally generated and has $h^{0}(A)=2$. Consider the
bundle $E$ whose dual fits into the exact sequence

$0\to{E}^{\vee}\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to A_{|_{C}}\to 0.$

Then $E$ is simple, has invariants has invariants $\mathrm{det}(E)\simeq L$,
$h^{0}(E)=k+2$, $h^{1}(E)=h^{2}(E)=0$ and does not depend on the choice of
$[A]\in W^{1}_{k+1}(C)$ or of $C\in|L|$.

###### Proof.

A very general polarized K3 has $\mathrm{Pic}(X)=\mathbb{Z}[L]$, which implies
any $C\in|L|$ is integral and also that $L$ is very ample, [Kn, Thm. 1.1]. For
any $C\in|L|$, restricting the embedding $\phi_{L}:X\to{\mathbf{P}}^{g}$ to
the hyperplane $C$ gives the canonical embedding
$\phi_{\omega_{C}}:C\to{\mathbf{P}}^{g-1}$, and so $\omega_{C}$ is very ample.
To show $(C,\omega_{C})$ is normally generated, it is equivalent to show
$K_{0,q}(C,\omega_{C})=0$ for $q\geq 2$. This is equivalent to
$K_{0,q}(X,L)=0$ for $q\geq 2$ by the Hyperplane Restriction Theorem, [Gr2].
The claim is now a (very special) case of Voisin’s Theorem, [V1].

As $C\in|L|$ is integral and lies on a smooth surface, we may construct the
Brill–Noether loci $W^{r}_{d}(C)\subseteq\bar{J}^{d}(C)$. Let $[A]\in
W^{1}_{k+1}(C)$ be globally generated with $h^{0}(A)=2$, and let $E^{\vee}$ be
defined by the exact sequence
$0\to{E}^{\vee}\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to A_{|_{C}}\to 0.$
By [L1], generalized in [Go, Lemma 2.2] to the torsion-free case, $E^{\vee}$
as defined in the lemma is a vector bundle of rank two. As in [L1, §1] the
condition $\mathrm{Pic}(X)=\mathbb{Z}[L]$ (satisfied for a very general
polarized K3), implies that $E$ is simple (also compare with [Ke1, Lemma 5.5]
and [CK, Thm. 3.1]). The bundle $E$ has invariants $\mathrm{det}(E)\simeq L$,
$h^{0}(E)=k+2$, $h^{1}(E)=h^{2}(E)=0$ as in [L1, §1]. Since the moduli space
of stable bundles on K3 surfaces with these invariants is zero dimensional,
the bundle $E$ does not depend on the choice of $[A]\in W^{1}_{k+1}(C)$ or of
$C\in|L|$.

We now show $[A]\in W^{1}_{k+1}(C)$ is globally generated and has
$h^{0}(A)=2$. Suppose $h^{0}(A)\geq 3$. Let $A^{\prime}\subseteq A$ be the
subsheaf of $A$ generated by the global sections of $A$, i.e. $A^{\prime}$ is
the image of the evaluation morphism

$\mathrm{H}^{0}(A)\otimes\mathcal{O}_{C}\to A.$

Then $A^{\prime}$ is a rank-one torsion free sheaf with
$\mathrm{H}^{0}(A^{\prime})=\mathrm{H}^{0}(A)$ which is generated by its
global sections. We have $[A^{\prime}]\in W^{2}_{d}(C)$ for $d\leq k+1$. Let
$i:C\hookrightarrow X$ be the natural inclusion. We have a Lazarsfeld–Mukai
bundle $F$ on $X$, whose dual fits into the exact sequence

$0\to{F}^{\vee}\to\mathrm{H}^{0}(A^{\prime})\otimes\mathcal{O}_{X}\to
i_{*}A^{\prime}\to 0.$

As above, the generality of $X$ implies that $F$ is simple. This in turn
implies $\rho(A^{\prime})=g-h^{0}(A^{\prime})(g-d+h^{0}(A^{\prime})-1)\geq 0$
by [L1, §1]. Thus
$g-d+h^{0}(A^{\prime})-1\leq\frac{g}{h^{0}(A^{\prime})}\leq\frac{2k}{3}$. But
$g-d+h^{0}(A^{\prime})-1\geq 2k-(k+1)+3-1=k+1$, which is a contradiction.
Similarly, if $A\in W^{1}_{k+1}(C)$ were not base-point free, the subsheaf
$A^{\prime}\subseteq A$ generated by the global sections of $A$ would have
$\rho(A^{\prime})<0$, where
$\rho(A^{\prime}):=g-h^{0}(A^{\prime})h^{1}(A^{\prime})$ is the Brill–Noether
number, which cannot happen by [L1, §1] as above. The same argument show that
$W^{1}_{d}(C)=\emptyset$ for $d\leq k$, since otherwise we would have torsion-
free sheaves with negative Brill–Noether number $\rho$.

∎

We now give a description of the sections of the vector bundle $E$ from Lemma
4.2.

######  Lemma 4.3.

Let $E$ be the vector bundle from Lemma 4.2. We have a natural isomorphism

$\mathrm{H}^{0}(E)\simeq\mathrm{Ker}\mu,$

where
$\mu:\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(L)\to\mathrm{H}^{0}(A\otimes\omega_{C}),$
is the multiplication map composed with restriction.

We further have an isomorphism

$\alpha:\mathrm{H}^{0}(A)\oplus\mathrm{H}^{0}(A^{-1}\otimes\omega_{C})\simeq\mathrm{Ker}\mu.$

Explicitly, if $t\in\mathrm{H}^{0}(L)$ defines $C$, if
$q:\mathrm{H}^{0}(\omega_{C})\to\mathrm{H}^{0}(L)$ is a section of the
restriction map $\mathrm{H}^{0}(L)\to\mathrm{H}^{0}(\omega_{C})$ and if
$\\{x,y\\}$ is a basis for $\mathrm{H}^{0}(A)$, we define $\alpha$ by the
formula

$\displaystyle\mathrm{H}^{0}(A)\oplus\mathrm{H}^{0}(A^{-1}\otimes\omega_{C})$
$\displaystyle\to\mathrm{Ker}\mu$ $\displaystyle(v,w)$
$\displaystyle\mapsto(v\otimes t,y\otimes q(xw)-x\otimes q(yw))$

###### Proof.

Twisting the defining sequence for $E^{\vee}$ by $\mathcal{O}_{X}(C)$ and
using the natural isomorphism $E\simeq E^{\vee}(C)$, we have a short exact
sequence

$0\to{E}\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}(C)\to A\otimes\omega_{C}\to
0.$

Thus $\mathrm{H}^{0}(E)\simeq\mathrm{Ker}\mu,$ where
$\mu:\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(L)\to\mathrm{H}^{0}(A\otimes\omega_{C}),$
is the multiplication map.

