# The Okubo-Weiss Criteria in Two-Dimensional Hydrodynamic and
Magnetohydrodynamic Flows

B. K. Shivamoggi111Permanent Address: University of Central Florida, Orlando,
FL 32816-1364, USA and G. J. F. van Heijst
J. M. Burgers Centre and Fluid Dynamics Laboratory
Department of Physics
Eindhoven University of Technology
NL-5600MB Eindhoven, The Netherlands

Abstract

The “slow-variation” restriction on the straining flow-velocity gradient field
used in the Okubo [1]-Weiss [2] criterion is quantified via the Beltrami
condition with the divorticity framework in 2D hydrodynamic flows. This turns
out to provide interesting interpretations of the Okubo-Weiss criterion in
terms of the topological properties of the underlying vorticity manifold.
These developments are then extended to 2D quasi-geostrophic flows (via the
potential divorticity framework) and magnetohydrodynamic flows and the Okubo-
Weiss criteria for these cases are considered.

1\. Introduction

A central question in the problem of transport in two-dimensional (2D)
turbulent flows is how to divide a vorticity field into hyperbolic (cascading
turbulence) and elliptic (coherent vortex) regions because the topology of 2D
turbulence is parameterized in terms of the relative dominance of flow
deformation or flow rotation. Okubo [1] and Weiss [2] gave a kinematic
criterion to serve as a diagnostic tool towards this goal which has been
widely used in numerical simulations (Brachet et al. [3], Ohkitani [4],
Babiano and Provenzale [5]) and laboratory experiments (Ouelette and Gollub
[6]) of 2D hydrodynamic flows.222The Okubo-Weiss parameter describing the
local strain-vorticity balance in the horizontal flow field of a shallow fluid
layer turns out also to quantify the deviations from two-dimensionality of
this flow (Cieslik et al. [7]). More specifically, the Okubo-Weiss parameter
turns out to be the source turn in the Poisson equation for the pressure (Kamp
[8]). A key assumption underlying the Okubo-Weiss criterion is that the
vorticity gradient field evolves quasi-adiabatically with respect to the
underlying straining flow-velocity gradient field. This issue was explored by
Basdevant and Philipovitch [9] who tried to improve on it by invoking the
topological properties of the pressure field, while Hua and Klein [10] tried
to include the strain-rate time evolution explicitly. The purpose of this
paper is to seek to quantify the “slow-variation” restriction on the straining
flow-velocity gradient field used in the Okubo-Weiss criterion via the
Beltrami condition with the divorticity framework in 2D hydrodynamic flows
(Shivamoggi et al. [11]). This also turns out to provide interesting
interpretations of the Okubo-Weiss criterion in terms of the topological
properties of the underlying vorticity manifold. These developments are then
extended to 2D quasi-geostrophic flows (via the potential divorticity
framework) and magnetohydrodynamic flows and the Okubo-Weiss criteria for
these cases are considered.

2\. Beltrami Condition Interpretation of the Okubo-Weiss Criterion

The vorticity dynamics in 2D hydrodynamic flows is governed by the following
equation (Kida [12], Kuznetsov et al. [13])

$\frac{\partial\boldsymbol{\mathscr{B}}}{\partial t}=\nabla\times\left({\bf
v}\times\boldsymbol{\mathscr{B}}\right)$ (1a)

or

$\frac{D\boldsymbol{\mathscr{B}}}{Dt}\equiv\frac{\partial\boldsymbol{\mathscr{B}}}{\partial
t}+\left({\bf
v}\cdot\nabla\right)\boldsymbol{\mathscr{B}}=\left(\boldsymbol{\mathscr{B}}\cdot\nabla\right){\bf
v}$ (1b)

where ${\bf v}=\left<u,v\right>$ is the flow velocity, $\boldsymbol{\omega}$
is the vorticity,

$\boldsymbol{\omega}\equiv\nabla\times{\bf v}$ (2a)

and $\boldsymbol{\mathscr{B}}$ is the divorticity,

$\boldsymbol{\mathscr{B}}\equiv\nabla\times\boldsymbol{\omega}.$ (2b)

Equation (1b) may be rewritten as

$\frac{D\boldsymbol{\mathscr{B}}}{Dt}=\boldsymbol{\mathscr{A}}\cdot\boldsymbol{\mathscr{B}}$
(1c)

where $\boldsymbol{\mathscr{A}}$ is the velocity gradient matrix,

$\boldsymbol{\mathscr{A}}\equiv\left[\begin{matrix}\partial u/\partial
x&\partial u/\partial y\\\ \partial v/\partial x&\partial v/\partial
y\end{matrix}\right]=\frac{1}{2}\left[\begin{matrix}s_{1}&s_{2}-\omega\\\
s_{2}+\omega&-s_{1}\end{matrix}\right]$ $s_{1}\equiv-2\frac{\partial
v}{\partial y},~{}s_{2}\equiv\frac{\partial v}{\partial x}+\frac{\partial
u}{\partial y},~{}\omega\equiv\frac{\partial v}{\partial x}-\frac{\partial
u}{\partial y}.$ (3)

If the straining flow velocity gradient tensor $\nabla{\bf v}$ is assumed,
following Okubo [1] and Weiss [2], to temporally evolve slowly so the
divorticity field evolves quasi-adiabatically with respect to the straining
flow-velocity gradient field, equation (1c) may be locally approximated by an
eigenvalue problem with eigenvalues given by

$\lambda^{2}={s_{1}}^{2}+{s_{2}}^{2}-\omega^{2}\equiv Q.$ (4)

The Okubo-Weiss parameter Q is a measure of the relative importance of flow
strain
(Q $>$ 0, hyperbolic) and vorticity (Q $<$ 0, elliptic). Numerical simulations
(Brachet et al. [3], Ohkitani [4], Babiano and Provenzale [5]) and laboratory
experiments (Ouellette and Gollub [6]) of 2D hydrodynamic flows confirmed that
coherent vortices are indeed located in elliptic regions while divorticity
sheets are located333It may be noted that divorticity sheets are also more
likely to occur near vorticity nulls due to selective rapid viscous decay of
vorticity in these layers (Shivamoggi et al. [11]), just as vortex sheets are
more likely to form near velocity nulls in 3D hydrodynamic flows. in
hyperbolic regions.

The “slow-variation” restriction on the straining flow-velocity gradient field
used above may be quantified via the Beltrami condition444A similar approach
was taken previously (Shivamoggi and van Heijst [14]) in the quantification of
the “slow variation” restriction used in Flierl-Stern-Whitehead [15] zero
angular momentum theorem for localized nonlinear structures in 2D quasi-
geostrophic flows on the $\beta$-plane. with the divorticity framework in 2D
hydrodynamic flows (Shivamoggi et al. [11]). Equations (1a-c) yield for the
Beltrami state (Shivamoggi et al. [11]),

$\boldsymbol{\mathscr{B}}=a{\bf v}$ (5)

$a$ being an arbitrary constant. Using (5), the Okubo-Weiss parameter Q
becomes

$Q=\frac{4}{a^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\frac{\partial^{2}\omega}{\partial y^{2}}\right].$ (6)

(6) implies that the Okubo-Weiss parameter also characterizes the topological
properties of the vorticity manifold - it is in fact the negative of the
Gaussian curvature of the vorticity surface. Thus, the character of the
ensuing time-dependent 2D flow behavior appears to be rooted in the local
topological properties of the underlying equilibrium vorticity manifold. It
may be mentioned that the above reduction was pointed out by Larcheveque [16]
on the premise of replacing streamlines by isovorticity lines which lacked, as
Larcheveque [16] admitted, any dynamical meaning - streamlines are actually
isomorphic to divorticity lines (as implied by the Beltrami condition (5)).

3\. The Okubo-Weiss Criterion for Quasi-geostrophic Flows

Consider a 2D quasi-geostrophic flow in which the baroclinic effects are
produced by the deformed free surface of the ocean. The governing equation (in
appropriate units) is (Charney [17])

$\frac{\partial{\bf q}}{\partial t}+\left({\bf v}\cdot\nabla\right){\bf q}=0$
(7)

where q is the potential vorticity vector,

${\bf q}\equiv\boldsymbol{\omega}-k^{2}\boldsymbol{\psi}+{\bf f}$ (8)

f is the Coriolis parameter, $k$ is the inverse Rossby radius of deformation,
$k\equiv\sqrt{{f_{0}}^{2}/gH}$, ($f_{0}$ being the local value of $|{\bf f}|$
and H the depth of the ocean, which is taken to be uniform), and

${\bf v}\equiv-\nabla\times\boldsymbol{\psi}$ (9)

Upon taking the curl of equation (7), we obtain

$\frac{\partial\boldsymbol{\mathscr{D}}}{\partial t}=\nabla\times\left({\bf
v}\times\boldsymbol{\mathscr{D}}\right)$ (10a)

where $\boldsymbol{\mathscr{D}}$ is the potential divorticity vector (in
analogy to the potential vorticity vector q),

$\boldsymbol{\mathscr{D}}\equiv\nabla\times{\bf
q}=\boldsymbol{\mathscr{B}}+k^{2}{\bf v}+{\bf h}$ (11)

and

${\bf h}=\nabla\times{\bf f}=\left<\beta,0,0\right>$

$\beta$ being the planetary vorticity gradient.

Equation (10a) may be rewritten as

$\frac{D\boldsymbol{\mathscr{D}}}{Dt}=\boldsymbol{\mathscr{A}}\cdot\boldsymbol{\mathscr{D}}.$
(10b)

Following Okubo [1] and Weiss [2], and assuming that the potential divorticity
field evolves quasi-adiabatically with respect to the straining flow velocity
gradient field, equation (10b) may again be locally approximated by an
eigenvalue problem with eigenvalues given by,

$\lambda^{2}=\frac{1}{4}\left(u_{y}v_{x}+{v_{y}}^{2}\right)\equiv Q.$ (12)

Equation (10a) yields for the Beltrami state,

$\boldsymbol{\mathscr{D}}=b{\bf v}$ (13)

$b$ being an arbitrary constant. Using (13), the Okubo-Weiss parameter Q
becomes

$Q\equiv\frac{1}{4b^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial
x\partial y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\frac{\partial^{2}\omega}{\partial y^{2}}\right]$ (14)

which is the same as (6) for 2D hydrodynamic case. This shows that the Okubo-
Weiss parameter is robust and remains intact under extension to 2D quasi-
geostrophic flows (in the $\beta$-plane approximation to the Coriolis
parameter). The inclusion of the nonlinear variation in the Coriolis parameter
(the so-called $\gamma$-effect) will, however, lead to changes in the Okubo-
Weiss parameter,

$Q=\frac{1}{4b^{2}}\left[\left(\frac{\partial^{2}\omega}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}\omega}{\partial
x^{2}}\left(\frac{\partial^{2}\omega}{\partial y^{2}}+\gamma\right)\right].$
(15)

4\. The Okubo-Weiss Criterion for Magnetohydrodynamic Flows

Consider a 2D incompressible magnetohydrodynamic (MHD) flow. The equation
governing the transport of the magnetic field ${\bf
B}=\left<B_{1},B_{2}\right>$ is (Goedbloed and Poedts [18])

$\frac{D{\bf B}}{Dt}=\left({\bf B}\cdot\nabla\right){\bf v}$ (16a)

which may be rewritten as

$\frac{D{\bf B}}{Dt}=\boldsymbol{\mathscr{A}}\cdot{\bf B}.$ (16b)

If we now assume that the magnetic field evolves quasi-adiabatically with
respect to the straining flow velocity gradient field, equation (16b) may
again be locally approximated by an eigenvalue problem with eigenvalues given
by,

$\lambda^{2}=\frac{1}{4}\left(u_{y}v_{x}+{v_{y}}^{2}\right)\equiv Q.$ (17)

Noting that the MHD Beltrami state (Shivamoggi [19]) corresponds to the so-
called Alfvénic state (Hasegawa [20])

${\bf v}=c{\bf B}$ (18)

$c$ being an arbitrary constant, (17) becomes

$Q=\frac{c^{2}}{4}\left({B_{1}}_{y}{B_{2}}_{x}+{B_{2y}}^{2}\right).$ (19)

In terms of the magnetic vector potential A given by

${\bf B}\equiv\nabla\times{\bf A},~{}{\bf A}=A{\bf\hat{i}}_{z}$ (20)

(19) becomes

$Q=\frac{c^{2}}{4}\left[\left(\frac{\partial^{2}A}{\partial x\partial
y}\right)^{2}-\frac{\partial^{2}A}{\partial
x^{2}}\frac{\partial^{2}A}{\partial y^{2}}\right].$ (21)

(21) implies that the Okubo-Weiss parameter Q for the MHD case characterizes
the topological properties of the magnetic flux surface - it is the negative
of the Gaussian curvature of the magnetic flux surface. As with the case of 2D
hydrodynamic flows, (21) can serve as a useful diagnostic tool to parameterize
the magnetic field topology in 2D MHD flows.

5\. Discussion

The “slow variation” restriction on the straining flow-velocity gradient field
used in the Okubo-Weiss criterion may be quantified via the Beltrami condition
with the divorticity framework in 2D hydrodynamic flows (Shivamoggi et al.
[11]). This also turns out to provide interesting interpretations of the
Okubo-Weiss criterion in terms of the topological properties of the underlying
vorticity manifold. Extension of these considerations to 2D quasi-geostrophic
flows (via the potential divorticity framework) shows the robustness of the
Okubo-Weiss parameter under varying 2D hydrodynamic flow situations. Extension
to 2D MHD flows, on the other hand, provides one again with a useful
diagnostic tool to parameterize the magnetic field topology in 2D MHD flows.

6\. Acknowledgments

The authors are thankful to Dr. Leon Kamp for helpful discussions. BKS would
like to thank The Netherlands Organization for Scientific Research (NWO) for
the financial support.

