11institutetext: Jinhyung Park 22institutetext: Department of Mathematical
Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Replubic of Korea,
22email<EMAIL_ADDRESS>
# Syzygies of algebraic varieties through symmetric products of algebraic
curves

Jinhyung Park0000-0002-9710-9728

###### Abstract

This is a survey paper on recent work on syzygies of algebraic varieties. We
discuss the gonality conjecture on weight-one syzygies of algebraic curves,
syzygies of secant varieties of algebraic curves, syzygies of tangent
developable surfaces and Green’s conjecture on syzygies of canonical curves,
and asymptotic syzygies of algebraic varieties. All results considered in this
paper were proven using the geometry of symmetric products of algebraic
curves.

##  1 Introduction

Throughout the paper, we work over an algebraically closed field $\mathbf{k}$
of arbitrary characteristic. In the late 1960s, Mumford Mumford  proved that
if $X$ is a smooth projective variety and $L$ is a sufficiently positive line
bundle, then $X\subseteq\mathbf{P}H^{0}(X,L)=\mathbf{P}^{r}$ is projectively
normal and $I_{X|\mathbf{P}^{r}}$ is generated by quadrics. In the early
1980s, Green realized that the classical results on equations defining
algebraic varieties should be seen as the first cases of a much more general
picture involving higher syzygies. After his pioneering work Green1 ; Green2
on Koszul cohomology, there has been a considerable amount of interest in
syzygies of algebraic varieties. In the early 2000s, Voisin Voisin1 ; Voisin2
resolved Green’s conjecture for general canonical curves using the Hilbert
scheme interpretation of Koszul cohomology. Voisin’s method has led to a great
deal of progress on syzygies of algebraic varieties over the past two decades.
Some of the recent developments were surveyed in EL3  and Farkas1 . We will
discuss further advances centered around the author’s work since those papers.

Let $X$ be a projective scheme of dimension $n$, and $L$ be a globally
generated line bundle on $X$ with $r:=h^{0}(X,L)-1$. We usually assume that
$L$ is sufficiently positive; in particular, $L$ is very ample, so it gives
rise to an embedding $X\subseteq\mathbf{P}H^{0}(X,L)=\mathbf{P}^{r}$. Let $B$
be a coherent sheaf on $X$ with $H^{0}(X,B\otimes L^{-m})=0$ for $m>0$. By
Hilbert syzygy theorem, the finitely generated graded section module

$R=R(X,B;L):=\bigoplus_{m\in\mathbf{Z}}H^{0}(X,B\otimes L^{m})$

over the polynomial ring $S:=\bigoplus_{m\in\mathbf{Z}}S^{m}H^{0}(X,L)$ has a
minimal free resolution

$0\longleftarrow R\longleftarrow E_{0}\longleftarrow
E_{1}\longleftarrow\cdots\longleftarrow E_{r}\longleftarrow 0,$

where

$E_{p}=\bigoplus_{q\in\mathbf{Z}}\underbrace{K_{p,q}(X,B;L)}_{\emph{Koszul
cohomology}}\otimes_{\mathbf{k}}S(-p-q).$

Here $K_{p,q}(X,B;L)$ is the cohomology of the Koszul-type complex

$\wedge^{p+1}H^{0}(L)\otimes H^{0}(B\otimes
L^{q+1})\\!\rightarrow\\!\wedge^{p}H^{0}(L)\otimes H^{0}(B\otimes
L^{q})\\!\rightarrow\\!\wedge^{p-1}H^{0}(L)\otimes H^{0}(B\otimes L^{q-1}),$

and it can be regarded as the space of $p$-th syzygies of weight $q$. Denote
by

$\kappa_{p,q}=\kappa_{p,q}(X,B;L):=\dim K_{p,q}(X,B;L)$

the _graded Betti numbers_ of $R=R(X,B;L)$. The _Betti table_ of $R$ is the
following:

| $0$ | $1$ | $2$ | $~{}\cdots~{}$
---|---|---|---|---
$0$ | $\kappa_{0,0}$ | $\kappa_{1,0}$ | $\kappa_{2,0}$ | $~{}\cdots~{}$
$1$ | $\kappa_{0,1}$ | $\kappa_{1,1}$ | $\kappa_{2,1}$ | $~{}\cdots~{}$
$2$ | $\kappa_{0,2}$ | $\kappa_{1,2}$ | $\kappa_{2,2}$ | $~{}\cdots~{}$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |

The width of the Betti table of $R$ is the _projective dimension_
$\operatorname{pd}(R)$. By Auslander–Buchsbaum formula,
$\operatorname{pd}(R)=r-\operatorname{depth}(R)$. The height of the Betti
table of $R$ is the _Castelnuovo–Mumford regularity_ $\operatorname{reg}(R)$
(see (positivity, , Subsection 1.8)).