By Lemma 2.1 we have an exact sequence

$0\to
A^{-1}\otimes\omega_{C}\xrightarrow{i}\mathrm{H}^{0}(A)\otimes\omega_{C}\xrightarrow{ev}A\otimes\omega_{C}\to
0,$

where $i(s)=y\otimes\rho(xs)-x\otimes\rho(ys),$ for $\\{x,y\\}$ a basis for
$\mathrm{H}^{0}(A)$. We have a commutative diagram

${0}$${0}$${\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}}$${\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}}$${0}$${E}$${\mathrm{H}^{0}(A)\otimes
L}$${A\otimes\omega_{C}}$${0}$${0}$${A^{-1}\otimes\omega_{C}}$${\mathrm{H}^{0}(A)\otimes\omega_{C}}$${A\otimes\omega_{C}}$${0}$${0}$${0}$$\scriptstyle{\sim}$$\scriptstyle{\otimes
t}$$\scriptstyle{\simeq}$$\scriptstyle{i}$

with exact rows and columns. The claimed isomorphism from
$\mathrm{H}^{0}(A)\oplus\mathrm{H}^{0}(A^{-1}\otimes\omega_{C})$ to
$\mathrm{Ker}\mu\simeq\mathrm{H}^{0}(E)$ follows. ∎

######  Remark 4.4.

From the first column of the diagram in the above proof, we have an exact
sequence

$0\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to E\to
A^{-1}\otimes\omega_{C}\to 0.$

Restricting to $C$, and using the isomorphism

$\mathrm{Ker}(\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to
A\otimes\mathcal{O}_{C})=:E^{\vee}\simeq E(-C),$

we also obtain an exact sequence $0\to A\to E_{|_{C}}\to
A^{-1}\otimes\omega_{C},$ of sheaves on $C$.

Let $E$ be the rank two Lazarsfeld–Mukai bundle from Lemma 4.2 and consider
the restriction $E_{C}$ to an integral curve $C\in|L|$. We have the natural
determinant map

$\mathrm{det}\;:\;\bigwedge^{2}\mathrm{H}^{0}(E_{C})\to\mathrm{H}^{0}(\omega_{C}),$

obtained as the composition
$\wedge^{2}\mathrm{H}^{0}(E_{C})\to\mathrm{H}^{0}(\wedge^{2}E_{C})\simeq\mathrm{H}^{0}(\omega_{C})$.

######  Lemma 4.5.

Choose any section $\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})\hookrightarrow\mathrm{H}^{0}(E_{C})$ to the natural map
$\mathrm{H}^{0}(E_{C})\twoheadrightarrow\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})$ from Lemma 4.2, inducing a splitting

$\mathrm{H}^{0}(E_{C})\simeq\mathrm{H}^{0}(A)\oplus\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}).$

Consider the inclusion
$\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})\subseteq\bigwedge^{2}\mathrm{H}^{0}(E_{C})$ resulting from the above
splitting. Then the restriction

$\mathrm{det}_{|_{\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})}}$

of the determinant map to this subspace coincides with the Petri map

$\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})\to\mathrm{H}^{0}(\omega_{C}).$

Moreover, the determinant map is zero on all other components:

$\mathrm{det}_{|_{\bigwedge^{2}\mathrm{H}^{0}(A)}}=0,\;\;\mathrm{det}_{|_{\bigwedge^{2}\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})}}=0.$

###### Proof.

Restriction induces an isomorphism
$\mathrm{H}^{0}(E)\simeq\mathrm{H}^{0}(E_{C})$ since
$\mathrm{H}^{0}(E(-C))=\mathrm{H}^{0}(E^{\vee})=0$. Let
$\det_{X}:\wedge^{2}\mathrm{H}^{0}(E)\to\mathrm{H}^{0}(L)$ denote the
determinant map (on $X$). We have the commutative diagram

${\bigwedge^{2}\mathrm{H}^{0}(E)}$${\bigwedge^{2}\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(L)}$${\mathrm{H}^{0}(L)}$${\mathrm{H}^{0}(L^{\otimes
2})}$$\scriptstyle{\mathrm{det}_{X}}$$\scriptstyle{\phi}$$\scriptstyle{j}$

where $\phi$ comes from the isomorphism

$\wedge^{2}(\mathrm{H}^{0}(A)\otimes
L)\simeq(\wedge^{2}\mathrm{H}^{0}(A))\otimes L^{\otimes 2}\simeq L^{\otimes
2},$

where the last isomorphism depends on a choice of ordered basis $\\{x,y\\}$ of
$\mathrm{H}^{0}(A)$, with $x\wedge y$ the generator for
$\wedge^{2}\mathrm{H}^{0}(A)$. Further, we have the formula
$\mathrm{det}=(\det_{X})_{|_{C}}:\wedge^{2}\mathrm{H}^{0}(E_{C})\to\mathrm{H}^{0}(\omega_{C}).$

We first show $\mathrm{det}_{|_{\bigwedge^{2}\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})}}=0$. From Lemma 4.3, it suffices to show

$\phi\left((y\otimes q(xw_{1})-x\otimes q(yw_{1}))\wedge(y\otimes
q(xw_{2})-x\otimes q(yw_{2}))\right)=0,$

where $w_{1},w_{2}\in\mathrm{H}^{0}(\omega_{C}\otimes A^{-1})$. In other
words, we need

$\alpha=q(xw_{1})q(yw_{2})-q(yw_{1})q(xw_{2})=0\in\mathrm{H}^{0}(L^{\otimes
2}).$

But $\alpha=\psi(\alpha^{\prime})$ lies in the image of
$\psi:\mathrm{Sym}^{2}\mathrm{H}^{0}(L)\to\mathrm{H}^{0}(L^{\otimes 2})$.
Cearly the restriction $\alpha^{\prime}_{|_{C}}$ of $\alpha^{\prime}$ to $C$
lies in the kernel of
$\psi_{|_{C}}:\mathrm{Sym}^{2}\mathrm{H}^{0}(\omega_{C})\to\mathrm{H}^{0}(\omega_{C}^{\otimes
2})$. Since restriction induces an isomorphism

$\mathrm{Ker}(\psi)=\mathrm{K}_{1,1}(X,L)\simeq\mathrm{K}_{1,1}(C,\omega_{C})=\mathrm{Ker}(\psi_{|_{C}}),$

see [Gr2], we have $\alpha^{\prime}\in\mathrm{Ker}(\psi)$ as required.