## References

  * [1] A. Okubo: Deep Sea Res. 17, 445, (1970).
  * [2] J. Weiss: Physica D 48, 273, (1991).
  * [3] M. E. Brachet, M. Meneguzzi, H. Politano and P. L. Sulem: J. Fluid Mech. 194, 333, (1988).
  * [4] K. Ohkitani: Phys. Fluids A 3, 1598, (1991).
  * [5] A. Babiano and A. Provenzale: J. Fluid Mech. 574, 429, (2007).
  * [6] N. T. Ouelette and J. P. Gollub: Phys. Rev. Lett. 99, 194502, (2007).
  * [7] A. R. Cieslik, L. P. J. Kamp, H. J. H. Clercx and G. J. F. van Heijst: Europhys. Lett. 85, 54001, (2009).
  * [8] L. P. J. Kamp: Phys. Fluids, Submitted, (2011).
  * [9] C. Basdevant and T. Philipovitch: Physica D 73, 17, (1994).
  * [10] B. L. Hua and P. Klein: Physica D 113, 98, (1998).
  * [11] B. K. Shivamoggi, G. J. F. van Heijst and J. Juul Rasmussen: Phys. Lett. A 374, 2309, (2010).
  * [12] S. Kida: J. Phys. Soc. Japan 54, 2840, (1985).
  * [13] E. A. Kuznetsov, V. Naulin, A. H. Nielsen and J. Juul Rasmussen: Phys. Fluids 19, 105110, (2007).
  * [14] B. K. Shivamoggi and G. J. F. van Heijst: Geophys. Astrophys. Fluid Dyn. 103, 293, (2009).
  * [15] G. R. Flierl, M. E. Stern and J. A. Whitehead: Dyn. Atmos. Oceans 7, 233, (1983).
  * [16] M. Larcheveque: Theor. Comp. Fluid Dyn. 5, 215, (1993).
  * [17] J. G. Charney: Geophys. Publ. Kosjones. Nor. Vidensk. Akad. Oslo 17, 1, (1947).
  * [18] H. Goedbloed and S. Poedts: Principles of Magnetohydrodynamics, Ch. 4, Cambridge Univ. Press, (2004).
  * [19] B. K. Shivamoggi: Euro. Phys. J. D 64, 393, (2011).
  * [20] A. Hasegawa: Adv. Phys. 34, 1, (1985).


# Syzygies of secant varieties of smooth projective curves and gonality
sequences

Junho Choe  School of Mathematics, KIAS, 85 Hoegiro Dongdaemun-gu, Seoul
02455, Republic of Korea<EMAIL_ADDRESS>,  Sijong Kwak  Department of
Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141,
Republic of Korea<EMAIL_ADDRESS>and  Jinhyung Park  Department of
Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to prove that one can read off the gonality
sequence of a smooth projective curve from syzygies of secant varieties of the
curve embedded by a line bundle of sufficiently large degree. More precisely,
together with Ein–Niu–Park’s theorem, our main result shows that the gonality
sequence of a smooth projective curve completely determines the shape of the
minimal free resolutions of secant varieties of the curve of sufficiently
large degree. This is a natural generalization of the gonality conjecture on
syzygies of smooth projective curves established by Ein–Lazarsfeld and
Rathmann to the secant varieties.

###### Key words and phrases:

algebraic curve, secant variety, syzygies, Koszul cohomology, gonality
sequence, symmetric product of a curve

###### 2020 Mathematics Subject Classification:

14N07, 14N05, 13D02

J. Choe was supported by a KIAS Individual Grant (MG083301) at Korea Institute
for Advanced Study. J. Park was partially supported by the National Research
Foundation (NRF) funded by the Korea government (MSIT) (NRF-2021R1C1C1005479
and NRF-2022M3C1C8094326).

##  1\. Introduction

Exploring the interplay between the geometric properties of algebraic
varieties and the algebraic properties of the equations defining algebraic
varieties is an important subject in algebraic geometry. Along this line,
Green [16] and Ein–Lazarsfeld [8] suggested studying syzygies of algebraic
varieties from a geometric point of view. The case of smooth projective curves
of large degree is now fairly well understood (see [9], [16], [17], [24]). On
the other hand, there has been a great deal of work on secant varieties of
projective varieties in the last three decades particularly because some
results on secant varieties have found some nontrivial applications to
algebraic statistics and algebraic complexity theory. In [4], Choe–Kwak
observed that there should be a “matryoshka structure” among secant varieties
of projective varieties. In this paper, we show that the gonality sequence of
a smooth projective curve gives a “matryoshka structure” for secant varieties
of the curve. Our theorem is a natural generalization of the results of
Ein–Lazarsfeld [10] and Rathmann [24] on the gonality conjecture of curves,
and complements the theorem of Ein–Niu–Park [11] on syzygies of secant
varieties of curves.

Throughout the paper, we work over an algebraically closed field $\mathbf{k}$
of characteristic zero. We start by recalling basic notions of syzygies of
algebraic varieties. Let $X$ be a projective variety, and $L$ be a very ample
line bundle on $X$ giving an embedding

$X\subseteq\mathbf{P}H^{0}(X,L)=\mathbf{P}^{r}.$

For a coherent sheaf $B$ on $X$, when $H^{0}(X,B\otimes L^{m})=0$ for all
$m\ll 0$, the section module
$R=R(X,B;L):=\bigoplus_{m\in\mathbf{Z}}H^{0}(X,B\otimes L^{m})$ is a finitely
generated graded module over the homogeneous coordinate ring
$S:=\bigoplus_{m\in\mathbf{Z}}S^{m}H^{0}(X,L)$ of $\mathbf{P}^{r}$. By
Hilbert’s syzygy theorem, $R$ is minimally resolved as

${0}$${R}$${F_{0}}$${F_{1}}$${~{}\cdots~{}}$${F_{r}}$${0,}$

where

$F_{p}=\bigoplus_{q\in\mathbf{Z}}K_{p,q}(X,B;L)\otimes S(-p-q).$

Here $K_{p,q}(X,B;L)$ can be regarded as the space of $p$-th syzygies of
weight $q$ of $R$ over $S$. This is called the _Koszul cohomology_ , and it is
the cohomology of the following Koszul-type complex

${\textstyle\bigwedge^{p+1}}H^{0}(X,L)\otimes H^{0}(X,B\otimes
L^{q-1})\longrightarrow{\textstyle\bigwedge^{p}}H^{0}(X,L)\otimes
H^{0}(X,B\otimes L^{q})\\\
\longrightarrow{\textstyle\bigwedge^{p-1}}H^{0}(X,L)\otimes H^{0}(X,B\otimes
L^{q+1}).$

For simplicity, we set $R(X,L):=R(X,\mathscr{O}_{X};L)$ and
$K_{p,q}(X,L):=K_{p,q}(X,\mathscr{O}_{X};L)$. For a globally generated vector
bundle $E$, let $M_{E}$ be the kernel bundle of the evaluation map
$\operatorname{ev}\colon H^{0}(X,E)\otimes\mathscr{O}_{X}\to E$. It is well
known (cf. [2, Section 2.1], [23, Proposition 2.1]) that if $H^{i}(X,B\otimes
L^{m})=0$ for $i>0,m>0$, then

$K_{p,q}(X,B;L)=H^{1}(X,{\textstyle\bigwedge^{p+1}}M_{L}\otimes B\otimes
L^{q-1})=\cdots=H^{q-1}(X,{\textstyle\bigwedge^{p+q-1}}M_{L}\otimes B\otimes
L)$

for $p\geq 0$ and $q\geq 2$. If furthermore $H^{i}(X,B)=0$ for
$\max\\{1,q-1\\}\leq i\leq q$ when $q\geq 1$ or $H^{0}(X,B\otimes L^{-1})=0$
when $q=0$, then

$K_{p,q}(X,B;L)=H^{q}(X,{\textstyle\bigwedge^{p+q}}M_{L}\otimes B)$

for $p\geq 0$ and $q\geq 0$. The geometric approach to vanishing or
nonvanishing of $K_{p,q}(X,B;L)$ is an active research subject in algebraic
geometry.

We now turn to the main objects of the paper – the secant varieties of curves.
Let $C$ be a smooth projective curve of genus $g$, and $L$ be a very ample
line bundle on $C$ giving an embedding

$C\subseteq\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}.$

For an integer $k\geq 0$, assume that

$\deg L\geq 2g+2k+1,$

and consider the _$k$ -th secant variety_

$\Sigma_{k}:=\overline{\bigcup_{x_{i}\in C}\langle
x_{1},\ldots,x_{k+1}\rangle}\subseteq\mathbf{P}^{r}$

of $C$ in $\mathbf{P}^{r}$. In our case, $\Sigma_{k}$ is simply the union of
$(k+1)$-secant $k$-planes to $C$ in $\mathbf{P}^{r}$. We have the natural
inclusions

$C=\Sigma_{0}\subseteq\Sigma_{1}\subseteq\cdots\subseteq\Sigma_{k-1}\subseteq\Sigma_{k}.$

Note that $\dim\Sigma_{k}=2k+1$ and
$\operatorname{Sing}\Sigma_{k}=\Sigma_{k-1}$. By [11, Theorem 1.1],
$\Sigma_{k}$ has normal Du Bois singularities. The singularities of the first
secant varieties of smooth projective varieties were before studied in [5] and
[28].

A celebrated theorem of Green [16] asserts that if $\deg L\geq 2g+1+\ell$ for
$\ell\geq 0$, then $C\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay
and $L$ satisfies property $N_{2,\ell}$; in other words,

$K_{p,q}(C,L)=0~{}~{}\text{ for $0\leq p\leq\ell$ and $q\geq 2$}.$

A natural generalization of Green’s theorem to secant varieties of $C$ in
$\mathbf{P}^{r}$ was conjectured by Sidman–Vermeire [25, Conjecture 1.3], and
this was established by Ein, Niu, and the third author [11, Theorem 1.2]: If
$\deg L\geq 2g+2k+1+\ell$ for $\ell\geq 0$, then
$\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay, and
$\mathscr{O}_{\Sigma_{k}}(1)$ satisfies property $N_{k+2,\ell}$; in other
words,

$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=0~{}~{}\text{ for $0\leq
p\leq\ell$ and $q\geq k+2$}.$

By specializing Danila’s theorem [7] to the curve case (see [12]), we have

$H^{0}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(m))=H^{0}(\mathbf{P}^{r},\mathscr{O}_{\mathbf{P}^{r}}(m))~{}~{}\text{
for $0\leq m\leq k+1$}.$

This then implies that if $0\leq p\leq\ell$, then

$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq
0~{}~{}\Longleftrightarrow~{}~{}\text{$p=0,q=0$ or $1\leq p\leq\ell$ and
$q=k+1$}.$

Thus the shape of the first $\ell$ steps of the minimal free resolution of
$R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ is completely determined. Recall
from [11, Theorem 1.2] that the Castelnuovo–Mumford regularity of
$\mathscr{O}_{\Sigma_{k}}$ is $2k+2$ if $g\geq 1$ and $k+1$ if $g=0$. As
$\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay, the
projective dimension of $R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ is
$e:=\operatorname{codim}\Sigma_{k}$. Notice that if $\deg L=2g+2k+1+\ell$,
then $\ell=e-g$. To summarize the discussion, the Betti table of
$R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ is the following:

| $0$ | $1$ | $2$ | $~{}\cdots$ | $e-g-1$ | $e-g$ | $e-g+1$ | $\cdots$ | $~{}e$
---|---|---|---|---|---|---|---|---|---
$0$ | 1 | - | - | $~{}\cdots$ | - | - | - | $\cdots$ |  -
$1$ | - | - | - | $~{}\cdots$ | - | - | - | $\cdots$ |  -
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |  | $\vdots$ | $\vdots$ | $\vdots$ |  | $~{}\vdots$
$k$ | - | - | - | $~{}\cdots$ | - | - | - | $\cdots$ |  -
$k+1$ | - | * | * | $~{}\cdots$ | * | * | ? | $\cdots$ |  ?
$k+2$ | - | - | - | $~{}\cdots$ | - | - | ? | $\cdots$ |  ?
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |  | $\vdots$ | $\vdots$ | $\vdots$ |  | $~{}\vdots$
$2k+2$ | - | - | - | $~{}\cdots$ | - | - | ? | $\cdots$ |  ?

Table 1. The Betti table of $R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$

Here “-” indicates a zero entry, “*” indicates a nonzero entry, and “?”
indicates an entry not yet determined. We remark that the syzygies of
$\Sigma_{k}$ in $\mathbf{P}^{r}$ are not “asymptotic syzygies” considered in
[9] and [23].

It is natural to study the undetermined part of the Betti table of
$R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$. The problem is to determine
vanishing and nonvanishing of
$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ for $e-g+1\leq p\leq e$ and
$k+1\leq q\leq 2k+2$. The case of $g=0$ is a classical result, the case of
$g=1$ was done by Graf von Bothmer–Hulek [14] and Fisher [13], and the case of
$g=2$ was recently settled by Li [22]. For the case of $g\geq 3$, we assume
that $\deg L$ is sufficiently large. Consider the case of $k=0$.
Green–Lazarsfeld [17, Theorem 2] proved that

$K_{p,2}(C,L)\neq 0~{}~{}\text{ for $e-g+1\leq p\leq e$}.$

This was recently generalized by Taylor [27, Corollary 3.6] as

$K_{p,2k+2}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0~{}~{}\text{ for
$e-g+1\leq p\leq e$}.$

The gonality conjecture of Green–Lazarsfeld asserts that

$K_{p,1}(C,L)\neq 0~{}~{}\text{ for $1\leq
p\leq\operatorname{codim}(C)-\operatorname{gon}(C)+1$},$

where

$\operatorname{gon}(C):=\min\\{d\mid\text{$C$ carries a linear series
$g_{d}^{1}$}\\}$

is the _gonality_ of $C$. The gonality conjecture was established by
Ein–Lazarsfeld [10] and Rathmann [24]. This suggests that the geometry of $C$
is deeply related to vanishing and nonvanishing of $K_{p,1}(C,L)$ for large
$p$. Along this line, Lawrence Ein previously asked what kind of geometry of
$C$ is involved in the behavior of vanishing and nonvanishing of
$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ for large $p$. In [4], the
first and second authors proposed that one should consider the _gonality
sequence_ of $C$. For an integer $q\geq 0$, let

$\gamma^{q}=\gamma^{q}(C):=\min\\{d-q\mid\text{$C$ carries a linear series
$g_{d}^{q}$}\\}.$

Then $\gamma^{1}+1=\operatorname{gon}(C)$. The gonality sequence of $C$ is a
sequence $(\gamma^{0}+0,\gamma^{1}+1,\gamma^{2}+2,\ldots)$. The gonality
sequence was previously studied by several authors in the theory of algebraic
curves (see e.g., [6], [20]). The conjecture of the first and second authors
in [4] predicts that

$K_{p,k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq
0~{}~{}\Longleftrightarrow~{}~{}1\leq p\leq e-\gamma^{k+1}.$

However, the cases of $K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ for
$k+2\leq q\leq 2k+1$ remained a mystery in general.