Our main concern is the overall shape of the Betti table of $R(X,B;L)$ when
$L$ is sufficiently positive. Specifically, we investigate from a geometric
perspective when $\kappa_{p,q}(X,B;L)$ are vanishing and what their values are
if they are not vanishing. In this paper, we always assume that
$H^{i}(X,B\otimes L^{m})=0$ for $i>0$ and $m>0$. Then
$\operatorname{reg}(R)\leq n+1$, so $K_{p,q}(X,B;L)=0$ for $q\geq n+2$. It is
well-known that

$K_{p,q}(X,B;L)=H^{q-1}(X,\wedge^{p+q-1}M_{L}\otimes B\otimes L)~{}~{}\text{
for $p\geq 0$ and $q\geq 2$},$

where $M_{L}$ is the kernel bundle of the evaluation map
$H^{0}(X,L)\otimes\mathscr{O}_{X}\to L$. If $H^{i}(X,B\otimes L^{m})=0$ for
$1\leq i\leq n-1$ and $m\in\mathbf{Z}$ (i.e., $R(X,B;L)$ is Cohen–Macaulay),
then $K_{p,q}(X,B;L)=H^{q}(X,\wedge^{p+q}M_{L}\otimes B)$ and
$K_{p,q}(X,B;L)=K_{r-p-n,n+1-q}(X,\omega_{X}\otimes B^{-1};L)^{\vee}$ for
$p\geq 0$ and $q\geq 0$. The latter can be easily checked using Serre duality.
For more details, we refer to (AN, , Chapter 2), (EL1, , Section 3), (Park3, ,
Section 2).

For a moment, suppose that $B=\mathscr{O}_{X}$. We put
$R(X;L):=R(X,\mathscr{O}_{X};L)$ and
$K_{p,q}(X;L):=K_{p,q}(X,\mathscr{O}_{X};L)$. We know
$K_{0,0}(X;L)=\mathbf{k}$ and $K_{p,0}(X;L)=0$ for $p\geq 1$. For integers
$d\geq 2$ and $m\geq 0$, we say that $L$ satisfies _$N_{d,m}$ -property_ (when
$d=2$, we simply say _$N_{m}$ -property_) if $K_{p,q}(X;L)=0$ for $0\leq p\leq
m$ and $q\geq d$. This notion was introduced in EGHP  based on GL1 ; GL2  (our
definition is slightly different but the same when $X\subseteq\mathbf{P}^{r}$
is arithmetically normal, i.e., $R=S/I_{X|\mathbf{P}^{r}}$). As
$K_{0,1}(X;L)=0$, we see that $R=S/I_{X|\mathbf{P}^{r}}$ if and only if $L$
satisfies $N_{0}$-property. In this case, $\kappa_{1,d}(X;L)$ is the number of
minimal generators of $I_{X|\mathbf{P}^{r}}$ of degree $d+1$, so $L$ satisfies
$N_{1}$-property if and only if $I_{X|\mathbf{P}^{r}}$ is generated by
quadrics.

Generalizing classical results of Castelnuovo, Mattuck, Mumford, Saint-Donat,
Fujita on equations defining algebraic curves, Green (Green1, , Theorem 4.a.1)
proved that if $C$ is a smooth projective curve of genus $g$ and $\deg L\geq
2g+1+p$, then $L$ satisfies $N_{p}$-property. Green–Lazarsfeld (GL2, , Theorem
1.2) then showed that $K_{p+1,2}(C;L)\neq 0$ when $\deg L=2g+1+p$ and
$H^{0}(C,L\otimes\omega_{C}^{-1})\neq 0$. Thus we obtain the following:

######  Theorem 1.1 ($\operatorname{char}(\mathbf{k})=0$)

If $\deg L\geq 2g+1$ and $H^{0}(C,L\otimes\omega_{C}^{-1})\neq 0$, then

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

The vanishing and nonvanishing of $K_{p,1}(C;L)$ is predicted by
Green–Lazarsfeld’s gonality conjecture (GL1, , Conjecture 3.7). This was
verified by Ein–Lazarsfeld EL2  when $\deg L\gg 0$. Recently, Niu–Park NP
showed the following effective gonality theorem establishing a sharp effective
bound on $L$ to satisfies the gonality conjecture.