From Lemma 4.3 we have
$\phi_{|_{\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(A)}}\subseteq\mathbb{C}\langle
t^{2}\rangle\subseteq\mathrm{H}^{0}(L^{\otimes 2})$, where
$t\in\mathrm{H}^{0}(L)$ defines $C$ and

$\displaystyle\phi\left((x\otimes t)\wedge(y\otimes q(xw)-x\otimes
q(yw))\right)$ $\displaystyle=tq(xw)$ $\displaystyle\phi\left((y\otimes
t)\wedge(y\otimes q(xw)-x\otimes q(yw))\right)$ $\displaystyle=tq(yw).$

It follows that

$j\circ\mathrm{det}_{X}(\bigwedge^{2}\mathrm{H}^{0}(E))\subseteq\mathbb{C}\langle
t\rangle\subseteq\mathrm{H}^{0}(L^{\otimes 2}).$

Thus, up to a nonzero scalar, $j$ is multiplication by $t\in\mathrm{H}^{0}(L)$
and we have the formulae

$\mathrm{det}_{X}=\frac{1}{t}\phi,\;\;\mathrm{det}={\mathrm{det}_{X}}_{|_{C}}.$

Since $t$ vanishes on $C$ the above calculations for the map $\phi$ give that
$\mathrm{det}_{|_{\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(A)}}=0$ and
$\mathrm{det}_{|_{\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})}}$ is the Petri map. ∎

We now use the determinant map to define a morphism from the Grassmannian of
planes in $\mathrm{H}^{0}(E))$ to the linear system $|L|$ as in [V1].

######  Proposition 4.6.

$E$ be the rank two Lazarsfeld–Mukai bundle from Lemma 4.2. We have a natural,
finite morphism

$d\;:\;\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))\to|L|,$

which is surjective, flat and has degree $\frac{(2k)!}{k!(k+1)!}$. Further,
the fibre of $d$ over a curve $C\in|L|$ may be identified naturally with the
Brill–Noether space $W^{1}_{k+1}(C)$.

###### Proof.

We follow [V1]. The determinant map
$\mathrm{det}\;:\;\wedge^{2}\mathrm{H}^{0}(X,E)\to\mathrm{H}^{0}(X,L)$ does
not vanish on any element of the form $s\wedge t$ for
$s,t\in\mathrm{H}^{0}(X,E)$, see [V1], proof of equation $(3.18)$. We
therefore have a well-defined morphism

$\displaystyle d\;:\;\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$ $\displaystyle\to|L|$
$\displaystyle W$ $\displaystyle\mapsto[\mathrm{det}(\wedge^{2}W)].$

The pull-back of the hyperplane class of $|L|$ by $d$ is the Plücker
hyperplane class of $\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$, [O, Lemma 2.1].
Since the Plücker hyperplane class is very ample, $d$ is finite. Since the
Grassmannian $\mathrm{Gr}(2,n)$ has degree given by the Catalan number
$\frac{(2(n-2))!}{(n-2)!(n-1)!}$, $d$ has degree $\frac{(2k)!}{k!(k+1)!}$. As
$\dim\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))=\dim|L|=2g$, the morphism $d$ is
surjective and flat by [H1, Ex. III.9.3].

Let $\pi:\mathcal{W}^{1}_{k+1}\to|L|$ be the relative Brill–Noether variety,
which is constructed as in [ACG, CH. XXI]. For any $C\in|L|$,
$\pi^{-1}(C)=W^{1}_{k+1}(C)$. We have a morphism
$\phi:\mathcal{W}^{1}_{k+1}\to\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$ defined as
such. Given a torsion-free sheaf $A\in W^{1}_{k+1}(C)$, for$C\in|L|$, we have
an exact sequence

$0\to E\to\mathrm{H}^{0}(A)\otimes L\to A\otimes\omega_{C}\to 0.$

If $t\in\mathrm{H}^{0}(L)$ defines $C$, then the image of
$\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\xrightarrow{\otimes
t}\mathrm{H}^{0}(A)\otimes L$ lands in $E$, and therefore induces an inclusion
$i_{A}:\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\hookrightarrow E$ by Lemma 4.3.
We set $\phi([A])=[\mathrm{H}^{0}(i_{A})]$. Notice also that
$\mathrm{Cok}(i_{A})\simeq A^{-1}\otimes\omega_{C}$.

If $C\in|L|$ is general, then both $\pi^{-1}(C)$ and $d^{-1}(C)$ are zero-
dimensional and nonempty. Hence $\phi$ is surjective. Thus, if
$[V\hookrightarrow\mathrm{H}^{0}(E)]\in\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$,
the cokernel of the natural map $V\otimes\mathcal{O}_{X}\to E$ is of the form
$A^{-1}_{V}\otimes\omega_{C}$ for some $A_{V}\in W^{1}_{k+1}(C^{\prime})$ for
some $C^{\prime}\in|L|$. Thus we define an inverse
$\psi:\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))\to\mathcal{W}^{1}_{k+1}$ to $\phi$ by
$\psi([V]):=[A_{V}]$. Thus $\phi$ is an isomorphism
$\mathcal{W}^{1}_{k+1}\simeq\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$. This
completes the proof.

∎

######  Lemma 4.7.

With notation as in Proposition 4.6, let $A\in W^{1}_{k+1}(C)\simeq d^{-1}(C)$
for $C\in|L|$ and let $s\neq 0\in\mathrm{H}^{0}(C,A)$. The restriction map
$\mathrm{res}:\mathrm{H}^{0}(X,L)\twoheadrightarrow\mathrm{H}^{0}(C,\omega_{C})$
induces an isomorphism

$\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}))\simeq\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes I_{Z(s)})).$

###### Proof.

By the Lefschetz Theorem [Gr2], $\mathrm{res}$ induces an isomorphism
$\alpha:\mathrm{K}_{k-1,1}(X,L)\xrightarrow{\sim}\mathrm{K}_{k-1,1}(C,\omega_{C})$.
We have the exact sequence $0\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to
E\to A^{-1}\otimes\omega_{C}\to 0$ by the proof of Lemma 4.3. Now consider the
commutative diagram

${0}$${\mathcal{O}_{X}}$${0}$${\mathcal{O}_{X}}$${E}$${L\otimes
I_{Z(s)}}$${0}$${0}$${\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}}$${E}$${A^{-1}\otimes\omega_{C}}$${0}$${0}$$\scriptstyle{\otimes
t}$$\scriptstyle{\otimes s}$$\scriptstyle{\simeq}$

where $t\in\mathrm{H}^{0}(L\otimes I_{Z_{(}s)})$ defines $C$. It follows that
$\mathrm{res}^{-1}(\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})=\mathrm{H}^{0}(L\otimes I_{Z(s)})$ and $\alpha$ maps the subspace
$\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes
I_{Z(s)}))\subseteq\mathrm{K}_{k-1,1}(X,L)$ isomorphically to
$\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}))\subseteq\mathrm{K}_{k-1,1}(C,\omega_{C})$. ∎

In the following proposition, we verify the assumptions of Proposition 2.7 in
a special case.

######  Proposition 4.8.

Let $(X,L)$ be a very general, primitively polarized, K3 surface of even genus
$g=2k$ for $g\geq 2$. Let $C\in|L|$ be an integral curve. For any $[A]\in
W^{1}_{k+1}(C)$ and $s\neq 0\in\mathrm{H}^{0}(C,A)$, the map

$\delta_{s}\;:\;\bigwedge^{k}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}\to\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee}))$

from Lemma 2.2 is injective.