In this paper, we completely resolve all the aforementioned problems at least
when $L$ is sufficiently positive: We show that the gonality sequence of $C$
determines vanishing and nonvanishing of
$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ for $e-g+1\leq p\leq e$ and
$k+1\leq q\leq 2k+2$.

######  Theorem 1.1.

Let $C$ be a smooth projective curve of genus $g\geq 2$, and $L$ be a very
ample line bundle of sufficiently large degree on $C$. For an integer $k\geq
0$, consider the $k$-th secant variety $\Sigma_{k}$ of $C$ in
$\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$, and put
$e:=\operatorname{codim}\Sigma_{k}=r-2k-1$. For each $k+1\leq q\leq 2k+2$, if
$e-g+1\leq p\leq e$, then we have

$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq
0~{}~{}\Longleftrightarrow~{}~{}e-g+1\leq p\leq e-\gamma^{2k+2-q}(C).$

As we discussed before, together with [11], our main theorem completely
determines the shape of the Betti table of
$R(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ in Table 1. Our approach using
secant varieties gives an alternative proof of the gonality conjecture which
is nothing but the case of $k=0$ in Theorem 1.1. On the other hand, by
duality, we have

$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=K_{e-p,2k+2-q}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))^{\vee}.$

The nontrivial parts covered by Theorem 1.1 are
$K_{p,q}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))$ for
$0\leq p\leq e-1$ and $0\leq q\leq k+1$. The Betti table of
$R(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))$ in this range
– the reverse of the part marked with “?” in Table 1 – is the following:

$\overbrace{\texttt{* $\cdots$ * * $\cdots$ * * $\cdots$ * * $\cdots$ *}}^{g}$
---
$\underbrace{\texttt{- $\cdots$ -}}_{\gamma^{1}(C)}$ * $\cdots$ * * $\cdots$ *
* $\cdots$ *
$\underbrace{\texttt{- $\cdots$ - - $\cdots$ -}}_{\gamma^{2}(C)}$ * $\cdots$ *
* $\cdots$ *
$\vdots$
$\underbrace{\texttt{- $\cdots$ - - $\cdots$ - - $\cdots$
-}}_{\gamma^{k+1}(C)}$ * $\cdots$ *

Table 2. The Betti table of
$R(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))$

Notice that $\gamma^{q}(C)=g$ for $q\geq g$. Thus if $k\geq g-1$, then the
last $k-g+2$ rows of Table 2 are all vanishing; in particular,

$\displaystyle K_{p,k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=0~{}~{}\text{
for $e-g+1\leq p\leq e$};$ $\displaystyle
K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=0~{}~{}\text{ for all $p$ and
$k+2\leq q\leq 2k+2-g$}.$

Next, observe that the first $m+1$ rows of Table 2 have the same vanishing and
nonvanishing patterns as those of the Betti table of
$R(\Sigma_{m},\omega_{\Sigma_{m}};\mathscr{O}_{\Sigma_{m}}(1))$ for each
$0\leq m\leq k$. Thus the syzygies of secant varieties of $C$ have a
surprisingly uniform behavior governed by the gonality sequence of $C$. It
resembles a matryoshka doll, which repeats similar patterns over and over
again, so one may say that there is a “matryoshka structure” among secant
varieties of $C$ in the sense of [4].

To prove Theorem 1.1, we utilize Bertram’s construction [3] as in [11]. There
is a vector bundle $E_{k+2,L}$ on the symmetric product $C_{k+2}$ of $C$ such
that $\beta_{k+1}\colon\mathbf{P}(E_{k+2,L})\to\Sigma_{k+1}$ is a resolution
of singularities and $Z_{k}:=\beta_{k+1}^{-1}(\Sigma_{k})$ is an effective
divisor. It is worth noting that we are working with $\Sigma_{k+1}$ to prove
Theorem 1.1 for $\Sigma_{k}$ instead of going to $\Sigma_{k-1}$ as in [11].

### Vanishing

Using the Du Bois-type condition

$R^{i}\beta_{k+1,*}\mathscr{O}_{\mathbf{P}(E_{k+2,L})}(-Z_{k})=\begin{cases}\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}&\text{if
$i=0$}\\\ 0&\text{if $i>0$}\end{cases}$

established in [11] and proceeding by induction on $q-k-1$, we reduce the
vanishing part of Theorem 1.1 to

$H^{q^{*}+1}(C_{k+2},{\textstyle\bigwedge^{p^{*}+q^{*}}}M_{E_{k+2,L}}\otimes
S_{k+2,\omega_{C}})=0,$

where $p^{*}:=e-p$ and $q^{*}:=2k+2-q$. Here $M_{E_{k+2,L}}$ is the kernel of
the evaluation map $H^{0}(C,L)\otimes\mathscr{O}_{C_{k+2}}\to E_{k+2,L}$, and
$S_{k+2,\omega_{C}}$ is a line bundle with $q_{k+2}^{*}S_{k,L}=L^{\boxtimes
k+2}$, where $q_{k+2}\colon C^{k+2}\to C_{k+2}$ is the map given by
$(x_{1},\ldots,x_{k+2})\mapsto x_{1}+\cdots+x_{k+2}$. We then show that

$\displaystyle
H^{q^{*}+1}(C_{k+2},{\textstyle\bigwedge^{p^{*}+q^{*}}}M_{E_{k+2,L}}\otimes
S_{k+2,\omega_{C}})$ $\displaystyle=H^{q^{*}+1}(C_{p^{*}+q^{*}}\times
C_{k+2},(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})).$

Here $N_{p^{*}+q^{*},L}$ is a line bundle with
$q_{p^{*}+q^{*}}^{*}N_{p^{*}+q^{*},L}=L^{\boxtimes p^{*}+q^{*}}(-\Delta)$,
where $\Delta$ is the sum of all pairwise diagonal on $C^{p^{*}+q^{*}}$. Let
$\operatorname{pr}_{1}\colon C_{p^{*}+q^{*}}\times C_{k+2}\to C_{p^{*}+q^{*}}$
be the projection map, and
$D_{p^{*}+q^{*},k+2}:=\\{(\xi_{1}+x,\xi_{2}+x)\mid\xi_{1}\in
C_{p^{*}+q^{*}-1},\xi_{2}\in C_{k+1},x\in C\\}$ be an effective divisor on
$C_{p^{*}+q^{*}}\times C_{k+2}$. Then it is enough to check that

($\star$)
$H^{i}(C_{p^{*}+q^{*}},R^{q^{*}+1-i}\operatorname{pr}_{1,*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2}))=0~{}~{}\text{ for $0\leq i\leq
q^{*}+1$}.$

When $i>0$, the cohomology vanishing ($\star$ ‣ 1) follows from Fujita–Serre
vanishing since $N_{p^{*}+q^{*},L}$ is sufficiently positive. When $i=0$, the
fiber of $R^{q^{*}+1}\operatorname{pr}_{1,*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})$ over $\xi\in C_{p^{*}+q^{*}}$ is

$H^{q^{*}+1}(C_{k+2},S_{k+2,\omega_{C}(-\xi)})=S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))\otimes{\textstyle\bigwedge^{q^{*}+1}}H^{1}(C,\omega_{C}(-\xi)).$

However, $h^{1}(C,\omega_{C}(-\xi))\leq q^{*}$ thanks to the “gonality
sequence condition” $\gamma^{q^{*}}(C)\geq p^{*}+1$, so

$R^{q^{*}+1}\operatorname{pr}_{1,*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})=0.$

Thus ($\star$ ‣ 1) holds for $i=0$.

### Nonvanishing

For the nonvanishing part of Theorem 1.1, it suffices to see that the map

$H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}(1))\longrightarrow
H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))$

is nonzero. Arguing as in the proof of the vanishing part, we reduce the
problem to showing that the map

(1.1) $R^{q^{*}+1}\operatorname{pr}_{1,*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})\\\ \longrightarrow
R^{q^{*}}\operatorname{pr}_{1,*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+1,\omega_{C}})(-D_{p^{*}+q^{*},k+1})\otimes H^{1}(C,\omega_{C})$

is nonzero. By the “gonality sequence condition” $\gamma^{q^{*}}(C)\leq
p^{*}$, we can find an effective divisor $\xi\in C_{p^{*}+q^{*}}$ with
$h^{0}(C,\omega_{C}(-\xi))=g-p^{*}$ and $h^{1}(C,\omega_{C}(-\xi))=q^{*}+1$.
The map (1.1) looks like
$\operatorname{id}_{S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))}\otimes\delta$
fiberwisely over $\xi\in C_{p^{*}+q^{*}}$, where $\delta$ is the Koszul-like
map

${\textstyle\bigwedge^{q^{*}+1}}H^{1}(C,\omega_{C}(-\xi))\longrightarrow{\textstyle\bigwedge^{q^{*}}}H^{1}(C,\omega_{C}(-\xi))\otimes
H^{1}(C,\omega_{C}).$

Since $\delta$ is clearly nonzero, it follows that the map (1.1) is nonzero.

The paper is organized as follows. We begin with collecting basic relevant
facts on the gonality sequence of a curve in Section 2. Section 3 provides a
review of basic properties of symmetric products and secant varieties of
curves. Section 4 is devoted to the proof of Theorem 1.1. Finally, in Section
5, we present some complementary results, and we also discuss some open
problems.

### Acknowledgments.

We would like to thank Lawrence Ein and Wenbo Niu for valuable and interesting
discussions. We are also grateful to Daniele Agostini, Marian Aprodu, Daniel
Erman, Robert Lazarsfeld, Frank-Olaf Schreyer, Jessica Sidman for their
interests.

##  2\. Gonality Sequences

In this section, we recall the definition and basic properties of the gonality
sequence of a smooth projective curve $C$ of genus $g\geq 2$, and we show some
relevant facts.

######  Definition 2.1.

For any integer $q\geq 0$, we define

$\gamma^{q}(C):=\min\\{d-q\mid\textup{$C$ carries a linear series
$g_{d}^{q}$}\\}.$

A sequence $(\gamma^{0}(C)+0,\gamma^{1}(C)+1,\gamma^{2}(C)+2,\ldots)$ is
called the _gonality sequence_ of $C$.

Note that $\gamma^{0}(C)=0$ and $\gamma^{1}(C)+1=\operatorname{gon}(C)$ is the
gonality of $C$. The following is an easy consequence of the Riemann–Roch
theorem, the Clifford theorem, and Brill–Noether theory.

######  Lemma 2.2 ([20, Lemmas 3.1 and 3.2]).

We have the following:

  1. (1)

$\gamma^{q}(C)\leq\gamma^{q+1}(C)$ for $q\geq 0$.

  2. (2)

$\min\\{q,g\\}\leq\gamma^{q}(C)\leq g-\lfloor g/(q+1)\rfloor$ for $q\geq 0$.
In particular, $\gamma^{g-1}(C)=g-1$ and $\gamma^{q}(C)=g$ for $q\geq g$.

If $C$ is hyperelliptic, then $\gamma^{q}(C)=q$ for $q\leq g$. However, as was
remarked in [20], it is not easy to compute the gonality sequence of a curve
in general. We refer to [20] for more details.

Next, we introduce a new positivity notion for a line bundle on $C$.

######  Definition 2.3.

Let $B$ be a line bundle on $C$. For integers $w,p\geq 0$, we say that $B$ is
_$w$ -weakly $p$-very ample_ if

$\operatorname{corank}\big{(}H^{0}(C,B)\longrightarrow
H^{0}(C,B|_{\xi})\big{)}\leq w$

for every effective divisor $\xi$ of degree $p+w+1$ on $C$.

Note that $B$ is $0$-weakly $p$-very ample if and only if $B$ is $p$-very
ample. Recall that $\gamma^{1}(C)\geq p+1$ (i.e., $\operatorname{gon}(C)\geq
p+2$) if and only if $\omega_{C}$ is $p$-very ample. The next proposition is a
generalization of this fact.

######  Proposition 2.4.

Let $q\geq 1$ be an integer. Then the following are equivalent:

  1. (1)

$\gamma^{q}(C)\geq p+1$.

  2. (2)

$h^{0}(C,\mathscr{O}_{C}(\xi))=h^{1}(C,\omega_{C}(-\xi))\leq q$ for every
effective divisor $\xi$ of degree $p+q$ on $C$.

  3. (3)

$\omega_{C}$ is $(q-1)$-weakly $p$-very ample.

In particular,

$\displaystyle\gamma^{q}(C)$ $\displaystyle=\max\\{p\geq
0\mid\omega_{C}\textup{ is $(q-1)$-weakly $p$-very ample}\\}+1$
$\displaystyle=\min\\{p\geq 0\mid\omega_{C}\textup{ fails to be $(q-1)$-weakly
$p$-very ample}\\}.$

###### Proof.

It is clear from the definitions. ∎

######  Lemma 2.5.

If $\omega_{C}$ is not $w$-weakly $p$-very ample with $0\leq p\leq g$, then
there is an effective divisor $\xi$ of degree $p+w+1$ such that

$\operatorname{corank}\big{(}H^{0}(C,\omega_{C})\longrightarrow
H^{0}(C,\omega_{C}|_{\xi})\big{)}=w+1,$

i.e., $h^{0}(C,\omega_{C}(-\xi))=g-p$ and $h^{1}(C,\omega_{C}(-\xi))=w+2$.