######  Theorem 1.2 ($\operatorname{char}(\mathbf{k})=0$)

If $\deg L\geq 2g+\operatorname{gon}(C)$, then

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

Here $\operatorname{gon}(C)$ is the _gonality_ of $C$, the minimal degree of a
branch covering $C\to\mathbf{P}^{1}$. As Castryck Castryck  observed that the
gonality conjecture may not hold for a plane curve when $\deg
L=2g+\operatorname{gon}(C)-1$, this theorem is the best possible and thus
gives a complete answer to the effective gonality problem. By Theorems 1.1 and
1.2, if $\deg L\geq 3g$, then the Betti table of $R(C;L)$ is the following:

$\begin{array}[]{c|ccccccccccc}&~{}0{}&~{}1{}&~{}2{}&~{}\cdots{}&r-g-1&r-g&\cdots&r-\operatorname{gon}(X)&r+1-\operatorname{gon}(X)&\cdots&r-1\\\
\hline\cr~{}0{}&1&-&-&\cdots&-&-&\cdots&-&-&\cdots&-\\\
~{}1{}&-&*&*&\cdots&*&*&\cdots&*&-&\cdots&-\\\
~{}2{}&-&-&-&\cdots&-&*&\cdots&*&*&\cdots&*\end{array}$

Here “$*$” indicates nonzero and “$-$” indicates zero. The number of nonzero
entries in the last row is $g$, and the number of last zero entries in the row
above is $\operatorname{gon}(C)-1$. This reveals the surprising connection
between the intrinsic geometric properties of the algebraic curve $C$ and the
extrinsic algebraic properties of the embedded curve
$C\subseteq\mathbf{P}^{r}$. The syzygies of algebraic curves will be discussed
in Section 2.

Sidman–Vermeire SV  proposed a generalization of Green’s $(2g+1+p)$-theorem to
secant varieties. Suppose that $\deg L\geq 2g+2k+1$. The _$k$ -th secant
variety_ $\Sigma_{k}$ of $C\subseteq\mathbf{P}^{r}$ is the union of
$(k+1)$-secant $k$-planes to $C$ in $\mathbf{P}^{r}$. Note that
$\Sigma_{0}=C$. We have $\dim\Sigma_{k}=2k+1$ and
$\operatorname{Sing}\Sigma_{k}=\Sigma_{k-1}$ except when $C=\mathbf{P}^{1}$
and $\deg L=2k+1$ (in this case $\Sigma_{k}=\mathbf{P}^{2k+1}$). Ein–Niu–Park
ENP1  proved the following theorem resolving the Sidman–Vermeire conjecture
(SV, , Conjecture 1.3) (the case of $k=0$ is Green’s $(2g+1+p)$-theorem, and
the case of $k=1$ was proved by Chou–Song CS , Sidman–Vermeire SV , Ullery
Ullery , Vermeire Vermeire , Yang RuijieYang ).

######  Theorem 1.3 ($\operatorname{char}(\mathbf{k})=0$)

The $k$-th secant variety $\Sigma_{k}$ has normal Du Bois singularities,
$\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay, and
$\mathscr{O}_{\Sigma_{k}}(1)$ satisfies $N_{k+2,p}$-property as soon as $\deg
L\geq 2g+2k+1+p$.