###### Proof.

Let $E$ be the rank-two Lazarsfeld–Mukai bundle from Lemma 4.2. The dual
$E^{\vee}$ fits into the exact sequence
$0\to{E}^{\vee}\to\mathrm{H}^{0}(A)\otimes\mathcal{O}_{X}\to i_{*}A\to 0.$ Let
$Z=Z(s)\subseteq C$ denote the zero locus of $s$. From the surjective
restriction map $\mathrm{H}^{0}(X,L)\to\mathrm{H}^{0}(C,\omega_{C})$, we have
isomorphisms

$\displaystyle\mathrm{Coker}\left(\mathrm{H}^{0}(X,L)\to\mathrm{H}^{0}(Z,L_{|_{Z}})\right)$
$\displaystyle\simeq\mathrm{Coker}\left(\mathrm{H}^{0}(C,\omega_{C})\to\mathrm{H}^{0}(Z,L_{|_{Z}})\right)$
$\displaystyle\simeq\mathrm{Ker}\left(\mathrm{H}^{1}(C,\omega_{C}\otimes
A^{\vee})\to\mathrm{H}^{1}(C,\omega_{C})\right),$

where the given maps are the natural ones, using that $A^{\vee}\simeq
I_{Z/C}$. The map $\mathrm{H}^{1}(C,\omega_{C}\otimes
A^{\vee})\to\mathrm{H}^{1}(C,\omega_{C})$ is surjective and has one-
dimensional kernel by Serre duality. Thus,
$\mathrm{H}^{0}(X,L)\to\mathrm{H}^{0}(Z,L_{|_{Z}})$ has one dimensional
cokernel. Further, $W^{1}_{d}(C)=\emptyset$ for all $d<k+1$ by Lemma 4.2.
Applying the above argument to any proper subscheme $Z^{\prime}\subsetneq Z$,
and using that $\mathrm{H}^{0}(I_{Z^{\prime}/C}^{\vee})<2$ (as
$W^{1}_{d(Z^{\prime}))}(C)=\emptyset$) shows that the natural restriction map

$\mathrm{H}^{0}(X,L)\to\mathrm{H}^{0}(Z^{\prime},L_{|_{Z^{\prime}}})$

is surjective. By Serre’s construction, [L2, §3], we can construct a rank two
vector bundle on $X$ associated to $Z\subseteq X$. This bundle has the same
invariants as $E$ and is stable, which implies that it must be isomorphic to
$E$, see [V1, §2]. Thus we have an exact sequence

$0\to\mathcal{O}_{X}\xrightarrow{s^{\prime}}E\to L\otimes I_{Z/X}\to 0,$

where the section $s^{\prime}\in\mathrm{H}^{0}(E)$ is uniquely determined up
to a nonzero scalar multiple. Since $\mathrm{H}^{1}(\mathcal{O}_{X})=0$, we
obtain an exact sequence
$0\to\mathrm{H}^{0}(\mathcal{O}_{X})\xrightarrow{s^{\prime}}\mathrm{H}^{0}(E)\xrightarrow{\pi}\mathrm{H}^{0}(L\otimes
I_{Z/X})\to 0.$

Following [AN2, §3.4.1], we have a natural, nonzero map

$\gamma\;:\;\bigwedge^{k+1}\mathrm{H}^{0}(X,L\otimes
I_{Z/X})\to\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(X,L\otimes I_{Z/X}))$

defined as such: let $U\subseteq\mathrm{H}^{0}(E)$ be any subspace with
$\pi_{|_{U}}:U\to\mathrm{H}^{0}(L\otimes I_{Z/X})$ an isomorphism. Then, since
$\rm{rank}(U)=k+1$, we have natural isomorphisms

$\displaystyle\mathrm{Hom}_{k}(\wedge^{k+1}\mathrm{H}^{0}(L\otimes
I_{Z/X}),\wedge^{k-1}\mathrm{H}^{0}(L\otimes
I_{Z/X})\otimes\mathrm{H}^{0}(L))$
$\displaystyle\simeq\wedge^{k-1}U\otimes\wedge^{k+1}U^{\vee}\otimes\mathrm{H}^{0}(L)$
$\displaystyle\simeq\wedge^{2}U^{\vee}\otimes\mathrm{H}^{0}(L)$
$\displaystyle\simeq\mathrm{Hom}_{k}(\wedge^{2}U,\mathrm{H}^{0}(L)).$

Let $\hat{\gamma}\in\mathrm{Hom}_{k}(\wedge^{k+1}\mathrm{H}^{0}(L\otimes
I_{Z/X}),\wedge^{k-1}\mathrm{H}^{0}(L\otimes
I_{Z/X})\otimes\mathrm{H}^{0}(L))$ denote the element corresponding to
$\mathrm{det}_{U}\in\mathrm{Hom}_{k}(\wedge^{2}U,\mathrm{H}^{0}(L))$, where
$\mathrm{det}$ is the natural map
$\mathrm{det}:\wedge^{2}\mathrm{H}^{0}(E)\to\mathrm{H}^{0}(\wedge^{2}E)\simeq\mathrm{H}^{0}(L).$
It is shown in [AN2, §3.4.1] that $\hat{\gamma}$ induces a nonzero map
$\gamma:\wedge^{k+1}\mathrm{H}^{0}(L\otimes
I_{Z/X})\to\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes I_{Z/X}))$, and that
this map does not depend on the choice of $U$.

Since $\gamma$ is nonzero and $h^{0}(L\otimes I_{Z/X}))=k+1$, the map $\gamma$
is injective. To complete the proof, it remains to show that we may identify
$\gamma$ with $\delta_{s}$. From the exact sequence

$0\to I_{C/X}\to I_{Z/X}\to I_{Z/C}\to 0,$

we obtain, after twisting by $L=I_{C/X}^{\vee}$, a short exact sequence

$0\to\mathcal{O}_{X}\xrightarrow{t}L\otimes I_{Z/X}\to\omega_{C}\otimes
A^{\vee}\to 0,$

where $t\in\mathrm{H}^{0}(L\otimes I_{Z/X})$ is a section defining the curve
$C$. Taking global sections we obtain a (non-canonical) isomorphism
$\wedge^{k+1}\mathrm{H}^{0}(L\otimes
I_{Z/X})\simeq\wedge^{k}\mathrm{H}^{0}(\omega_{C}\otimes
A^{\vee})\otimes{\mathbb{C}\langle t\rangle}.$

We will now show that there is a natural isomorphism ${\mathbb{C}\langle
t\rangle}\simeq\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle s\rangle}$. By
Remark 4.4, we have a commutative diagram