###### Proof.

Since $\omega_{C}$ is not $w$-weakly $p$-very ample, there is an effective
divisor $\xi_{0}$ of degree $p+w+1$ on $C$ such that

$h^{1}(C,\omega_{C}(-\xi_{0}))\geq w+2.$

If $h^{1}(C,\omega_{C}(-\xi_{0}))=w+2$, then we are done by taking
$\xi=\xi_{0}$. Suppose that $h^{1}(C,\omega_{C}(-\xi_{0}))\geq w+3$. The
Riemann–Roch theorem yields

$h^{0}(C,\omega_{C}(-\xi_{0}))=g-p+h^{1}(C,\omega_{C}(-\xi_{0}))-w-2\geq 1.$

It is elementary to see that if $B$ is a line bundle on $C$ with
$H^{0}(C,B)\neq 0$ and $H^{1}(C,B)\neq 0$, then

$h^{0}(C,B(-x_{0}+x_{1}))=h^{0}(C,B)-1~{}~{}\text{ and
}~{}~{}h^{1}(C,B(-x_{0}+x_{1}))=h^{1}(C,B)-1$

for general points $x_{0},x_{1}\in C$. Thus we find

$h^{0}(C,\mathscr{O}_{C}(\xi_{0}+x_{0}-x_{1}))=h^{1}(C,\omega_{C}(-\xi_{0}-x_{0}+x_{1}))=h^{1}(C,\omega_{C}(-\xi_{0}))-1\geq
w+2,$

so we can choose an effective divisor $\xi_{1}\in|\xi_{0}+x_{0}-x_{1}|$ of
degree $p+w+1$. Then

$h^{1}(C,\omega_{C}(-\xi_{1}))=h^{1}(C,\omega_{C}(-\xi_{0}))-1.$

Continuing this process, we finally reach an effective divisor $\xi$ of degree
$p+w+1$ such that $h^{1}(C,\omega_{C}(-\xi))=w+2$. ∎

##  3\. Symmetric Products and Secant Varieties of Curves

In this section, we review basic properties of symmetric products and secant
varieties of smooth projective curves, and we show some useful lemmas for the
proof of Theorem 1.1. We refer to [3] and [11] for a more detailed account.

Let $C$ be a smooth projective curve of genus $g$. For an integer $k\geq 1$,
we write the $k$-th symmetric product of the curve $C$ as $C_{k}$ and the
$k$-th ordinary product of the curve $C$ as $C^{k}$. The symmetric group
$\mathfrak{S}_{k}$ naturally acts on $C^{k}$, and
$C_{k}=C^{k}/\mathfrak{S}_{k}$. We have the quotient morphism

$q_{k}\colon C^{k}\longrightarrow C_{k},~{}~{}(x_{1},\ldots,x_{k})\longmapsto
x_{1}+\cdots+x_{k},$

which is a finite flat surjective morphism of degree $k!$. For a line bundle
$L$ on $C$, there are two line bundles $S_{k,L}$ and $N_{k,L}$ on $C_{k}$ such
that

$q_{k}^{*}S_{k,L}=L^{\boxtimes k}=\underbrace{L\boxtimes\cdots\boxtimes
L}_{k~{}\text{times}}~{}~{}\text{ and }~{}~{}q_{k}^{*}N_{k,L}=L^{\boxtimes
k}\big{(}-\sum_{1\leq i<j\leq k}\Delta_{i,j}\big{)},$

where $\Delta_{i,j}:=\\{(x_{1},\ldots,x_{k})\in C^{k}\mid x_{i}=x_{j}\\}$ is a
pairwise diagonal. Let $\delta_{k}$ be a divisor on $C_{k}$ such that
$\mathscr{O}_{C_{k}}(\delta_{k})=S_{k,\mathscr{O}_{C}}\otimes
N_{k,\mathscr{O}_{C}}^{-1}$. Then $N_{k,L}=S_{k,L}(-\delta_{k})$ for any line
bundle $L$ on $C$. It is well known that

$H^{0}(C_{k},S_{k,L})=S^{k}H^{0}(C,L)~{}~{}\text{ and
}~{}~{}H^{0}(C_{k},N_{k,L})={\textstyle\bigwedge^{k}}H^{0}(C,L).$

Furthermore, we have the following.

######  Lemma 3.1 ([1, Lemma 2.4], [11, Lemma 3.7]).

We have

$\displaystyle
H^{i}(C_{k},S_{k,L})=S^{k-i}H^{0}(C,L)\otimes{\textstyle\bigwedge^{i}}H^{1}(C,L)~{}~{}\text{
for $i\geq 0$};$ $\displaystyle
H^{i}(C_{k},N_{k,L})={\textstyle\bigwedge^{k-i}}H^{0}(C,L)\otimes
S^{i}H^{1}(C,L)~{}~{}\text{ for $i\geq 0$.}$

Let $D_{k,m}$ be the effective divisor on $C_{k}\times C_{m}$ given by the
image of the map

$C_{k-1}\times C\times C_{m-1}\longrightarrow C_{k}\times
C_{m},~{}~{}(\xi_{1},x,\xi_{2})\longmapsto(\xi_{1}+x,\xi_{2}+x),$

and $\operatorname{pr}_{1}\colon C_{k}\times C_{m}\to
C_{k},~{}\operatorname{pr}_{2}\colon C_{k}\times C_{m}\to C_{m}$ be the
projections. For $\xi\in C_{k}$, we set
$C_{m,\xi}:=\operatorname{pr}_{1}^{-1}(\xi)=\\{\xi\\}\times C_{m}$. Then
$\mathscr{O}_{C_{m}}(C_{m,\xi}\cap D_{k,m})=S_{m,\mathscr{O}_{C}(\xi)}$. For a
coherent sheaf $\mathscr{F}$ on $C_{m}$, we put

$M_{k}^{i}\mathscr{F}:=R^{i}\operatorname{pr}_{1,*}(\mathscr{O}_{C_{k}}\boxtimes\mathscr{F})(-D_{k,m})~{}~{}\text{
for $i\geq 0$}.$

It is a coherent sheaf on $C_{k}$. Identifying $C_{m,\xi}=C_{m}$ for each
$\xi\in C_{k}$, we have a natural map

$\rho^{i}(\xi)\colon M_{k}^{i}\mathscr{F}\otimes\mathbf{k}(\xi)\longrightarrow
H^{i}(C_{m},\mathscr{F}\otimes S_{m,\mathscr{O}_{C}(-\xi)}).$

Suppose that $\mathscr{F}$ is flat over $C_{k}$. By Grauert’s theorem, when
$h^{i}(C_{m},\mathscr{F}\otimes S_{m,\mathscr{O}_{C}(-\xi)})$ is constant for
all $\xi\in C_{k}$, $M_{k}^{i}\mathscr{F}$ is a vector bundle and
$\rho^{i}(\xi)$ is an isomorphism . By the cohomology and base change, when
$\rho^{i+1}(\xi)$ is surjective for $\xi\in C_{k}$, $\rho^{i}(\xi)$ is an
isomorphism if and only if $M_{k}^{i+1}\mathscr{F}$ is locally free in a
neighborhood of $\xi\in C_{k}$.

######  Lemma 3.2 (cf. [10, Lemma 1.2]).

For a given coherent sheaf $\mathscr{F}$ on $C_{m}$, if $\deg L$ is
sufficiently large, then

$H^{i}(C_{k},M_{k}^{j}\mathscr{F}\otimes N_{k,L})=0~{}~{}\text{ for $i>0$ and
$j\geq 0$}.$

###### Proof.

As $\deg L\gg 0$, we may write $L=L^{\prime}\otimes\mathscr{O}_{C}(mx)$ for a
point $x\in C$ and a sufficiently large integer $m\gg 0$ such that
$N_{k,L^{\prime}}$ is nef. Then $N_{k,L}=N_{k,L^{\prime}}\otimes
S_{k,\mathscr{O}_{C}(x)}^{m}$. Since $S_{k,\mathscr{O}_{C}(x)}$ is ample and
$m$ is sufficiently large, the lemma follows from Fujita–Serre vanishing [21,
Theorem 1.4.35]. ∎

In the above situation, we now consider the case $m=1$. Then $D_{k,1}$ is the
image of the injective map

$C_{k-1}\times C\longrightarrow C_{k}\times
C,~{}~{}(\xi,x)\longmapsto(\xi+x,x).$

Let $\sigma_{k}:=\operatorname{pr}_{1}|_{D_{k,1}}$. Identifying $D_{k,1}$ with
$C_{k-1}\times C$, we obtain a map

$\sigma_{k}\colon C_{k-1}\times C\longrightarrow
C_{k},~{}~{}(\xi,x)\longmapsto\xi+x,$

which is a finite flat surjective morphism of degree $k$. If we view $C_{k}$
as the Hilbert scheme of $k$ points on $C$, then $\sigma_{k}$ is the universal
family. The _tautological bundle_ on $C_{k}$ associated to $L$ is defined to
be

$E_{k,L}:=\sigma_{k,*}(\mathscr{O}_{C_{k-1}}\boxtimes L).$

It is a vector bundle of rank $k$ on $C_{k}$. Note that
$H^{0}(C,E_{k,L})=H^{0}(C,L)$ and $\det E_{k,L}=N_{k,L}$. Suppose that $L$ is
$(k-1)$-very ample. Then $E_{k,L}$ is globally generated. Applying
$\operatorname{pr}_{1,*}$ to the short exact sequence

${0}$${(\mathscr{O}_{C_{k}}\boxtimes
L)(-D_{k,1})}$${\mathscr{O}_{C_{k}}\boxtimes
L}$${\mathscr{O}_{C_{k-1}}\boxtimes L}$${0,}$$\scriptstyle{\cdot D_{k,1}}$

we get a short exact sequence

${0}$${M_{E_{k,L}}}$${H^{0}(C,L)\otimes\mathscr{O}_{C_{k}}}$${E_{k,L}}$${0.}$$\scriptstyle{\operatorname{ev}}$

Notice that $M_{E_{k,L}}=M_{k}^{0}L$ is a vector bundle of rank $h^{0}(C,L)-k$
on $C_{k}$. This short exact sequence looks like

${0}$${H^{0}(C,L(-\xi))}$${H^{0}(C,L)}$${H^{0}(C,L|_{\xi})}$${0}$

over $\xi\in C_{k}$ fiberwisely.

######  Lemma 3.3.

Suppose that $\deg L\geq 2g+k-1$. Then

$M^{i}_{k}N_{m,L}=\begin{cases}{\textstyle\bigwedge^{m}}M_{E_{k,L}}&\text{if
$i=0$}\\\ 0&\text{if $i>0$}.\end{cases}$

In particular, for any line bundle $B$ on $C_{k}$, we have

$H^{i}(C_{k}\times C_{m},(B\boxtimes
N_{m,L})(-D_{k,m}))=H^{i}(C_{k},{\textstyle\bigwedge^{m}}M_{E_{k,L}}\otimes
B)~{}~{}\text{ for $i\geq 0$}.$

###### Proof.

By Lemma 3.1,

$H^{i}(C_{m},N_{m,L(-\xi)})={\textstyle\bigwedge^{m-i}}H^{0}(C,L(-\xi))\otimes
S^{i}H^{1}(C,L(-\xi)).$

for any $\xi\in C_{k}$. Since $\deg L(-\xi)\geq 2g-1$, it follows that
$H^{1}(C,L(-\xi))=0$. Thus we get $H^{i}(C_{m},N_{m,L(-\xi)})=0$ for $i>0$, so
we obtain $M^{i}_{k}N_{m,L}=0$ for $i>0$. Note that $M^{0}_{k}N_{m,L}$ is a
vector bundle on $C_{k}$ whose fiber is
${\textstyle\bigwedge^{m}}H^{0}(C,L(-\xi))$ over $\xi\in C_{k}$. Applying
$\operatorname{pr}_{1,*}$ to the injective map

$(\mathscr{O}_{C_{k}}\boxtimes
N_{m,L})(-D_{k,m})\longhookrightarrow\mathscr{O}_{C_{k}}\boxtimes N_{m,L},$

we get an injective map

$M^{0}_{k}N_{m,L}\longhookrightarrow{\textstyle\bigwedge^{m}}H^{0}(C,L)\otimes\mathscr{O}_{C_{k}},$

which looks like

${\textstyle\bigwedge^{m}}H^{0}(C,L(-\xi))\longhookrightarrow{\textstyle\bigwedge^{m}}H^{0}(C,L)$

over $\xi\in C_{k}$ fiberwisely. On the other hand, notice that $L$ is
$(k-1)$-very ample. The injective map

$M_{E_{k,L}}\longhookrightarrow H^{0}(C,L)\otimes\mathscr{O}_{C_{k}}$

induces an injective map

${\textstyle\bigwedge^{m}}M_{E_{k,L}}\longhookrightarrow{\textstyle\bigwedge^{m}}H^{0}(C,L)\otimes\mathscr{O}_{C_{k}},$

which looks like

${\textstyle\bigwedge^{m}}H^{0}(C,L(-\xi))\longhookrightarrow{\textstyle\bigwedge^{m}}H^{0}(C,L)$

over $\xi\in C_{k}$ fiberwisely. Thus we can conclude that
$M^{0}_{k}N_{m,L}={\textstyle\bigwedge^{m}}M_{E_{k,L}}$. Now, the second
statement follows from the projection formula and the Leray spectral sequence
for $\operatorname{pr}_{1}$. ∎

From now on, as in [3] and [11], suppose that

$\deg L\geq 2g+2k+1.$

For an integer $k\geq 0$, let

$B_{k}=B_{k}(L):=\mathbf{P}(E_{k+1,L})$

with the canonical projection $\pi_{k}\colon B_{k}\to C_{k+1}$, and $H_{k}$ be
a tautological divisor so that
$\mathscr{O}_{B_{k}}(H_{k})=\mathscr{O}_{\mathbf{P}(E_{k+1,L})}(1)$. As
$E_{k+1,L}$ is globally generated, $H_{k}$ is base point free. Note that

$H^{0}(B_{k},H_{k})=H^{0}(C_{k+1},E_{k+1,L})=H^{0}(C,L).$

The image of the morphism given by the complete linear system $|H_{k}|$ is the
_$k$ -th secant variety_ $\Sigma_{k}$ of $C$ in
$\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$. Denote the induced map by

$\beta_{k}\colon B_{k}\longrightarrow\Sigma_{k},$

which is a resolution of singularities. By [11, Theorem 1.1], $\Sigma_{k}$ has
normal Du Bois singularities, and in particular,