If $e:=\operatorname{codim}\Sigma_{k}=r-2k-1$, then the Betti table of
$R(\Sigma_{k};\mathscr{O}_{\Sigma_{k}}(1))$ is the following:

$\begin{array}[]{c|ccccccccc}&0&1&2&~{}\cdots&e-g-1&e-g&e-g+1&\cdots&~{}e\\\
\hline\cr 0&1&-&-&~{}\cdots&-&-&-&\cdots&~{}-\\\
1&-&-&-&~{}\cdots&-&-&-&\cdots&~{}-\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots&&~{}\vdots\\\
k&-&-&-&~{}\cdots&-&-&-&\cdots&~{}-\\\ k+1&-&*&*&~{}\cdots&*&*&?&\cdots&?\\\
k+2&-&-&-&~{}\cdots&-&-&?&\cdots&?\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots&&~{}\vdots\\\
2k+2&-&-&-&~{}\cdots&-&-&?&\cdots&?\end{array}$

When $\deg L\gg 0$, Choe–Kwak–Park CKP  completely determined the remaining
part of the Betti table, which is a natural generalization of the gonality
conjecture.

######  Theorem 1.4 ($\operatorname{char}(\mathbf{k})=0$)

For each $k+1\leq q\leq 2k+2$, if $e-g+1\leq p\leq e$, then

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

Here $(\gamma^{0}(C)+0,\gamma^{1}(C)+1,\gamma^{2}(C)+2,\ldots)$ is the
_gonality sequence_ of $C$ with $\gamma^{1}(C)+1=\operatorname{gon}(C)$. This
was originally conjectured in CK . Theorem 1.4 says that the part marked with
“$?$” in the above Betti table is the following:

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

The syzygies of secant varieties will be discussed in Section 3.

The _tangent developable surface_ of $C\subseteq\mathbf{P}^{r}$ is the union
of tangent lines to $C$ in $\mathbf{P}^{r}$. There has been a lot of interest
in the syzygies of the tangent developable surface of a rational normal curve
of degree $g$ because they are the same to the syzygies of general canonical
curves of genus $g$. This was independently observed by O’Grady and
Buchweitz–Schreyer in 1980s. After 35 years,
Aprodu–Farkas–Papadima–Raicu–Weyman AFPRW  finally proved the following:

######  Theorem 1.5 ($\operatorname{char}(\mathbf{k})=0$ or $\geq(g+2)/2$)

If $T$ is the tangent developable surface of a rational normal curve of degree
$g$, then $\mathscr{O}_{T}(1)$ satisfies $N_{\lfloor(g-3)/2\rfloor}$-property.

This implies generic Green’s conjecture (Green1, , Conjecture 5.6), which
asserts that the canonical line bundle of a general curve of genus $g$
satisfies $N_{\lfloor(g-3)/2\rfloor}$-property. Recently, it was realized by
Park Park2  that $T$ is a Weil divisor linearly equivalent to $-K_{\Sigma}$,
where $\Sigma$ is the first secant variety of the rational normal curve of
degree $g$. This leads to a substantialy simple geometric proof of Theorem
1.5. Furthermore, it was also shown that if $C$ is a smooth projective curve
of genus $g\geq 1$ and $L$ is a line bundle on $C$ with $\deg L\geq 4g+3$,
then the tangent developable surface $T$ of $C$ in $\mathbf{P}H^{0}(C,L)$ is
arithmetically normal ((Park2, , Theorem 1.3)). The syzygies of tangent
developable surfaces and Green’s conjecture will be discussed in Section 4.

Green’s $(2g+1+p)$-theorem has stimulated further work on $N_{p}$-property,
and several analogous statements for higher dimensional algebraic varieties
have been established. On the other hand, in 2001, Ottaviani–Paoletti OP
called attention to the failure of $N_{m}$-property. Putting
$r_{d}:=h^{0}(\mathbf{P}^{2},\mathscr{O}_{\mathbf{P}^{2}}(d))-1$, they proved
that

$K_{p,2}(\mathbf{P}^{2};\mathscr{O}_{\mathbf{P}^{2}}(d))\neq 0~{}~{}\text{ for
$3d-2\leq p\leq r_{d}-3$}.$

As $r_{d}\approx d^{2}/2$, this shows $N_{m}$-property for
$\mathscr{O}_{\mathbf{P}^{2}}(d)$ describes only a small fraction of the
syzygies of the $d$-th Veronese embedding of $\mathbf{P}^{2}$. It is an
interesting problem to describe the overall asymptotic behaviors of
$K_{p,q}(X,B;L)$ as the positivity of $L$ grows. To set the stage, assume that
$X$ is a smooth projective variety of dimension $n$ and $B$ is a line bundle
on $X$, and let $L_{d}:=\mathscr{O}_{X}(dA+P)$ for an integer $d\gg 0$ and
$r_{d}:=h^{0}(X,L_{d})-1$, where $A$ is an ample divisor on $X$ and $P$ is an
arbitrary divisor on $X$. By (EL2, , Proposition 5.1 and Corollary 5.2), we
have