${0}$${0}$${\mathrm{H}^{0}(\mathcal{O}_{X})}$${0}$${\mathrm{H}^{0}(\mathcal{O}_{C})\simeq\mathrm{H}^{0}(\mathcal{O}_{X})}$${\mathrm{H}^{0}(E)}$${\mathrm{H}^{0}(L\otimes
I_{Z/X})}$${0}$${0}$${\mathrm{H}^{0}(A)}$${\mathrm{H}^{0}(E_{C})}$${\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})}$${0}$${\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}}$${0}$${0}$$\scriptstyle{t}$$\scriptstyle{s^{\prime}}$$\scriptstyle{s}$$\scriptstyle{\pi}$$\scriptstyle{\simeq}$$\scriptstyle{q}$$\scriptstyle{\widetilde{\pi}}$

with exact rows and columns, which gives the claimed isomorphism
${\mathbb{C}\langle t\rangle}\simeq\frac{\mathrm{H}^{0}(A)}{\mathbb{C}\langle
s\rangle}$. Now let
$\\{s^{\prime},t^{\prime},v^{\prime}_{1},\ldots,v^{\prime}_{k}\\}$ be a basis
for $\mathrm{H}^{0}(E)$ with $\pi(t^{\prime})=t$, and $v_{i}:=\pi(v_{i})$,
$1\leq i\leq k$. By [AN2, §3.4.1], we have the formula (up to sign)

$\displaystyle\hat{\gamma}(v_{1}\wedge\ldots\wedge v_{k}\wedge t)$
$\displaystyle=\sum_{i<j}(-1)^{i+j}v_{1}\wedge\ldots\wedge\hat{v_{i}}\wedge\ldots\wedge\hat{v_{j}}\ldots\wedge
v_{k}\wedge t\otimes\mathrm{det}(v^{\prime}_{i}\wedge v_{j})$
$\displaystyle+\sum_{i\leq
k}(-1)^{i+k+1}v_{1}\wedge\ldots\wedge\hat{v_{i}}\wedge\ldots\wedge
v_{k}\otimes\mathrm{det}(v^{\prime}_{i}\wedge t).$

We have a natural isomorphism $\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes
I_{Z/X}))\simeq\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(C\otimes
A^{-1}))$ induced by restriction to $C$, [Ke3, §2]. Let $\widetilde{\gamma}$
be the composition of $\gamma$ with this identification. Since $t$ lies in the
kernel of the restriction map $q:\mathrm{H}^{0}(L\otimes
I_{Z/X})\to\mathrm{H}^{0}(C\otimes A^{-1}))$, $\widetilde{\gamma}$ takes
$v_{1}\wedge\ldots\wedge v_{k}\wedge t$ to the equivalence class of

$\displaystyle\sum_{i\leq
k}(-1)^{i+k+1}q(v_{1})\wedge\ldots\wedge\widehat{q(v_{i})}\wedge\ldots\wedge
q(v_{k})\otimes\mathrm{det}_{C}(v^{\prime}_{i}\wedge t^{\prime})$
$\displaystyle=\sum_{i\leq
k}(-1)^{i+k+1}q(v_{1})\wedge\ldots\wedge\widehat{q(v_{i})}\wedge\ldots\wedge
q(v_{k})\otimes\widetilde{\pi}(v^{\prime}_{i})\otimes t^{\prime},$

where we treat
$t^{\prime}\in\mathrm{Ker}(H^{0}(E_{C})\to\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}))$ as an element of $\mathrm{H}^{0}(A)$ and use that $\mathrm{det}_{C}$
coincides with the natural map
$\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1})\to\mathrm{H}^{0}(\omega_{C})$ on the component
$\mathrm{H}^{0}(A)\otimes\mathrm{H}^{0}(\omega_{C}\otimes A^{-1})$, and is
zero on all other components, by Lemma 4.5. Comparing with Lemma 2.2, we see
that we may identify $\gamma$ with $\delta_{s}$, as required. ∎

By Lemma 4.8 and Proposition 2.7, if $(X,L)$ is a very general, primitively
polarized, K3 surface of even genus $g=2k$ for $g\geq 2$ and $C\in|L|$ be an
integral curve, then we have a well-defined morphism
$S:X^{1}_{k+1}(C)\to{\mathbf{P}}(\mathrm{K}_{k-1,1}(C,\omega_{C}))$.

######  Proposition 4.9.

Set ${\mathbf{P}}:={\mathbf{P}}(\mathrm{K}_{k-1,1}(C,\omega_{C}))$. With the
above notation, the restriction map

$S^{*}\;:\;\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(1))\to\mathrm{H}^{0}(S^{*}\mathcal{O}_{{\mathbf{P}}}(1))$

is injective.

###### Proof.

Let $G:=\mathrm{Gr}_{2}(\mathrm{H}^{0}(E))$, where $E$ is the vector bundle
from Lemma 4.2. We have the tautological subbundle
$F\hookrightarrow\mathrm{H}^{0}(E)\otimes\mathcal{O}_{G}$ on the Grassmannian
$G$, with $F\otimes k(p)\simeq V\subseteq\mathrm{H}^{0}(E)$ for
$p=[V\subseteq\mathrm{H}^{0}(E)]\in G$. Let

$p\;:\;{\mathbf{P}}(F)\to|L|$

be the composite of the projection ${\mathbf{P}}(F)\to G$ with the morphism
$d:G\to|L|$ from Proposition 4.6. We may naturally identify $p^{-1}(C)$ with
$X^{1}_{k+1}(C)$. We have the natural map,
$q:{\mathbf{P}}(F)\to{\mathbf{P}}(\mathrm{H}^{0}(E))$ as well as a morphism
$\psi:{\mathbf{P}}(\mathrm{H}^{0}(E))\to{\mathbf{P}}(\mathrm{K}_{k-1,1}(X,L))$
defined by

$\psi([s]):=[\mathrm{K}_{k-1,1}(X,L,\mathrm{H}^{0}(L\otimes I_{Z(s)}))],$

[Ke3, Thm. 2]. By Lemma 4.7, for $s\in\mathrm{H}^{0}(A)\in W^{1}_{k+1}(C)$, we
have an isomorphism

$\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}))\simeq\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes I_{Z(s)})),$

where we view $\mathrm{H}^{0}(A)$ as a subspace of $\mathrm{H}^{0}(E)$ via
Proposition 4.6. In particular,

$\dim_{\mathbb{C}}\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
A^{-1}))=\dim_{\mathbb{C}}\mathrm{K}_{k-1,1}(X,L;\mathrm{H}^{0}(L\otimes
I_{Z(s)}))=1,$

so the injective map $\delta_{s}$ from Proposition 4.8 is an isomorphism.
Furthermore, $S$ is identified with $\psi\circ q_{|_{p^{-1}(C)}}.$ By [Ke3,
Thm. 2], the map
$\psi^{*}:\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{K}_{k-1,1}(X,L))}(1))\to\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{H}^{0}(E))}(k-2)),$
is an isomorphism. We further have the isomorphism