$\beta_{k,*}\mathscr{O}_{B_{k}}=\mathscr{O}_{\Sigma_{k}}.$

By [12, Theorem 1.2], $\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically
Cohen–Macaulay, and $H^{2k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(m))=0$ for
$m>0$. Note that
$\mathscr{O}_{B_{k}}(H_{k})=\beta^{*}\mathscr{O}_{\Sigma_{k}}(1)$. Put
$M_{H_{k}}:=\beta_{k}^{*}M_{\mathscr{O}_{\Sigma_{k}}(1)}$, which fits into a
short exact sequence

${0}$${M_{H_{k}}}$${H^{0}(C,L)\otimes\mathscr{O}_{B_{k}}}$${\mathscr{O}_{B_{k}}(H_{k})}$${0.}$$\scriptstyle{\operatorname{ev}}$

Set $Z_{k-1}:=\beta_{k}^{-1}(\Sigma_{k-1})$, which is an irreducible effective
divisor on $B_{k}$. Then
$\beta_{k,*}\mathscr{O}_{B_{k}}(-Z_{k-1})=\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}$.
We have a commutative diagram

${B_{k}}$${Z_{k-1}}$${C_{k+1}}$${\Sigma_{k}}$${\Sigma_{k-1}.}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\beta_{k}}$

Notice that $\omega_{C_{k+1}}=N_{k+1,\omega_{C}}$. Then we have

$\omega_{B_{k}}=\mathscr{O}_{B_{k}}(-(k+1)H_{k})\otimes\pi_{k}^{*}(\omega_{C_{k+1}}\otimes\det
E_{k+1,L})=\mathscr{O}_{B_{k}}(-(k+1)H_{k})\otimes\pi_{k}^{*}S_{k+1,\omega_{C}\otimes
L}(-2\delta_{k+1}).$

We will compute $\omega_{\Sigma_{k}}$ in Proposition 3.6. On the other hand,
the map $\sigma_{k+1}\colon C_{k}\times C\to C_{k+1}$ provides a morphism
$\alpha_{k}\colon B_{k-1}\times C\to B_{k}$ birational onto its image (see [3,
p. 432]). By [3, Lemma 1.1 $(a)$] (see [11, Subsection 3.2]), we have a
commutative diagram

(3.1) ${B_{k-1}\times
C}$${B_{k}}$${\Sigma_{k-1}}$${\Sigma_{k},}$$\scriptstyle{\alpha_{k}}$$\scriptstyle{\beta_{k}}$

where the left vertical map is the first projection followed by $\beta_{k-1}$.

######  Proposition 3.4 ([11]).

We have the following:

  1. (1)

$\mathscr{O}_{B_{k}}(Z_{k-1})=\mathscr{O}_{B_{k}}((k+1)H_{k})\otimes\pi_{k}^{*}S_{k+1,L}(-2\delta_{k+1})^{-1}$
and $\omega_{B_{k}}(Z_{k-1})=\pi_{k}^{*}S_{k+1,\omega_{C}}$.

  2. (2)

$R^{i}\beta_{k,*}\mathscr{O}_{B_{k}}(-Z_{k-1})=\begin{cases}\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}&\text{if
$i=0$}\\\ 0&\text{if $i>0$}.\end{cases}$

  3. (3)

$R^{i}\pi_{k,*}{\textstyle\bigwedge^{j}}M_{H_{k}}=\begin{cases}{\textstyle\bigwedge^{j}}M_{E_{k+1,L}}&\text{if
$i=0$}\\\ 0&\text{if $i>0$}.\end{cases}$

###### Proof.

$(1)$ The first assertion is [11, Proposition 3.5 $(2)$]. Note that $\det
E_{k+1,L}=S_{k+1,L}(-\delta_{k+1})$ and
$\omega_{C_{k+1}}=S_{k+1,\omega_{C}}(-\delta_{k+1})$. Since
$\omega_{B_{k}}=\mathscr{O}_{B_{k}}(-(k+1)H_{k+1})\otimes\pi_{k}^{*}S_{k+1,\omega_{C}\otimes
L}(-2\delta_{k+1})$, the second assertion follows.

$(2)$ It is [11, Theorem 5.2 $(2)$].

$(3)$ It is shown in [11, Proof of Lemma 5.1]. For reader’s convenience, we
give a sketch of the proof. We have a short exact sequence

${0}$${\pi_{k}^{*}M_{E_{k+1,L}}}$${M_{H_{k}}}$${K}$${0,}$

where $K|_{\pi_{k}^{-1}(\xi)}=M_{\mathscr{O}_{\mathbf{P}^{k}}(1)}$ for $\xi\in
C_{k+1}$. By Bott vanishing,

$R^{i}\pi_{k,*}{\textstyle\bigwedge^{j}}K=\begin{cases}\mathscr{O}_{C_{k+1}}&\text{if
$i=0$ and $j=0$}\\\ 0&\text{if $i>0$ or $j>0$}.\end{cases}$

Considering the filtration of ${\textstyle\bigwedge^{j}}M_{H_{k}}$ associated
to the above short exact sequence, we obtain the assertion. ∎

######  Lemma 3.5.

We have

$\begin{array}[]{l}H^{i}(\Sigma_{k},{\textstyle\bigwedge^{j}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}(1))\\\\[5.0pt]
=H^{i}(B_{k},{\textstyle\bigwedge^{j}}M_{H_{k}}\otimes\mathscr{O}_{B_{k}}(H_{k}-Z_{k-1}))\\\\[5.0pt]
=H^{2k+1-i}(B_{k},{\textstyle\bigwedge^{r-j}}M_{H_{k}}\otimes\omega_{B_{k}}(Z_{k-1}))^{\vee}\\\\[5.0pt]
=H^{2k+1-i}(C_{k+1},{\textstyle\bigwedge^{r-j}}M_{E_{k+1,L}}\otimes
S_{k+1,\omega_{C}})^{\vee}\\\\[5.0pt] =H^{2k+1-i}(C_{r-j}\times
C_{k+1},(N_{r-j,L}\boxtimes
S_{k+1,\omega_{C}})(-D_{r-j,k+1}))^{\vee}.\end{array}$

###### Proof.

The first equality follows from Proposition 3.4 $(2)$ and the projection
formula. Note that $\operatorname{rank}M_{H_{k}}=h^{0}(C,L)-1=r$ and $\det
M_{H_{k}}=\mathscr{O}_{B_{k}}(-H_{k})$. It follows that
${\textstyle\bigwedge^{j}}M_{H_{k}}^{\vee}={\textstyle\bigwedge^{r-j}}M_{H_{k}}\otimes\mathscr{O}_{B_{k}}(H_{k})$.
Then the second equality follows from Serre duality. The third equality
follows from Proposition 3.4 $(1)$ and $(3)$. The final equality follows from
Lemma 3.3. ∎

Finally, we show some useful facts on the dualizing sheaf
$\omega_{\Sigma_{k}}$. The following proposition will not be used in the proof
of Theorem 1.1 but will be used for some additional results.

######  Proposition 3.6 (Ein111This was shown to the third author by Lawrence
Ein in personal communication.).

We have the following:

  1. (1)

$\beta_{k,*}\omega_{B_{k}}(Z_{k-1})=\omega_{\Sigma_{k}}$.

  2. (2)

There is a short exact sequence

${0}$${\beta_{k,*}\omega_{B_{k}}}$${\omega_{\Sigma_{k}}}$${\beta_{k,*}\omega_{Z_{k-1}}}$${0.}$

  3. (3)

$H^{0}(\Sigma_{k},\omega_{\Sigma_{k}}(\ell))=H^{0}(C_{k+1},S^{\ell}E_{k+1,L}\otimes
S_{k+1,\omega_{C}})$ for all $\ell\geq 0$.

  4. (4)

If $k\geq 2$, then
$H^{0}(\Sigma_{k},\omega_{\Sigma_{k}}(\ell))=H^{0}(\Sigma_{k-1},\beta_{k,*}\omega_{Z_{k-1}}(\ell))$
for each $0\leq\ell\leq k$.

###### Proof.

$(1)$ By [11, Proposition 3.15], there is a log resolution
$b_{k}\colon\operatorname{bl}_{k}(B_{k})\to B_{k}$ of $(B_{k},Z_{k-1})$
constructed by Bertram in [3] such that

$b_{k}^{*}\omega_{B_{k}}(Z_{k-1})=\omega_{\operatorname{bl}_{k}(B_{k})}(E_{0}+E_{1}+\cdots+E_{k-1}),$

where $E_{0},E_{1},\ldots,E_{k-2}$ are $\operatorname{bl}_{k}$-exceptional
divisors and $E_{k-1}=\operatorname{bl}_{k,*}^{-1}Z_{k-1}$. We have

$(\beta_{k}\circ
b_{k})_{*}\omega_{\operatorname{bl}_{k}(B_{k})}(E_{0}+E_{1}+\cdots+E_{k-1})=\beta_{k,*}\omega_{B_{k}}(Z_{k-1}).$

Note that $\beta_{k}\circ
b_{k}\colon\operatorname{bl}_{k}(B_{k})\to\Sigma_{k}$ is a log resolution of
$\Sigma_{k}$. Since $\Sigma_{k}$ has normal Cohen-Macaulay Du Bois
singularities by [11, Theorems 1.1 and 1.2], it follows from [19, Theorem 1.1]
that

$(\beta_{k}\circ
b_{k})_{*}\omega_{\operatorname{bl}_{k}(B_{k})}(E_{0}+E_{1}+\cdots+E_{k-1})=\omega_{\Sigma_{k}}.$

Thus $\beta_{k,*}\omega_{B_{k}}(Z_{k-1})=\omega_{\Sigma_{k}}$.

$(2)$ We have a short exact sequence

${0}$${\omega_{B_{k}}}$${\omega_{B_{k}}(Z_{k-1})}$${\omega_{Z_{k-1}}}$${0.}$

By Grauert–Riemenschneider vanishing, $R^{i}\beta_{k,*}\omega_{B_{k}}=0$ for
$i>0$. Applying $\beta_{k,*}$ to the above short exact sequence, we obtain the
assertion $(2)$.

$(3)$ We have
$H^{0}(\Sigma_{k},\omega_{\Sigma_{k}}(\ell))=H^{0}(B_{k},\omega_{B_{k}}(Z_{k-1}+\ell
H_{k}))$. Recall from Proposition 3.4 $(1)$ that
$\omega_{B_{k}}(Z_{k-1})=\pi_{k}^{*}S_{k+1,\omega_{C}}$. Thus
$H^{0}(B_{k},\omega_{B_{k}}(Z_{k-1}+\ell
H_{k}))=H^{0}(C_{k+1},S^{\ell}E_{k+1,L}\otimes S_{k+1,\omega_{C}})$.

$(4)$ Since $R^{i}\beta_{k,*}\omega_{B_{k}}=0$ for $i>0$, we have

$H^{i}(\Sigma_{k},\beta_{k,*}\omega_{B_{k}}(\ell))=H^{i}(B_{k},\omega_{B_{k}}(\ell
H_{k}))~{}~{}\text{ for $i\geq 0$.}$

If $k\geq 2$ and $0\leq\ell\leq k$, we have $H^{i}(B_{k},\omega_{B_{k}}(\ell
H_{k}))=0$ for each $i=0,1$. Thus the assertion $(4)$ follows. ∎

##  4\. Proof of Main Theorem

In this section, we prove Theorem 1.1. First, we recall the setting. Let $C$
be a smooth projective curve of genus $g\geq 2$, and $L$ be a very ample line
bundle on $C$. Consider the $k$-th secant variety $\Sigma_{k}$ of $C$ in
$\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$. Assume that $\deg L\gg 0$. When $k=0$
(i.e., $\Sigma_{0}=C$), Theorem 1.1 is the gonality conjecture established by
Ein–Lazarsfeld [10] and Rathmann [24]. Thus we assume that $k\geq 1$.222By a
small modification, our proof works for the case of $k=0$. The vanishing part
gives an alternative proof of the gonality conjecture. Indeed, when $k=0$ and
$q=1$, we only need to verify (4.3b). Put
$e:=\operatorname{codim}\Sigma_{k}=r-2k-1$ and $\gamma^{i}:=\gamma^{i}(C)$ for
$i\geq 0$. Fix an index $k+1\leq q\leq 2k+2$.

### Vanishing

We show that

(4.1)
$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))=0~{}~{}\text{
for $p\geq e-\gamma^{2k+2-q}+1$}.$

Consider a short exact sequence

${0}$${\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}}$${\mathscr{O}_{\Sigma_{k+1}}}$${\mathscr{O}_{\Sigma_{k}}}$${0.}$

This induces an exact sequence

(4.2)
$H^{q-1}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1))\longrightarrow
H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))\\\
\longrightarrow
H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))\longrightarrow
H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1)).$

It suffices to prove that

(4.3a) $\displaystyle
H^{q-1}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1))=0;$
(4.3b) $\displaystyle
H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))=0.$

First, we check (4.3b). By Lemma 3.5,

$\begin{array}[]{l}H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))\\\\[5.0pt]
=H^{q^{*}+1}(C_{p^{*}+q^{*}}\times C_{k+2},(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})),\end{array}$

where $p^{*}:=e-p\leq\gamma^{q^{*}}-1$ and $0\leq q^{*}:=2k+2-q\leq k+1$. By
the Leray spectral sequence for $\operatorname{pr}_{1}\colon
C_{p^{*}+q^{*}}\times C_{k+2}\to C_{p^{*}+q^{*}}$, it is enough to confirm
that

$H^{i}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{q^{*}+1-i}S_{k+2,\omega_{C}}\otimes
N_{p^{*}+q^{*},L})=0~{}~{}\text{ for $0\leq i\leq q^{*}+1$}.$

When $i>0$, this follows from Lemma 3.2. For the case $i=0$, we apply Lemma
3.1 to see that

$H^{q^{*}+1}(C_{k+2},S_{k+2,\omega_{C}(-\xi)})=S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))\otimes{\textstyle\bigwedge^{q^{*}+1}}H^{1}(C,\omega_{C}(-\xi))$

for any $\xi\in C_{p^{*}+q^{*}}$. Proposition 2.4 says that
$H^{1}(C,\omega_{C}(-\xi))\leq q^{*}$ since $\gamma^{q^{*}}\geq p^{*}+1$, so
we obtain

$H^{q^{*}+1}(C_{k+2},S_{k+2,\omega_{C}(-\xi)})=0.$

Thus $M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}=0$, and we obtain (4.3b). To
finish the proof of (4.1), we proceed by induction on $q-k-1$. If $q=k+1$,
then clearly

$H^{k}(\Sigma_{k+1},{\textstyle\bigwedge^{p+k}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1))=K_{p,k+1}(\Sigma_{k+1},\mathscr{O}_{\Sigma_{k+1}}(1))=0,$

i.e., (4.3a) holds. Thus (4.1) follows in this case. Suppose that $q\geq k+2$.
Lemma 2.2 $(1)$ implies that $e-\gamma^{2k+2-q}+1\geq(e-2)-\gamma^{2k+4-q}+1$.
By induction hypothesis,

$H^{q-1}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))=K_{p,q}(\Sigma_{k+1},\mathscr{O}_{\Sigma_{k+1}}(1))=0,$

i.e., (4.3a) holds. Thus (4.1) follows.