$\begin{array}[]{l}K_{p,0}(X,B;L_{d})\neq 0~{}\Longleftrightarrow~{}0\leq
p\leq h^{0}(X,B)-1;\\\ K_{p,n+1}(X,B;L_{d})\neq
0~{}\Longleftrightarrow~{}r_{d}-n-h^{0}(X,\omega_{X}\otimes B^{-1})+1\leq
p\leq r_{d}-n.\end{array}$ (1)

The main issue is then to study vanishing and nonvanishing of
$K_{p,q}(X,B;L_{d})$ for $1\leq q\leq n$. Ein–Lazarsfeld EL2  and Park Park1 ;
Park3  established the following:

######  Theorem 1.6 ($\operatorname{char}(\mathbf{k})\geq 0$)

For each $1\leq q\leq n$, there exist functions $c_{q}(d)$ and
$c_{q}^{\prime}(d)$ depending on $q,d$ and $X,B,A,P$ with

$c_{q}(d)=\Theta(d^{q-1})~{}~{}\text{ and
}~{}~{}c_{q}^{\prime}(d)=\begin{cases}\Theta(d^{n-q})&\text{if
$H^{q-1}(X,B)=0$ or $q=1$}\\\ q-1&\text{if $H^{q-1}(X,B)\neq 0$ and $q\geq
2$}\end{cases}$

such that if $d$ is sufficiently large, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\Longleftrightarrow~{}~{}c_{q}(d)\leq p\leq
r_{d}-c_{q}^{\prime}(d).$

For a nonnegative function $f(d)$, we write $f(d)=\Theta(d^{k})$ if there are
constants $C_{1},C_{2}>0$ such that $C_{1}d^{k}\leq f(d)\leq C_{2}d^{k}$ for
any sufficiently large positive integer $d$ (for convention, we write
$f(d)=\Theta(1)$ even when $f(d)=0$ for all $d\gg 0$). A widely-accepted
intuition derived from the case of curves was that syzygies become simpler as
the positivity of the embedding line bundle is increasing. However, as was
noticed by Ein–Lazarsfeld EL2 , this intuition was misleading since Theorem
1.6 says that almost all possibly nonzero syzygies are actually nonzero.
Instead, one may say that the syzygies have a surprisingly uniform asymptotic
behavior as the positivity of the embedding line bundle is increasing.
Although Theorem 1.6 is for higher dimensional algebraic varieties, the
geometry of symmetric products of algebraic curves plays a crucial role in an
important step of the proof. The asymptotic syzygies of algebraic varieties
will be discussed in Section 5.

##  2 Gonality conjecture on weight-one syzygies of algebraic curves

In this section, we assume that $\operatorname{char}(\mathbf{k})=0$. Let $C$
be a smooth projective curve of genus $g\geq 0$, and $B,L$ be line bundles on
$C$. We begin with the preparation for the necessary techniques relating
$K_{p,1}(C,B;L)$ to vector bundles on symmetric products of $C$. For more
details, we refer the reader to EL2 , ENP1 , NP .

For an integer $m\geq 0$, the $m$-th symmetric product $C_{m}$ of $C$
parameterizes effective divisors of degree $m$ on $C$. It is well-known that
$C_{m}$ is a smooth projective variety of dimension $m$. For integers $m,n\geq
1$, there is an addition map

$\sigma_{m,n}\colon C_{m}\times C_{n}\longrightarrow
C_{m+n},~{}~{}(\xi,\xi^{\prime})\longmapsto\xi+\xi^{\prime},$

which is a finite morphism. The incidence subvariety

$D_{m,n}=\\{(\xi,\eta)\in C_{m}\times
C_{n}\mid\operatorname{Supp}(\xi)\cap\operatorname{Supp}(\eta)\neq\emptyset\\}\subseteq
C_{m}\times C_{n}$

is an effective divisor on $C_{m}\times C_{n}$. If $n=1$, then
$D_{m,1}=C_{m-1}\times C$ (the map is given by $(\xi,x)\mapsto(\xi-x,x)$) and
$\sigma_{m-1,1}=\operatorname{pr}_{1}|_{D_{m,1}}\colon C_{m-1}\times C\to
C_{m}$ is the universal family over $C_{m}$, where
$\operatorname{pr}_{1}\colon C_{m}\times C\to C_{m}$ is the projection. Let