$q^{*}\;:\;\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(\mathrm{H}^{0}(E))}(k-2))\simeq\mathrm{Sym}^{k-2}\mathrm{H}^{0}(E)^{\vee}\to\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}(F)}(k-2))\simeq\mathrm{H}^{0}(G,\mathrm{Sym}^{k-2}F^{\vee}).$

We further have a natural identification
$\mathrm{H}^{0}(p^{-1}(C),\mathcal{O}_{{\mathbf{P}}(F)}(k-2))\simeq\mathrm{H}^{0}(d^{-1}(C),\mathrm{Sym}^{k-2}F^{\vee})$.
So it suffices to show that the restriction map

$\mathrm{H}^{0}(G,\mathrm{Sym}^{k-2}F^{\vee})\to\mathrm{H}^{0}(d^{-1}(C),\mathrm{Sym}^{k-2}F^{\vee})$

is injective. This is proven in [V1], proof of Proposition 7, by a Koszul
complex computation, noting that $d^{-1}(C)$ is a complete intersection of
hyperplane sections of $G$. ∎

We now deform to a _general_ nodal curve, not necessarily lying on a K3
surface. Set $g=2k\geq 2$, fix $0\leq n\leq 2k$, and let $C$ be a general,
integral, $n$-nodal curve, i.e. $C$ has precisely $n$ nodes and no other
singularities. By Proposition 2.9, $C$ has gonality $k+1$. Further,
$W^{1}_{k+1}(C)$ is zero-dimensional and reduced, and any closed point $[A]\in
W^{1}_{k+1}(C)$ corresponds to a _locally free sheaf_ $A$ on $C$. We
additionally have $W^{2}_{k+1}(C)=\emptyset$. By Proposition 1.9, the natural
map

$\delta_{s}\;:\;\wedge^{k}\mathrm{H}^{0}(\omega_{C}\otimes
M^{\vee})\otimes\frac{\mathrm{H}^{0}(M)}{\mathbb{C}\langle
s\rangle}\to\mathrm{K}_{k-1,1}(C,\omega_{C};\mathrm{H}^{0}(\omega_{C}\otimes
M^{\vee}))$

for any line bundle $[M]\in W^{1}_{k+1}(C)$ and $s\neq
0\in\mathrm{H}^{0}(C,M)$ is an isomorphism. By Proposition 2.7, we have a
well-defined morphism
$S:X^{1}_{k+1}(C)\to{\mathbf{P}}(\mathrm{K}_{k-1,1}(C,\omega_{C})).$ We first
need a lemma.

######  Lemma 4.10.

Let $(B,0)$ be a smooth, integral, pointed variety of dimension one. Let
$\pi:\mathcal{C}\to B$ be a flat, projective family of integral, nodal curves,
with $\mathcal{C}_{b}:=\pi^{-1}(b)$ for $b\in B$. Set $C:=\mathcal{C}_{0}$.
Suppose $\omega_{C}$ is globally generated and
$\mathrm{K}_{k,1}(C,\omega_{C})=0$. Then, after replacing $(B,0)$ by an open
set about $0$, we have a vector bundle $\mathcal{K}$ on $B$, with natural
isomorphisms
$\mathcal{K}_{b}\simeq\mathrm{K}_{k-1,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$
for all $b\in B$.

###### Proof.

By Proposition 1.2 we may assume that $\mathcal{C}_{b}$ is globally generated
and that $\mathrm{K}_{k,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})=0$ for
all $b\in B$. In particular, this implies that
$(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$ has the Betti table of a general
canonical curve, for all $b\in B$. In particular,
$\beta_{p,1}:=\dim\mathrm{K}_{p,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$
is constant and independent of $b\in B$ for any $p$. Let $M_{b}$ denote the
kernel bundle $0\to
M_{b}\to\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\otimes\mathcal{O}_{\mathcal{C}_{b}}\to\omega_{\mathcal{C}_{b}}\to
0.$ We have a natural isomorphism

$\displaystyle\text{Ker}\left(d:\wedge^{k-1}\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\otimes\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\to\wedge^{k-2}\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\otimes\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}}^{\otimes
2})\right)\simeq\mathrm{H}^{0}(\wedge^{k-1}M_{b}\otimes\omega_{\mathcal{C}_{b}}),$

see [AF, §2.3]. Further, we may naturally identify
$\mathrm{K}_{k-1,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$ with

$\mathrm{Coker}\left(\wedge^{k}\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\to\mathrm{H}^{0}(\wedge^{k-1}M_{b}\otimes\omega_{\mathcal{C}_{b}})\right),$

where the map
$\wedge^{k}\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\to\mathrm{H}^{0}(\wedge^{k-1}M_{b}\otimes\omega_{\mathcal{C}_{b}})$
is injective. Since $h^{0}(\omega_{\mathcal{C}_{b}})$ and $\beta_{p,1}$ are
independent of $b\in B$,
$h^{0}(\wedge^{k-1}M_{b}\otimes\omega_{\mathcal{C}_{b}})$ is constant for
$b\in B$. We have a vector bundle $\mathcal{M}$ on $\mathcal{C}$ fitting into
the exact sequence
$0\to\mathcal{M}\to\pi^{*}\pi_{*}\omega_{\pi}\to\omega_{\pi}\to 0,$ with fibre
$\mathcal{M}_{b}\simeq M_{b}$ on $\mathcal{C}_{b}$. Consider now the vector
bundles $\mathcal{A}:=\wedge^{k}\pi_{*}\omega_{\pi}$,
$\mathcal{B}:=\pi_{*}(\wedge^{k-1}\mathcal{M}\otimes\omega_{\pi})$. We have an
injective morphism $\mathcal{A}\to\mathcal{B}$ relativizing
$\wedge^{k}\mathrm{H}^{0}(\omega_{\mathcal{C}_{b}})\to\mathrm{H}^{0}(\wedge^{k-1}M_{b}\otimes\omega_{\mathcal{C}_{b}})$,
for $b\in B$. By Lemma 1.1, we have a vector bundle
$\mathcal{K}:=\mathrm{Coker}(\mathcal{A}\to\mathcal{B})$. Then
$\mathcal{K}_{b}\simeq\mathrm{K}_{k-1,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$
for all $b\in B$, as required. ∎

We now generalize Proposition 4.9 to the situation of a general nodal curve.

######  Proposition 4.11.

Let $C$ be a general, integral, $m$-nodal curve of even arithmetic genus
$g=2k\geq 2$ for $0\leq m\leq k-1$. The pull-back map on global sections

$S^{*}\;:\;\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(1))\to\mathrm{H}^{0}(S^{*}\mathcal{O}_{{\mathbf{P}}}(1)),$

with ${\mathbf{P}}:={\mathbf{P}}(\mathrm{K}_{k-1,1}(C,\omega_{C}))$, is
injective.