### Nonvanishing

We show that

(4.4)
$K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))\neq
0~{}~{}\text{ for $e-g+1\leq p\leq e-\gamma^{2k+2-q}$}.$

We have a commutative diagram with exact rows

${0}$${\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}}$${\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k+1}}}$${\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}}$${0}$${0}$${\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}}$${\mathscr{O}_{\Sigma_{k+1}}}$${\mathscr{O}_{\Sigma_{k}}}$${0.}$

This gives a commutative diagram

${H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}(1))}$${H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))}$${H^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))}$${H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1)).}$$\scriptstyle{\varphi}$

It is enough to prove that the map $\varphi$ is nonzero. For this purpose,
considering the commutative diagram (3.1), we regard $\varphi$ as a map

$\varphi\colon
H^{q-1}(B_{k},{\textstyle\bigwedge^{p+q-1}}M_{H_{k}}\otimes\mathscr{O}_{B_{k}}(H_{k}-Z_{k-1}))\otimes
H^{0}(C,\mathscr{O}_{C})\\\ \longrightarrow
H^{q}(B_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{H_{k+1}}\otimes\mathscr{O}_{B_{k+1}}(H_{k+1}-Z_{k}))$

In view of Lemma 3.5, the map $\varphi$ is dual to the map

$\varphi^{\vee}\colon H^{q^{*}+1}(C_{p^{*}+q^{*}}\times
C_{k+2},(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2}))\\\ \longrightarrow
H^{q^{*}}(C_{p^{*}+q^{*}}\times C_{k+1},(N_{p^{*}+q^{*},L}\boxtimes
S_{k+1,\omega_{C}})(-D_{p^{*}+q^{*},k+1}))\otimes H^{1}(C,\omega_{C}),$

where $\gamma^{q^{*}}\leq p^{*}:=e-p\leq g-1$ and $0\leq q^{*}:=2k+2-q\leq
k+1$. Notice that this map is induced from an injective map

$(\operatorname{id}_{C_{p^{*}+q^{*}}}\times\sigma_{k+2})^{*}(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2})\longhookrightarrow(N_{p^{*}+q^{*},L}\boxtimes
S_{k+1,\omega_{C}})(-D_{p^{*}+q^{*},k+1})\boxtimes\omega_{C}$

of line bundles on $C_{p^{*}+q^{*}}\times C_{k+1}\times C$. Lemma 3.2 says
that

$H^{i}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{j}(S_{\ell,\omega_{C}})\otimes
N_{p^{*}+q^{*},L})=0~{}~{}\text{ for $i>0,~{}j\geq 0,~{}\ell=k+1$ or $k+2$}.$

By the Leray spectral sequences for $\operatorname{pr}_{1}$, we may think that
$\varphi^{\vee}$ is a map

$\varphi^{\vee}\colon
H^{0}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes
N_{p^{*}+q^{*},L})\\\ \longrightarrow
H^{0}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{q^{*}}S_{k+1,\omega_{C}}\otimes
N_{p^{*}+q^{*},L})\otimes H^{1}(C,\omega_{C}).$

Notice that this map is induced from a map

$\psi\colon M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\longrightarrow
M_{p^{*}+q^{*}}^{q^{*}}S_{k+1,\omega_{C}}\otimes H^{1}(C,\omega_{C})$

of coherent sheaves on $C_{p^{*}+q^{*}}$ tensoring by $N_{p^{*}+q^{*},L}$. As
$N_{p^{*}+q^{*},L}$ is sufficiently positive, to prove that the map
$\varphi^{\vee}$ is nonzero, it suffices to confirm that the map $\psi$ is
nonzero. To this end, we apply Proposition 2.4 to see that $\omega_{C}$ fails
to be $(q^{*}-1)$-weakly $p^{*}$-very ample since $p^{*}\geq\gamma^{q^{*}}$.
Then Lemma 2.5 gives an effective divisor $\xi\in C_{p^{*}+q^{*}}$ on $C$ such
that

$h^{0}(C,\omega_{C}(-\xi))=g-p^{*}\geq 1~{}~{}\text{ and
}~{}~{}h^{1}(C,\omega_{C}(-\xi))=q^{*}+1.$

By Lemma 3.1,

$H^{i}(C_{\ell},S_{\ell,\omega_{C}(-\xi)})=S^{\ell-i}H^{0}(C,\omega_{C}(-\xi))\otimes{\textstyle\bigwedge^{i}}H^{1}(C,\omega_{C}(-\xi)),$

so this cohomology vanishes when $i\geq q^{*}+2$. By semicontinuity,
$h^{1}(C,\omega_{C}(-\xi^{\prime}))\leq q^{*}+1$ (and hence
$H^{q^{*}+2}(C_{\ell},S_{\ell,\omega_{C}(-\xi^{\prime})})=0$) for
$\xi^{\prime}$ in a neighborhood of $\xi$ in $C_{p^{*}+q^{*}}$. By the
cohomology and base change,

$\rho(\xi)^{q^{*}+1}\colon
M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes\mathbf{k}(\xi)\longrightarrow
H^{q^{*}+1}(C_{k+2},S_{k+2,\omega_{C}(-\xi)})$

is an isomorphism. We have a commutative diagram

${M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes\mathbf{k}(\xi)}$${M_{p^{*}+q^{*}}^{q^{*}}S_{k+1,\omega_{C}}\otimes
H^{1}(C,\omega_{C})\otimes\mathbf{k}(\xi)\
}$${H^{q^{*}+1}(C_{k+2},S_{k+2,\omega_{C}(-\xi)})}$${H^{q^{*}}(C_{k+1},S_{k+1,\omega_{C}(-\xi)})\otimes
H^{1}(C,\omega_{C}).}$$\scriptstyle{\psi\otimes\mathbf{k}(\xi)}$$\scriptstyle{\rho(\xi)^{q^{*}+1}}$

$\scriptstyle\simeq$

We reduce the problem to checking that the bottom map is nonzero. To this end,
note that the bottom map can be identified with the map

$\operatorname{id}_{S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))}\otimes\delta\colon
S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))\otimes{\textstyle\bigwedge^{q^{*}+1}}H^{1}(C,\omega_{C}(-\xi))\\\
\longrightarrow
S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))\otimes{\textstyle\bigwedge^{q^{*}}}H^{1}(C,\omega_{C}(-\xi))\otimes
H^{1}(C,\omega_{C}),$

where $\delta$ is a Koszul-like map. For a surjective map

$\eta\colon
H^{1}(C,\omega_{C}(-\xi))\xrightarrow{~{}\cdot\xi~{}}H^{1}(C,\omega_{C}),$

let $s_{1},\ldots,s_{q^{*}+1}$ be a basis of $H^{1}(C,\omega_{C}(-\xi))$ with
$\eta(s_{1})=\cdots=\eta(s_{q^{*}})=0$ but $\eta(s_{q^{*}+1})\neq 0$. Then

$\delta(s_{1}\wedge\cdots\wedge s_{q^{*}}\wedge
s_{q^{*}+1})=(-1)^{q^{*}}s_{1}\wedge\cdots\wedge
s_{q^{*}}\otimes\eta(s_{q^{*}+1})\neq 0.$

Thus the bottom map
$\operatorname{id}_{S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))}\otimes\delta$ in
the above commutative diagram is nonzero. Therefore, the map $\varphi^{\vee}$
(and hence $\varphi$) is nonzero, so (4.4) follows.

##  5\. Complements and Questions

In this section, we present some additonal results and problems. We keep using
the notations in the previous section. Let $C$ be a smooth projective curve of
genus $g\geq 2$, and $L$ be a line bundle on $C$ with $\deg L\geq 2g+2k+1$. We
denote by $\Sigma_{k}$ the $k$-th secant variety of $C$ in
$\mathbf{P}^{2k+1+e}=\mathbf{P}H^{0}(C,L)$. Consider the case that $k=0$.
Recall from [24, Theorem 1.1] that if $H^{1}(C,L\otimes\omega_{C}^{-1})=0$,
then

$K_{p,1}(C,L)\neq 0~{}~{}\Longleftrightarrow~{}~{}1\leq p\leq
e-\operatorname{gon}(C)+1.$

Recall from [16, Theorem (4.a.1)], [17, Theorem 2] that if
$H^{0}(C,L\otimes\omega_{C}^{-1})\neq 0$, then

$K_{p,2}(C,L)\neq 0~{}~{}\Longleftrightarrow~{}~{}e-g+1\leq p\leq e.$

Thus Theorem 1.1 holds for $k=0$ as soon as $\deg L\geq 4g-3$.

######  Problem 5.1.

Find an effective bound for $\deg L$ such that the conclusion of Theorem 1.1
holds.

We do not attempt to make a conjecture for what the best bound for $\deg L$
should be, but we expect that it would be linear in $g$. Here we give answers
for some partial cases.

### Effective Nonvanishing for $q=k+1$

Recall from Lemma 2.2 $(2)$ that $\gamma^{k+1}(C)=g$ for $k\geq g-1$. If
$k\geq g-1$ and $\deg L\geq 2g+2k+1$, then [11, Theorem 1.2] implies that

$K_{p,k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0~{}~{}\text{ for
$1\leq p\leq e-\gamma^{k+1}(C)$}.$

Thus we assume that $k\leq g-2$. On the other hand, Sidman–Vermeire [25,
Theorem 1.2] proved that if $L=L_{1}\otimes L_{2}$, where $L_{1},L_{2}$ are
line bundles on $C$ with $s+1:=h^{0}(C,L_{1})\geq k+2$ and
$t+1:=h^{0}(C,L_{2})\geq k+2$, then

$K_{p,k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0~{}~{}\text{ for
$1\leq p\leq s+t-2k-1$}.$

This yields the following effective nonvanishing statement:

######  Proposition 5.2.

Assume that $k\leq g-2$ and $\deg L\geq 2g+\gamma^{k+1}(C)+k$. Then

$K_{p,k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0~{}~{}\text{ for
$1\leq p\leq e-\gamma^{k+1}(C)$}.$

###### Proof.

By Lemma 2.2 $(2)$, $\gamma^{k+1}(C)\geq k+1$, so $\deg L\geq 2g+2k+1$. We
write $\deg L=2g+\gamma^{k+1}(C)+k+\ell$ for some integer $\ell\geq 0$. Then
$e=g+\gamma^{k+1}(C)+\ell-k-1$. Lemma 2.5 gives a line bundle $L_{1}$ on $C$
with $\deg L_{1}=\gamma^{k+1}(C)+k+1$ and $s+1:=h^{0}(C,L_{1})=k+2$. Let
$L_{2}:=L\otimes L_{1}^{-1}$ so that $L=L_{1}\otimes L_{2}$. Then $\deg
L_{2}=2g-1+\ell$ and $t+1:=h^{0}(C,L_{2})=g+\ell$. Note that

$s+t-2k-1=g+\ell-k-1=e-\gamma^{k+1}(C).$

Thus the proposition follows from [25, Theorem 1.2]. ∎

######  Remark 5.3.

Assume that $k\leq g-2$. By Lemma 2.2 $(2)$, $\gamma^{k+1}(C)\leq g-1$. Then
Proposition 5.2 holds when $\deg L\geq 4g-3$.

### Effective Nonvanishing for $q=2k+2$

Assume that $\deg L\geq 2g+2k+1$. By duality, we have

$K_{p,2k+2}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=K_{e-p,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))^{\vee}.$

Note that if
$K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))\neq 0$,
then $K_{p,2k+2}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0$ for $e-g+1\leq
p\leq e$. We need to find an effective bound on $\deg L$ for

$K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))=H^{0}(\Sigma_{k},{\textstyle\bigwedge^{g-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\omega_{\Sigma_{k}})\neq
0.$

Notice that
$K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))$ is the
kernel of the Koszul differential

$\delta\colon{\textstyle\bigwedge^{g-1}}H^{0}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\otimes
H^{0}(\Sigma_{k},\omega_{\Sigma_{k}})\longrightarrow{\textstyle\bigwedge^{g-2}}H^{0}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\otimes
H^{0}(\Sigma_{k},\omega_{\Sigma_{k}}(1)).$

In view of Proposition 3.6, $\delta$ can be identified with the map

$\delta\colon{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
S^{k+1}H^{0}(C,\omega_{C})\longrightarrow{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C})\otimes S^{k}H^{0}(C,\omega_{C})$

given by

$\delta(s_{1}\wedge\cdots\wedge s_{g-1}\otimes
f)=\sum_{i=1}^{g-1}\sum_{j=1}^{g}(-1)^{i-1}s_{1}\wedge\cdots\wedge\widehat{s_{i}}\wedge\cdots\wedge
s_{g-1}\otimes s_{i}x_{j}\otimes\frac{\partial f}{\partial x_{j}},$

where $x_{1},\ldots,x_{g}$ is a basis of $H^{0}(C,\omega_{C})$. The following
gives an answer to a question of Sidman–Vermeire in [26, p.164].