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

be the _tautological bundle_ , which is a vector bundle of rank $m$ on $C_{m}$
with $H^{0}(C_{m},E_{m,L})=H^{0}(C,L)$. Put $N_{m,L}:=\det(E_{m,L})$. If we
write $N_{m,\mathscr{O}_{C}}=\mathscr{O}_{C_{m}}(-\delta_{m})$, then
$\sigma_{m-1,1}^{*}\delta_{m}=\operatorname{pr}_{1}^{*}\delta_{m-1}+D_{m-1,1}\subseteq
C_{m-1}\times C$. Note that $\Omega_{C_{m}}^{1}=E_{m,\omega_{C}}$ and
$\omega_{C_{m}}=N_{m,\omega_{C}}$. Now, consider the group action of the
symmetric group $\mathfrak{S}_{m}$ on the $m$-th ordinary product $C^{m}$ of
$C$ and the quotient map $q_{m}\colon C^{m}\to C_{m}$. There is a line bundle
$S_{m,L}$ on $C_{m}$ which is the $\mathfrak{S}_{m}$-invariant descent of

$L^{\boxtimes m}:=\underbrace{L\boxtimes\cdots\boxtimes L}_{\text{$m$
times}}.$

Note that $q_{m}^{*}S_{m,L}=L^{\boxtimes m}$ and
$N_{m,L}=S_{m,L}(-\delta_{m})$. By (ENP1, , Lemma 3.7) (see also (Agostini2, ,
Lemma 2.4)), we have

$\begin{array}[]{l}H^{i}(C_{m},N_{m,L})=\wedge^{m-i}H^{0}(C,L)\otimes
S^{i}H^{1}(C,L);\\\
H^{i}(C_{m},S_{m,L})=S^{m-i}H^{0}(C,L)\otimes\wedge^{i}H^{1}(C,L).\end{array}$

Notice that the fiber of $E_{p+1,B}$ over $\xi\in C_{p+1}$ is
$H^{0}(\xi,B|_{\xi})$. Thus $B$ is $p$-very ample (i.e., the restriction map
$H^{0}(C,B)\to H^{0}(\xi,B|_{\xi})$ is surjective for every $\xi\in C_{p+1}$)
if and only if $\operatorname{ev}_{p+1,B}$ is surjective if and only if
$E_{p+1,B}$ is globally generated. From now on, suppose that $B$ is $p$-very
ample and $L$ is globally generated. Let $M_{p+1,B}$ be the kernel bundle of
the evaluation map $H^{0}(C,B)\otimes\mathscr{O}_{C_{p+1}}\to E_{p+1,B}$. The
key observation of Voisin (see e.g., (EL2, , Lemma 1.1)) is that
$K_{p,1}(C,B;L)$ is the cokernel of the multiplication map

$H^{0}(C,B)\otimes H^{0}(C_{p+1},N_{p+1,L})\longrightarrow
H^{0}(C_{p+1},E_{p+1,B}\otimes N_{p+1,L}).$

Thus $H^{1}(C_{p+1},M_{p+1,B}\otimes N_{p+1,L})=0$ implies $K_{p,1}(C,B;L)=0$,
and the converse also holds when $H^{1}(C,L)=0$.

We now turn to the gonality conjecture of Green–Lazarsfeld (GL1, , Conjecture
3.7) asserting that $K_{p,1}(C;L)=0$ for $p\geq r-\operatorname{gon}(C)+1$
when $\deg L\gg 0$. By the duality theorem ((Green1, , Theorem 2.c.1)),

$K_{p,1}(C;L)=K_{r(L)-p-1,1}(C,\omega_{C};L)^{\vee},$

where $r(L):=h^{0}(C,L)-1$. It is well-known that $\operatorname{gon}(C)-2\geq
p$ if and only if $\omega_{C}$ is $p$-very ample. Thus the gonality conjecture
can be restated as if $\omega_{C}$ is $p$-very ample, then
$K_{p,1}(C,\omega_{C};L)=0$. Niu–Park (NP, , Theorem 1.3) established a
general effective vanishing theorem for weight-one syzygies of algebraic
curves.