###### Proof.

A very general, primitively polarized K3 surface $(X,L)$ of genus $g=2k$
contains $n$-nodal curves in the linear system $|L|$ for any $n$ by Chen’s
Theorem, [C] (see also [Ta, Theorem 3.8]). So let $(X,L)$ be a very general,
primitively polarized K3 surface and let $D\in|L|$ be an integral, $n$-nodal
curve. Let $\pi:\mathcal{C}\to B$ be flat family of integral, $n$-nodal curves
of genus $2k$ over a smooth, integral, pointed variety $(B,0)$ of dimension
one, with $D\simeq\pi^{-1}(0)$ the special fibre and
$\mathcal{C}_{b}:=\pi^{-1}(b)$ a general, integral $n$-nodal curve for $b\neq
0\in B$. We may also assume that $\pi$ admits a section $B\to\mathcal{C}$,
landing in the smooth locus of each fibre. We have a moduli space
$\mathcal{W}^{1}_{k+1}$ and a projective morphism $q:\mathcal{W}^{1}_{k+1}\to
B$ such that $q^{-1}(b)\simeq W^{1}_{k+1}(\mathcal{C}_{b})$ by relativizing
the construction of the Brill–Noether loci as in [ACG, Ch. XXI].

We first show that the morphism $q:\mathcal{W}^{1}_{k+1}\to B$ is flat. The
claim will then follow easily from Proposition 4.9. Note that $q$ is finite by
Proposition 2.9 and Proposition 4.6. For $b\neq 0$,
$W^{1}_{k+1}(\mathcal{C}_{b})$ is smooth of dimension zero by Proposition 2.9.
After replacing the curve $B$ by an open set about the origin $0$, we may
assume that $q^{-1}(U)\to U$ is étale, for $U=B\setminus\\{0\\}$, and that
$h^{0}(\mathcal{O}_{W^{1}_{k+1}(\mathcal{C}_{b})})=c$ is constant for $b\neq
0\in B$. It thus suffices to show $h^{0}(\mathcal{O}_{W^{1}_{k+1}(D)})=c$, by
[H1, Thm. III.9.9]. Since the closure of $q^{-1}(U)$ in
$\mathcal{W}^{1}_{k+1}$ is flat over $B$ by [H1, Prop. III.9.7], we have
$h^{0}(\mathcal{O}_{W^{1}_{k+1}(D)})\geq c$. By Proposition 4.6,
$h^{0}(\mathcal{O}_{W^{1}_{k+1}(D)})=\frac{(2k)!}{k!(k+1)!}$. So it suffices
to show that $c\geq\frac{(2k)!}{k!(k+1)!}$. But, if $Y$ is a general, smooth
curve of genus $2k$, then $W^{1}_{k+1}(Y)$ is reduced and zero-dimensional of
length $\frac{(2k)!}{k!(k+1)!}$, [GH]. It follows that
$c\geq\frac{(2k)!}{k!(k+1)!}$, by degenerating a general smooth curve to the
$n$-nodal curve $\mathcal{C}_{b}$ (for some $b\neq 0$) and applying [H1, Prop.
III.9.7] again.

So $q:\mathcal{W}^{1}_{k+1}\to B$ is finite and flat. By Proposition 2.9 and
Lemma 4.2, for any $b\in B$ and any $[A]\in W^{1}_{k+1}(\mathcal{C}_{b})$,
$h^{0}(A)=2$. We thus have a ${\mathbf{P}}^{1}$ bundle $\mathcal{X}^{1}_{k+1}$
over $\mathcal{W}^{1}_{k+2}$, with a morphism $r:\mathcal{X}^{1}_{k+1}\to B$,
with fibre $r^{-1}(b)\simeq X^{1}_{k+1}(\mathcal{C}_{b})$. By Lemma 4.10 we
have, after shrinking $B$, a vector bundle $\mathcal{K}$ with natural
isomorphisms
$\mathcal{K}_{b}\simeq\mathrm{K}_{k-1,1}(\mathcal{C}_{b},\omega_{\mathcal{C}_{b}})$
for all $b\in B$. Proposition 2.7 relativizes to give a morphism

$\mathcal{S}:\mathcal{X}^{1}_{k+1}\to{\mathbf{P}}(\mathcal{K})$

over $B$. By Proposition 4.9, $\mathcal{S}_{b}^{*}$ is injective for all $b\in
B$, after shrinking $B$, which gives the claim. ∎

######  Proposition 4.12.

Let $D$ be a integral $n$-nodal curve of even arithmetic genus $g=2k$ with
$\omega_{D}$ very ample and $(D,\omega_{D})$ normally generated. Let $A\in
W^{1}_{k+1}(D)$ be a base-point free line bundle with $h^{0}(A)=2$. Let
$\alpha\neq 0\in K_{k-1,1}(D,\omega_{D};\mathrm{H}^{0}(\omega_{D}\otimes
I_{Z(s)}))$ for $s\neq 0\in\mathrm{H}^{0}(A)$. Then the associated syzygy
scheme $X_{\alpha}$ is a scroll and $D$ lies in the smooth locus of
$X_{\alpha}$, with associated line bundle

$A\simeq L_{\alpha}:=\mathcal{O}_{D}(R),$

where $R$ is the ruling on $X_{\alpha}$.

###### Proof.

Let $X_{A}\subseteq{\mathbf{P}}^{g-1}$ be the scroll induced by $A$, [Sch,
§2]. Then $X_{A}$ is a rational normal scroll of degree $h^{1}(A)=k$ and
dimension $k$ containing $D\subseteq{\mathbf{P}}^{g-1}$ and further $D$ lies
in the smooth locus of $X_{\alpha}$ since $A$ is base-point free, [FK2, §4].
We have the injective restriction map

$i\;:\mathrm{K}_{k-1,1}(X_{A},H)\hookrightarrow\mathrm{K}_{k-1,1}(D,\omega_{D})$

where $H$ is the hyperplane. Notice that
$(\mathcal{O}_{X_{A}}(R))_{|_{D}}\simeq A$ and that the $k$ dimensional vector
space $\mathrm{H}^{0}(X_{A},H-R)$ is map isomorphically to
$\mathrm{H}^{0}(D,\omega_{D}-A)$ under the natural identification
$\mathrm{H}^{0}(X_{A},H)\simeq\mathrm{H}^{0}(D,\omega_{D})$. Let
$s^{\prime}\in\mathrm{H}^{0}(X,R)$ be the unique section with
$s^{\prime}_{|_{D}}=s$. The space
$\mathrm{K}_{k-1,1}(X_{A},H;\mathrm{H}^{0}(H\otimes I_{Z(s^{\prime})}))$ is
one dimensional from Section 1 and injects into the one dimensional space
$\mathrm{K}_{k-1,1}(D,\omega_{D};\mathrm{H}^{0}(\omega_{D}\otimes I_{Z(s)}))$.
Hence $i$ induces an isomorphism

$\mathrm{K}_{k-1,1}(X_{A},H;\mathrm{H}^{0}(H\otimes
I_{Z(s^{\prime})}))\simeq\mathrm{K}_{k-1,1}(D,\omega_{D};\mathrm{H}^{0}(\omega_{D}\otimes
I_{Z(s)})).$

In particular, for $\alpha\neq
0\in\mathrm{K}_{k-1,1}(D,\omega_{D};\mathrm{H}^{0}(\omega_{D}\otimes
I_{Z(s)}))$ there exists $\alpha^{\prime}\in\mathrm{K}_{k-1,1}(X_{A},H)$ with
$i(\alpha^{\prime})=\alpha$. By definition of the syzygy scheme $X_{\alpha}$
we therefore have $X_{A}\subseteq X_{\alpha}$. But
$X_{\alpha}\subseteq{\mathbf{P}}^{g-1}$ is also a rational normal scroll of
degree $k$, [vB3, Cor. 5.2] and hence $X_{\alpha}=X_{A}$. This completes the
proof. ∎

We may now prove the main result of this paper.