######  Proposition 5.4.

We have the following:

  1. (1)

If $k$ is even, then there is an injective map

$S^{g-1}H^{0}(C,L\otimes\omega_{C}^{-k-1})\longhookrightarrow
K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1)).$

  2. (2)

If $k$ is odd, then there is an injective map

${\textstyle\bigwedge^{g-1}}H^{0}(C,L\otimes\omega_{C}^{-k-1})\longhookrightarrow
K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1)).$

In particular, if

$h^{0}(C,L\otimes\omega_{C}^{-k-1})\geq\begin{cases}1&\textup{when $k$ is
even}\\\ g-1&\textup{when $k$ is odd,}\end{cases}$

then

$K_{p,2k+2}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))\neq 0~{}~{}\text{ for
$e-g+1\leq p\leq e$}.$

###### Proof.

First, we recall some notations from multilinear algebra. Let $V$ be a vector
space over $\mathbf{k}$, and

$T^{m}V:=\underbrace{V\otimes\cdots\otimes V}_{m~{}\text{times}}~{}~{}\text{
for any integer $m\geq 0$}.$

Since $\operatorname{char}(\mathbf{k})=0$, there are natural splitting
injective $\mathbf{k}$-linear maps

$\displaystyle\textup{alt}\colon{\textstyle\bigwedge^{m}}V\longhookrightarrow
T^{m}V,~{}~{}v_{1}\wedge\cdots\wedge
v_{m}\longmapsto\sum_{\sigma\in\mathfrak{S}_{m}}\textup{sign}(\sigma)v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(m)};$ $\displaystyle\textup{sym}\colon S^{m}V\longhookrightarrow
T^{m}V,~{}~{}v_{1}\cdots
v_{m}\longmapsto\sum_{\sigma\in\mathfrak{S}_{m}}v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(m)}.$

Put $\textup{Alt}^{m}V:=\textup{alt}({\textstyle\bigwedge^{m}}V)$ and
$\textup{Sym}^{m}V:=\textup{sym}(S^{m}V)$.

Now, write $L_{0}:=L\otimes\omega_{C}^{-k-1}$, and let

$R:=\bigoplus_{i,j\geq 0}H^{0}(C,L_{0}^{i}\otimes\omega_{C}^{j}).$

We have the following commutative diagram

${{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
S^{k+1}H^{0}(C,\omega_{C})}$${{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C})\otimes S^{k}H^{0}(C,\omega_{C})}$${T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C})}$${T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C}),}$$\scriptstyle{\delta}$$\scriptstyle{\textup{alt}\otimes\operatorname{id}_{S^{k+1}H^{0}(C,\omega_{C})}}$$\scriptstyle{\textup{alt}\otimes\operatorname{id}_{H^{0}(C,L\otimes\omega_{C})}\otimes\operatorname{id}_{S^{k}H^{0}(C,\omega_{C})}}$$\scriptstyle{d}$

where the bottom map $d$ is defined by

$d(s_{1}\otimes\cdots\otimes s_{g-2}\otimes s_{g-1}\otimes
f)=\sum_{i=1}^{g}s_{1}\otimes\cdots\otimes s_{g-2}\otimes
s_{g-1}x_{i}\otimes\frac{\partial f}{\partial x_{i}}.$

Notice that there is a canonical ring structure on $T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C})$ and the operator $d$ on $T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C})$ satisfies the chain rule. Consider the
alternating tensor

$\textup{alt}(x_{1}\otimes\cdots\otimes
x_{g})\in\textup{Alt}^{g}H^{0}(C,\omega_{C})\subseteq T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C}).$

Suppose that $k$ is even. We may assume that
$H^{0}(C,L\otimes\omega_{C}^{-k-1})=H^{0}(C,L_{0})\neq 0$. Let $\alpha_{0}\in
S^{g-1}H^{0}(C,L_{0})$ be any nonzero element, and

$\alpha:=(\textup{sym}(\alpha_{0})\otimes
1)(\textup{alt}(x_{1}\otimes\cdots\otimes x_{g}))^{k+1}\in T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C}).$

On the factor $T^{g-1}R$, the element $\textup{sym}(\alpha_{0})\otimes 1$ is
symmetric, and the element $\textup{alt}(x_{1}\otimes\cdots\otimes x_{g})$ is
alternating. Thus $\alpha$ is alternating, that is,
$\alpha\in\textup{Alt}^{g-1}H^{0}(C,L)\otimes S^{k+1}H^{0}(C,\omega_{C})$. On
the other hand, by the chain rule, $d\alpha=0$ since
$d(\textup{sym}(\alpha_{0})\otimes 1)=0$ and
$d(\textup{alt}(x_{1}\otimes\cdots\otimes x_{g}))=0$. As $T^{g-1}R\otimes
S^{\ast}H^{0}(C,\omega_{C})$ is an integral domain, $\alpha$ is a nonzero
element. We have shown that there is an element
$\alpha^{\prime}\in{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
S^{k+1}H^{0}(C,\omega_{C})$ such that $\delta(\alpha^{\prime})=0$. By sending
$\alpha_{0}$ to $\alpha^{\prime}$, we obtain the injective map in $(1)$.
Suppose that $k$ is odd. Replacing $\textup{sym}(\alpha_{0})$ with
$\textup{alt}(\alpha_{0})$ in the definition of $\alpha$, we obtain the
injective map in $(2)$. ∎

If $C$ is a hyperelliptic curve, then there is a morphism $\tau\colon
C\to\mathbf{P}^{1}$ of degree two such that
$\tau^{*}\mathscr{O}_{\mathbf{P}^{1}}(g-1)=\omega_{C}$. Let
$P:=\tau^{*}\mathscr{O}_{\mathbf{P}^{1}}(1)$ so that $\omega_{C}=P^{g-1}$. In
this case, we can improve the previous proposition as follows.

######  Proposition 5.5.

Assume that $C$ is a hyperelliptic curve. If $H^{0}(C,L\otimes P^{-g-k+1})\neq
0$, then

$K_{p,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))\neq
0~{}~{}\text{ for $0\leq p\leq g-1$}.$

###### Proof.

Let $A:=\mathscr{O}_{\mathbf{P}^{1}}(g-1)$ and
$B:=\mathscr{O}_{\mathbf{P}^{1}}(g+k-1)$. Then
$H^{0}(C,L\otimes\tau^{*}B^{-1})\neq 0$. We have a commutative diagram

${{\textstyle\bigwedge^{g-1}}H^{0}(\mathbf{P}^{1},B)\otimes
S^{k+1}H^{0}(\mathbf{P}^{1},A)}$${{\textstyle\bigwedge^{g-2}}H^{0}(\mathbf{P}^{1},B)\otimes
H^{0}(\mathbf{P}^{1},B\otimes A)\otimes
S^{k}H^{0}(\mathbf{P}^{1},A)}$${{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
S^{k+1}H^{0}(C,\omega_{C})}$${{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C})\otimes
S^{k}H^{0}(C,\omega_{C}).}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{\delta}$

It suffices to show that the upper horizontal map $\delta^{\prime}$ is not
injective. To this end, notice that $\delta^{\prime}$ can be identified with
the map

$\delta^{\prime}\colon{\textstyle\bigwedge^{g-1}}H^{0}(\mathbf{P}^{1},B)\otimes
H^{0}(\mathbf{P}^{k+1},S_{k+1,A})\longrightarrow{\textstyle\bigwedge^{g-2}}H^{0}(\mathbf{P}^{1},B)\otimes
H^{0}(\mathbf{P}^{k+1},E_{k+1,B}\otimes S_{k+1,A})$

by regarding $\mathbf{P}^{k+1}=(\mathbf{P}^{1})_{k+1}$. Thus we obtain

$\ker(\delta^{\prime})=H^{0}(\mathbf{P}^{k+1},{\textstyle\bigwedge^{g-1}}M_{E_{k+1,B}}\otimes
S_{k+1,A}).$

Since

$M_{E_{k+1,B}}=H^{0}(\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{1}}(g-2))\otimes\mathscr{O}_{\mathbf{P}^{k+1}}(-1)~{}~{}\text{
and }~{}~{}S_{k+1,A}=\mathscr{O}_{\mathbf{P}^{k+1}}(g-1),$

it follows that
$\ker(\delta^{\prime})=H^{0}(\mathbf{P}^{k+1},\mathscr{O}_{\mathbf{P}^{k+1}})\neq
0$. ∎

######  Remark 5.6.

The last part of Proposition 5.4 holds as soon as

$\deg L\geq\begin{cases}g+(k+1)(2g-2)&\textup{when $k$ is even}\\\
2g-2+(k+1)(2g-2)&\textup{when $k$ is odd}.\end{cases}$

When $C$ is hyperelliptic, Proposition 5.5 holds as soon as $\deg L\geq
3g+2k-2$.

######  Example 5.7.

Suppose that $C$ is nonhyperelliptic and $L:=\omega_{C}(D)$, where $D$ is a
general divisor of degree $g-1$ so that
$h^{0}(C,\mathscr{O}_{C}(D))=h^{1}(C,\mathscr{O}_{C}(D))=0$. Then $\deg
L=3g-3$, and $K_{g-1,0}(C,\omega_{C};L)=0$ by [17, Theorem 2]. For an integer
$1\leq k\leq(g-4)/2$, we have $\deg L\geq 2g+2k+1$. Consider the commutative
diagram

${{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
S^{k+1}H^{0}(C,\omega_{C})}$${{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C})\otimes
S^{k}H^{0}(C,\omega_{C})}$${{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
H^{0}(C,\omega_{C})\otimes
S^{k}H^{0}(C,\omega_{C})}$${{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C})\otimes
S^{k}H^{0}(C,\omega_{C}),}$$\scriptstyle{\delta}$$\scriptstyle{\operatorname{id}_{\wedge^{g-1}H^{0}(C,L)}\otimes
m}$$\scriptstyle{\delta\otimes\operatorname{id}_{S^{k}H^{0}(C,\omega_{C})}}$

where $m\colon S^{k+1}H^{0}(C,\omega_{C})\to H^{0}(C,\omega_{C})\otimes
S^{k}H^{0}(C,\omega_{C})$ is given by $m(f)=\sum_{j=1}^{g}x_{j}\otimes\partial
f/\partial x_{j}$. As $K_{g-1,0}(C,\omega_{C};L)$ is the kernel of the Koszul
differential

$\delta\colon{\textstyle\bigwedge^{g-1}}H^{0}(C,L)\otimes
H^{0}(C,\omega_{C})\longrightarrow{\textstyle\bigwedge^{g-2}}H^{0}(C,L)\otimes
H^{0}(C,L\otimes\omega_{C}),$

we see that
$K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))\subseteq
K_{g-1,0}(C,\omega_{C};L)\otimes S^{k}H^{0}(C,\omega_{C})$. Thus we obtain
$K_{g-1,0}(\Sigma_{k},\omega_{\Sigma_{k}};\mathscr{O}_{\Sigma_{k}}(1))=0$ in
this case.

### Effective Vanishing for $q=2k+1$

Let $c:=\gamma^{1}(C)=\operatorname{gon}(C)-1$. Then $\omega_{C}$ is
$(c-1)$-very ample. For any $1\leq p\leq c$, as
$h^{0}(C,\omega_{C}(-\xi))=g-p$ for all $\xi\in C_{p}$, we see that
$M_{E_{p,\omega_{C}}}=\operatorname{pr}_{1,*}(\mathscr{O}_{C_{p}}\boxtimes\omega_{C})(-D_{p,1})$
is a vector bundle on $C_{p}$. First, we prove the following vanishing result:

######  Proposition 5.8 (cf. [10, Proposition 2.1]).

Assume that $\deg L\geq(c^{2}+kc+k+1)(g-1)+1$. Then

$H^{i}(C_{p},S^{k}M_{E_{p,\omega_{C}}}\otimes N_{p,L})=0~{}~{}\text{ for $i>0$
and $1\leq p\leq c$}.$

###### Proof.

Let $V\subseteq H^{0}(C,\omega_{C})$ be a general subspace of dimension $2p$
so that the evaluation map $\operatorname{ev}\colon V\otimes C_{p}\to
E_{p,\omega_{C}}$ is surjective, and $M_{V}$ be the kernel of the evaluation
map. Then $M_{V}$ is a vector bundle of rank $p$ on $C_{p}$. We have a short
exact sequence

${0}$${M_{V}}$${M_{E_{p,\omega_{C}}}}$${(H^{0}(C,\omega_{C})/V)\otimes\mathscr{O}_{C_{p}}}$${0.}$

By considering the filtration of $S^{k}M_{E_{p,\omega_{C}}}$ associated to
this short exact sequence, we reduce the problem to proving that

(5.1) $H^{i}(C_{p},S^{j}M_{V}\otimes N_{p,L})=0~{}~{}\text{ for $i>0$ and
$0\leq j\leq k$}.$

Notice that $M_{V}\otimes N_{p,\omega_{C}}$ is globally generated and
$A_{j}:=N_{p,L}\otimes N_{p,\omega_{C}}^{-(p+j)}$ is ample for $0\leq j\leq k$
(see [10, Proof of Proposition 2.1]). Then

$S^{j}M_{V}\otimes N_{p,L}=N_{p,\omega_{C}}\otimes S^{j}(M_{V}\otimes
N_{p,\omega_{C}})\otimes\det(M_{V}\otimes N_{p,\omega_{C}})\otimes A_{j}.$

As $\omega_{C_{p}}=N_{p,\omega_{C}}$, the required cohomology vanishing (5.1)
follows from Griffiths vanishing [21, Variant 7.3.2]. ∎

######  Proposition 5.9.

Assume that $\deg L\geq\big{(}c^{2}+(c+1)(k+1+\lfloor
c/2\rfloor)+1\big{)}(g-1)+1$. Then

$K_{p,2k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=0~{}~{}\text{ for $p\geq
e-c+1$}.$

###### Proof.

Arguing as in the proof of Theorem 1.1, we reduce the problem to proving that

$H^{2k+1}(\Sigma_{k+i},{\textstyle\bigwedge^{p+2k}}M_{\mathscr{O}_{\Sigma_{k+i}}(1)}\otimes\mathscr{I}_{\Sigma_{k+i-1}|\Sigma_{k+i}}(1))=0~{}~{}\text{
for $i\geq 1$},$

which is equivalent to

$H^{2i}(C_{p^{*}}\times C_{k+1+i},(N_{p^{*},L}\boxtimes
S_{k+1+i,\omega_{C}})(-D_{p^{*},k+1+i}))=0~{}~{}\text{ for $i\geq 1$},$

where $p^{*}:=e-p+1\leq c$, by Lemma 3.5. Using Lemma 3.1, a similar argument
of the proof of Lemma 3.3 yields that

$M_{p^{*}}^{j}S_{k+1+i,\omega_{C}}=\begin{cases}S^{k+1+i}M_{E_{p^{*},\omega_{C}}}&\text{if
$j=0$}\\\ S^{k+i}M_{E_{p^{*},\omega_{C}}}&\text{if $j=1$}\\\ 0&\text{if $j\geq
2$}.\end{cases}$

Then it is enough to show that

$H^{2i-1+j}(C_{p^{*}},S^{k+i+j}M_{E_{p^{*},\omega_{C}}}\otimes
N_{p^{*},L})=0~{}~{}\text{ for $i\geq 1$ and $j=0,1$},$

but this follows from Proposition 5.8. ∎

######  Remark 5.10.

In view of [11, Theorem 4.1] and [24, Theorem 3.1], we expect that
Propositions 5.8 and 5.9 hold under a much weaker assumption.