###### Theorem 2.1

Suppose that $B$ is $p$-very ample. If

$h^{1}(C,L\otimes B^{-1})\leq r(B)-p-1,$ (2)

then $L$ is $p$-very ample and $K_{p,1}(C,B;L)=0$.

This theorem is a significant improvement of Rathmann’s theorem (Rathmann1, ,
Theorem 1.2) saying if $h^{1}(C,L)=h^{1}(C,L\otimes B^{-1})=0$, then
$K_{p,1}(C,B;L)=0$. This implies that if $\deg L\geq 4g-3$, then the gonality
conjecture holds. By Riemann–Roch, the condition (2) is equivalent to

$\deg L\geq 2g+p+h^{0}(C,L\otimes B^{-1})-h^{1}(C,B).$

It is worth noting that Theorem 2.1 immediately implies Green’s
$(2g+1+p)$-theorem since $K_{p,2}(C;L)=K_{p,1}(C,L;L)$. Using Theorem 2.1 and
Green–Lazarsfeld’s nonvanishing theorem (Green1, , Appendix) (see also
Proposition 1), Niu–Park proved the following effective gonality theorem (NP,
, Corollary 5.3), which implies Theorem 1.2 saying that the gonality
conjecture holds when $\deg L\geq 2g+\operatorname{gon}(C)$.

###### Theorem 2.2

Assume that $g\geq 2$ and $\deg L\geq 2g+p$. Then $K_{p,1}(C,\omega_{C};L)\neq
0$ if and only if one of the following holds:

  1. $(a)$

$\omega_{C}$ is not $p$-very ample.

  2. $(b)$

$C\subseteq\mathbf{P}H^{0}(C,H)=\mathbf{P}^{2}$ is a plane curve of degree
$p+3\geq 4$ and $L=\omega_{C}\otimes H$. In this case, $\omega_{C}$ is
$p$-very ample and $\deg L=2g+p+1$.

  3. $(c)$

$C$ is arbitrary and $L=\omega_{C}(\xi)$ for some $\xi\in C_{p+2}$ with
$\dim|\xi|=1$. In this case, $\omega_{C}$ is $p$-very ample and $\deg L=2g+p$.

It was originally expected by Green–Lazarsfeld (GL1, , page 87) and explicitly
conjectured by Farkas–Kemeny (FK2, , page 3) that $K_{p,1}(C,\omega_{C};L)\neq
0$ if and only if $\omega_{C}$ is not $p$-very ample when $\deg L\geq 2g+p+1$.
However, Castryck Castryck  noticed the example in $(b)$ and suggested that
this conjecture should be modified. Theorem 2.2 completely resolves the
effective gonality problem.

To prove Theorem 2.1, we need to show that $H^{1}(C_{p+1},M_{p+1,B}\otimes
N_{p+1,L})=0$. By Serre duality, it is equivalent to

$H^{p}(C_{p+1},\wedge^{r(B)-p-1}M_{p+1,B}\otimes N_{p+1,\omega_{C}\otimes
B\otimes L^{-1}})=0.$

By (NP, , Lemma 2.3) (cf. (CKP, , Lemma 3.3)), we have

$\displaystyle
R^{i}\operatorname{pr}_{2,*}(N_{r(B)-p-1,B}\boxtimes\mathscr{O}_{C_{p+1}})(-D_{r(B)-p-1,p+1})=S^{i}H^{1}(B)\otimes\wedge^{r(B)-p-1-i}M_{p+2,B}$

for $i\geq 0$, where $\operatorname{pr}_{2}\colon C_{r(B)-p-1}\times
C_{p+1}\to C_{p+1}$ is the projection map. By the Leary spectral sequence for
$\operatorname{pr}_{2}$, the desired cohomology vanishing is implied by