######  Theorem 4.13.

Let $C$ be a general curve of genus $g\geq 8$. Then, for any $p$,
$\mathrm{K}_{p,1}(C,\omega_{C})$ is spanned by syzygies of rank $p+1$.

###### Proof.

By Voisin’s Theorem [V2] (see also [Ke3] for a simpler proof),
$\mathrm{K}_{p,1}(C,\omega_{C})=0$ for $p\geq\lfloor\frac{g}{2}\rfloor$. Thus
the result is trivial unless $p\leq\lfloor\frac{g-2}{2}\rfloor$. Fix such a
$p$ and set $m=g-2p-2$. Let $x_{i},y_{i}\in C$ be general points for $1\leq
i\leq m$.

For any $1\leq j\leq m$, we let $D_{j}$ be the $j$-nodal curve obtained by
identifying $x_{i}$ to $y_{i}$ for $1\leq i\leq j$. We let $p_{i}\in D_{j}$
denote the node over $x_{i},y_{i}$ for $1\leq i\leq j$. Set $D_{0}:=C$. We let

$\mu_{j}\;:\;D_{j-1}\to D_{j}$

be the partial normalization map at the node $p_{j}$. By Proposition 4.1 it
suffices to show that $\mathrm{K}_{p+m,1}(D_{m},\omega_{D_{m}})$ is spanned by
syzygies of minimal rank $p+m+1$ such that $D_{m}$ lies in the smooth locus of
the syzygy scheme $X_{\alpha}:=\mathrm{Syz}(\alpha)$ and such that we have
$\deg(L_{\alpha})=g-p$, $h^{0}(C,\mu^{*}L_{\alpha})=2$, where $\mu:C\to D_{m}$
is the normalization map (note that this implies that all pullbacks of
$L_{\alpha}$ along partial-normalizations at any number of nodes have two
sections).

Set $\ell=g-p-1$, then $D_{m}$ has arithmetic genus $2\ell$. By Proposition
4.11,

$S^{*}\;:\;\mathrm{H}^{0}(\mathcal{O}_{{\mathbf{P}}}(1))\to\mathrm{H}^{0}(S^{*}\mathcal{O}_{{\mathbf{P}}}(1)),$

with
${\mathbf{P}}:={\mathbf{P}}(\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}}))$, is
injective since $m=g-2p-2\leq\ell-1$. Note that $W^{1}_{\ell+1}(D_{m})$ is
zero-dimensional and reduced, and each element $[A]\in W^{1}_{\ell+1}(D_{m})$
is a line bundle with exactly two sections by Proposition 2.9. The morphism
$S$ is the map

$\displaystyle S\;:\;X^{1}_{\ell+1}(D_{m})$
$\displaystyle\to{\mathbf{P}}(\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}})),$
$\displaystyle[(A,s)]$
$\displaystyle\mapsto[\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}};\mathrm{H}^{0}(\omega_{D_{m}}\otimes
I_{Z(s)})].$

By Proposition 4.11, the image of $S$ does not lie in any hyperplane and hence
the image $S(X^{1}_{\ell+1}(D_{m}))$ spans
${\mathbf{P}}(\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}}))$. Thus
$\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}})$ is spanned by syzygies $\alpha$
which lie in
$\mathrm{K}_{\ell-1,1}(D_{m},\omega_{D_{m}};\mathrm{H}^{0}(\omega_{D_{m}}\otimes
I_{Z(s)})$ for $s\in\mathrm{H}^{0}(A)$, $A\in W^{1}_{\ell+1}(D_{m})$. By
Proposition 4.12, such syzygies $\alpha$ are of minimal rank, $D_{m}$ lies in
the smooth locus of the syzygy scheme $X_{\alpha}$, and the associated ruling
is the line bundle $A\in W^{1}_{\ell+1}(D_{m})$.

It remains to show that, for any $A\in W^{1}_{\ell+1}(D_{m})$, we have
$h^{0}(C,\mu^{*}A)=2$. Assume that $h^{0}(C,\mu^{*}A)\geq 3$ for some $A\in
W^{1}_{\ell+1}(D_{m})$. Notice that, for each $1\leq i\leq m$, there exists
$s_{i}\in\mu^{*}\mathrm{H}^{0}(D_{m},A)$ with

$s(x_{i})=s(y_{i})=0.$

Consider the space $G^{1}_{\ell+1}(C)$ of $g^{r}_{d}$’s, i.e. pairs
$V\subseteq\mathrm{H}^{0}(C,L)$, where $L\in W^{1}_{\ell+1}(C)$ and $V$ is a
two dimensional, base-point free subspace of the space of global sections of
$L$. As $C$ is general, $G^{1}_{\ell+1}(C)$ is smooth of dimension
$\rho(g,1,\ell+1)=m$, [ACG, XXI, Prop. 6.8]. If $m=0$ then $D_{m}=C$ and there
is nothing to prove, so assume $m>0$, in which case $G^{1}_{\ell+1}(C)$ is
further irreducible, [FuLa]. Let $\mathcal{Z}\subseteq G^{1}_{\ell+1}(C)$ be
the closed subscheme of pairs $V\subseteq\mathrm{H}^{0}(C,L)$ with $L\in
W^{2}_{\ell+1}(C)$. Then $\mathcal{Z}$ has codimension at least one in
$G^{1}_{\ell+1}(C)$, [ACG, IV, Lemma 3.5]. On the other hand, if
$(x_{i},y_{i})$ are general for $1\leq i\leq m$, then the locus $\mathcal{T}$
of pairs $[V\subseteq\mathrm{H}^{0}(C,L)]\in\mathcal{Z}$ satisfying the
condition

there exist $s_{1},\ldots,s_{m}\in V$ with $s_{i}(x_{i})=s_{i}(y_{i})=0$ for
$1\leq i\leq m$

has codimension at least $m$. Thus $\mathcal{T}=\emptyset$ as required.

∎

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