######  Example 5.11.

Let $C$ be a smooth plane quartic curve. Then the genus $g$ of $C$ is $3$, and
$\gamma^{0}(C)=0,\gamma^{1}(C)=2,\gamma^{2}(C)=2$. Let
$L_{1}:=\omega_{C}^{3},L_{2}:=L_{1}(-x),L_{3}:=L_{1}(-x-y)$, where $x,y$ are
random points on $C$. Note that $\deg L_{1}=12,~{}\deg L_{2}=11,~{}\deg
L_{3}=10$. A Macaulay2 [15] computation shows that the Betti tables of
$R(\Sigma_{1},\mathscr{O}_{\Sigma_{1}}(1))$ for $L=L_{1},L_{2},L_{3}$ are the
following:

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---|---
$0$ | 1 | - | - | - | - | - | -
$1$ | - | - | - | - | - | - | -
$2$ | - | 38 | 108 | 102 | 10 | - | -
$3$ | - | - | - | - | 30 | - | -
$4$ | - | - | - | - | 3 | 18 | 6

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---|---
$0$ | 1 | - | - | - | - | -
$1$ | - | - | - | - | - | -
$2$ | - | 20 | 36 | 6 | - | -
$3$ | - | - | - | 20 | 1 | -
$4$ | - | - | - | 1 | 15 | 6

| $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$0$ | 1 | - | - | - | -
$1$ | - | - | - | - | -
$2$ | - | 8 | 3 | - | -
$3$ | - | - | 12 | 2 | -
$4$ | - | - | - | 12 | 6

Table 3. The Betti tables of $R(\Sigma_{1},\mathscr{O}_{\Sigma_{1}}(1))$

When $L=L_{3}$, we see that
$K_{g-1,0}(\Sigma_{1},\omega_{\Sigma_{1}};\mathscr{O}_{\Sigma_{1}}(1))=0$. In
this case,
$h^{0}(C,L\otimes\omega_{C}^{-2})=h^{0}(C,\omega_{C}(-x-y))=1<2=g-1$. This
shows that the condition in Proposition 5.4 is sharp. On the other hand,
notice that
$K_{1,1}(\Sigma_{1},\omega_{\Sigma_{1}};\mathscr{O}_{\Sigma_{1}}(1))\neq 0$
for $L=L_{2},L_{3}$; in other words, the conclusion of Proposition 5.9 does
not hold. However, $K_{1,1}(C,\omega_{C};L)=0$ for $L=L_{2},L_{3}$ by [24,
Theorem 1.1] since $\deg L\geq 9=4g-3$.

We now turn to the quantitative study of the nonzero Betti numbers

$\kappa_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1)):=\dim
K_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1)).$

It would be exceedingly interesting to know whether there is a uniform
asymptotic behavior of $\kappa_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$
as the positivity of $L$ grows. If so, one may further ask what kind of
geometry of $C$ is related to this asymptotic behavior.

For integers $m,\ell\geq 1$, we define

$\mathcal{L}_{m}^{\ell}=\mathcal{L}_{m}^{\ell}(C):=\\{\xi\in C_{m}\mid
h^{1}(C,\omega_{C}(-\xi))\geq\ell\\}.$

Let $e:=\operatorname{codim}\Sigma_{k}$.

######  Proposition 5.12.

Fix an integer $k+1\leq q\leq 2k+1$. Assume that
$L=L_{d}:=\mathscr{O}_{C}(dA+P)$ for an integer $d\gg 0$, where $A$ is an
ample divisor on $C$ and $P$ is any divisor on $C$. Then
$\kappa_{e-\gamma^{2k+2-q}(C),q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ is a
polynomial in $d$ of degree
$\dim\mathcal{L}_{2k+2-q+\gamma^{2k+2-q}(C)}^{2k+3-q}(C)$.

###### Proof.

Put $p:=e-\gamma^{2k+2-q}(C)$. By Theorem 1.1,

$\displaystyle
H^{q-1}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1))=K_{p,q}(\Sigma_{k+1},\mathscr{O}_{\Sigma_{k+1}}(1))=0;$
$\displaystyle
H^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{O}_{\Sigma_{k+1}}(1))=K_{p-1,q+1}(\Sigma_{k+1},\mathscr{O}_{\Sigma_{k+1}}(1))=0.$

Then the exact sequence (4.2) shows that

$\kappa_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=h^{q-1}(\Sigma_{k},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(1))=h^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1)).$

By Lemmas 3.2 and 3.5 and the Leray spectral sequence for
$\operatorname{pr}_{1}\colon C_{p^{*}+q^{*}}\times C_{k+2}\to
C_{p^{*}+q^{*}}$, we have

$h^{q}(\Sigma_{k+1},{\textstyle\bigwedge^{p+q-1}}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))=h^{0}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes
N_{p^{*}+q^{*},L}),$

where $p^{*}:=e-p$ and $q^{*}:=2k+2-q$. Note that
$\textup{Supp\,}M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}=\mathcal{L}_{p^{*}+q^{*}}^{q^{*}+1}(C)$.
As we may write $N_{p^{*}+q^{*},L}=N_{p^{*}+q^{*},\mathscr{O}_{C}(P)}\otimes
S_{p^{*}+q^{*},\mathscr{O}_{C}(A)}^{d}$ and
$S_{p^{*}+q^{*},\mathscr{O}_{C}(A)}$ is ample, we see that

$h^{0}(C_{p^{*}+q^{*}},M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes
N_{p^{*}+q^{*},L})=\chi(M_{p^{*}+q^{*}}^{q^{*}+1}S_{k+2,\omega_{C}}\otimes
N_{p^{*}+q^{*},\mathscr{O}_{C}(P)}\otimes
S_{p^{*}+q^{*},\mathscr{O}_{C}(A)}^{d})$

is a polynomial in $d$ of degree $\dim\mathcal{L}_{p^{*}+q^{*}}^{q^{*}+1}(C)$.
∎

In the situation of the above proposition, for $e-g+1\leq p\leq e$,
Ein–Lazarsfeld [10, Theorem C] proved that $\kappa_{p,1}(C,\omega_{C};L)$ is a
polynomial in $d$ (see [29] for a higher dimensional generalization). Thus it
is natural to ask the following.

######  Question 5.13.

For $e-g+1\leq p\leq e$ and $k+1\leq q\leq 2k+2$, is
$\kappa_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ a polynomial in
$d:=\deg L$ when $d\gg 0$?

In some cases, one can compute
$\kappa_{p,q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ exactly. For instance,
$\kappa_{e,2k+2}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))={g+k\choose k+1}$
(see [11, Theorem 1.2]). In the curve case, Kemeny [18, Theorem 1.1] proved
that if $C$ is a general curve of genus $g\geq 2k-1$ and gonality
$k=\gamma^{1}(C)+1\geq 4$ and $L$ is a line bundle on $C$ with $\deg L\geq
2g+k$, then

$\kappa_{e-\gamma^{1}(C),1}(C,L)=e-\gamma^{1}(C),$

where $e:=h^{1}(C,L)-2$ is the codimension of $C$ in
$\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$. This theorem can be geometrically
interpreted as follows. Let $\tau\colon C\to\mathbf{P}^{1}$ be a branched
covering of degree $k$. Then the linear spans of the fibers of $\tau$ in
$\mathbf{P}^{r}$ sweep out a $k$-dimensional scroll $S$ containing $C$. There
is a natural injective map $\iota_{p}\colon K_{p,1}(S,\mathscr{O}_{S}(1))\to
K_{p,1}(C,L)$. Kemeny’s theorem says that $\iota_{e-\gamma^{1}(C)}$ is in fact
an isomorphism. Along this line, one may ask the following:

######  Question 5.14.

Fix an integer $k+1\leq q\leq 2k+1$. Under what conditions, can one compute
$\kappa_{e-\gamma^{2k+2-q}(C),q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$
exactly? In this case, can one find some interesting geometric meaning of
spanning Koszul classes of
$K_{e-\gamma^{2k+2-q}(C),q}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$?

For an integer $k\geq 0$, suppose that $C$ is a general curve carrying a
unique $(k+1)$-dimensional linear system $|L_{1}|$ of degree
$\gamma^{k+1}(C)+k+1$. Then we expect that

$\kappa_{e-\gamma^{k+1}(C),k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))=\binom{e-\gamma^{k+1}(C)+k}{k+1}.$

Suppose that the expectation is true. Let $M$ be a matrix given by the
multiplication map

$H^{0}(C,L_{1})\otimes H^{0}(C,L\otimes L_{1}^{-1})\longrightarrow
H^{0}(C,L),$

and $X\subseteq\mathbb{P}^{r}$ be the projective variety cut out by
$(k+2)$-minors of $M$. Then the natural map

$K_{e-\gamma^{k+1}(C),k+1}(X,\mathscr{O}_{X}(1))\longrightarrow
K_{e-\gamma^{k+1}(C),k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$

is an isomorphism. We remark that $R(X,\mathscr{O}_{X}(1))$ is minimally
resolved by the Eagon–Northcott complex associated to $M$. Thus
$K_{e-\gamma^{k+1}(C),k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(1))$ is spanned
by Koszul classes of the smallest rank $e-\gamma^{k+1}(C)+k+1$ (see [4,
Corollary 4.3]).

## References

  * [1] Daniele Agostini, The Martens–Mumford theorem and the Green–Lazarsfeld secant conjecture, preprint (2021), arXiv:2110.03561.
  * [2] Marian Aprodu and Jan Nagel, Koszul cohomology and algebraic geometry, University Lecture Series, 52 (2010), Amer. Math. Soc., Providence, RI.
  * [3] Aaron Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35 (1992), 429–469.
  * [4] Junho Choe and Sijong Kwak, A matryoshka structure of higher secant varieties and the generalized Bronowski’s conjecture, Adv. Math. 406 (2022), Paper no.108526, 45 pp.
  * [5] Chih-Chi Chou and Lei Song, Singularities of secant varieties, Int. Math. Res. Not. 2018 no.9, 2844–2865.
  * [6] Marc Coppens, Changho Keem, and Gerriet Martens, Primitive linear series on curves, Manuscripta Math. 77 (1992), 237–264.
  * [7] Gentiana Danila, Sections de la puissance tensorielle du fibrè tautologique sur le schéma de Hilbert des points d’une surface, Bull. Lond. Math. Soc. 39 (2007), 311–316.
  * [8] Lawrence Ein and Robert Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), 51–67.
  * [9] Lawrence Ein and Robert Lazarsfeld, Asymptotic syzygies of algebraic varieties, Invent. Math. 190 (2012), 603–646.
  * [10] Lawrence Ein and Robert Lazarsfeld, The gonality conjecture on syzygies of algebraic curves of large degree, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 301–313.
  * [11] Lawrence Ein, Wenbo Niu, and Jinhyung Park, Singularities and syzygies of secant varieties of nonsingular projective curves, Invent. Math. 222 (2020), 615–665.
  * [12] Lawrence Ein, Wenbo Niu, and Jinhyung Park, A remark on global sections of secant bundles of curves, Boll. Unione Mat. Ital. 15 (2022), 163–171.
  * [13] Tom Fisher, The higher secant varieties of an elliptic normal curve, preprint (2006).
  * [14] Hans-Christian Graf von Bothmer and Klaus Hulek, Geometric syzygies of elliptic normal curves and their secant varieties, Manuscripta Math. 113 (2004), 35–68.
  * [15] Daniel Grayson and Michael Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
  * [16] Mark Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125–171.
  * [17] Mark Green and Robert Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), 301–314.
  * [18] Michael Kemeny, Projecting syzygies of curves, Algebraic Geometry 7 (2020), 561–580.
  * [19] Sándor J. Kovács, Karl Schwede, and Karen E. Smith, The canonical sheaf of Du Bois singularities, Adv. Math. 224 (2010), 1618–1640.
  * [20] Herbert Lange and Gerriet Martens, On the gonality sequence of an algebraic curve, Manuscripta Math. 137 (2012), 457–473.
  * [21] Robert Lazarsfeld, Positivity in algebraic geometry I $\&$ II, A Series of Modern Surveys in Math. 48 $\&$ 49 (2004), Springer-Verlag, Berlin.
  * [22] Li Li, Syzygies of secant varieties of curves of genus 2, preprint (2023), arXiv:2305.02479.
  * [23] Jinhyung Park, Asymptotic vanishing of syzygies of algebraic varieties, Comm. Amer. Math. Soc. 2 (2022), 133–148.
  * [24] Jürgen Rathmann, An effective bound for the gonality conjecture, preprint (2016), arXiv:1604.06072.
  * [25] Jessica Sidman and Peter Vermeire, Syzygies of the secant variety of a curve, Algebra Number Theory 3 (2009), 445–465.
  * [26] Jessica Sidman and Peter Vermeire, Equations defining secant varieties: geometry and computation, Combinatorial aspects of commutative algebra and algebraic geometry, Abel Symp. 6 (2011), Springer, Berlin, 155–174.
  * [27] Gregory Taylor, Asymptotic syzygies of secant varieties of curves, J. Pure Appl. Algebra 226 (2022), 107139.
  * [28] Brooke Ullery, On the normality of secant varieties, Adv. Math. 288 (2016), 631–647.
  * [29] David Yang, $S_{n}$-equivariant sheaves and Koszul cohomology, Res. Math. Sci. 1 (2014), Art. 10, 6 pp.