$H^{i}(C_{j}\times C_{p+1},(N_{j,B}\boxtimes N_{p+1,\omega_{C}\otimes B\otimes
L^{-1}})(-D_{r(B)-p-1,p+1}))=0$

for $0\leq j\leq r(B)-p-1$ and $0\leq i\leq p$. Taking the dual and
considering the Leray spectral sequence for the projection
$\operatorname{pr}_{1}\colon C_{j}\times C_{p+1}\to C_{j}$, we see that the
desired cohomology vanishing is implied by

$\dim\mathcal{L}_{j}^{\ell}(\omega_{C}\otimes B\otimes L^{-1})\leq
p+j-i-\ell~{}~{}\text{ for $\ell\geq 1$}.$

But this can be easily deduced from the condition (2). Here

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

which is the _secant space_ of $j$-secant $(r(\omega_{C}\otimes B\otimes
L^{-1})-\ell)$-planes to $C$ in $\mathbf{P}H^{0}(C,\omega_{C}\otimes B\otimes
L^{-1})$ when $\ell\geq 1$ and $j\geq r(\omega_{C}\otimes B\otimes
L^{-1})-\ell+1$.

Niu–Park (NP, , Corollary 4.7) also showed a general effective nonvanishing
theorem for weight-one syzygies of algebraic curves (see also (Agostini1, ,
Proposition 3.5)).

###### Proposition 1

Suppose that $B$ is $p$-very ample but not $(p+1)$-very ample. If $\deg L\geq
2g+p+1$ then $K_{p+1,1}(C,B;L)\neq 0$ except for the following:

  1. $(a)$

$g=0$, $B=\mathscr{O}_{\mathbf{P}^{1}}(p)$,
$L=\mathscr{O}_{\mathbf{P}^{1}}(p+1)$ with $p\geq 1$. In this case,
$K_{p+1,1}(\mathbf{P}^{1},B;L)=0$.

  2. $(b)$

$g=1$, $B=\mathscr{O}_{C}$, $\deg L=3$. In this case, $K_{1,1}(C;L)=0$.

Finally, we discuss Green–Lazarsfeld’s secant conjecture (GL1, , Conjecture
3.4): if $L$ is very ample and

$\deg L\geq 2g+1+p-2h^{1}(C,L)-\operatorname{Cliff}(C),$

then $L$ satisfies $N_{p}$-property if and only if $L$ is $(p+1)$-very ample.
This was verified by Green–Lazarsfeld for $p=0$ (GL1, , Theorem 1). Here
$\operatorname{Cliff}(C)$ is the _Clifford index_ of $C$ defined by the
minimum of $\operatorname{Cliff}(A):=\deg A-2h^{0}(C,A)+2$ for a line bundle
$A$ on $C$ with $h^{0}(C,A)\geq 2$ and $h^{1}(C,A)\geq 2$. In the situation of
the conjecture, $h^{1}(C,L)\leq 1$. The conjecture for $h^{1}(C,L)=1$ is
equivalent to Green’s conjecture (Green1, , Conjecture 5.1) (see (AN, ,
Proposition 4.30)). When $h^{1}(C,L)=0$, the conjecture can be restated as if
$\deg L=2g+1+p-c$ with $0\leq c\leq\operatorname{Cliff}(C)$ and $L$ is
$(p+1)$-very ample, then $L$ satisfies $N_{p}$-property (indeed, the converse
direction is known (see (AN, , Theorem 4.37))). This was shown by
Farkas–Kemeny FK1  when $C$ and $L$ are general. The case of $c=0$ is Green’s
$(2g+1+p)$-theorem, the case of $c=1$ was verified by Green–Lazarsfeld (GL2, ,
Theorem 1.2), and the case of $c=2$ was recently settled by Agostini Agostini2
when $C$ is not bielliptic.

We briefly explain Agostini’s approach in Agostini2 . We want to prove that

$K_{p,2}(C;L)=H^{1}(C,\wedge^{p+1}M_{L}\otimes
L)=H^{0}(C,\wedge^{g-c}M_{L}\otimes\omega_{C})^{\vee}=0.$

As $H^{0}(C,\wedge^{g-c}M_{L}\otimes\omega_{C})=H^{0}(C_{g-c}\times
C,(N_{g-c,L}\boxtimes\omega_{C})(-D_{g-c,1}))$, it suffices to show that

$H^{0}(C_{g-2c}\times C_{c}\times
C,(\sigma_{g-2c,c}\times\operatorname{id}_{C})^{*}(N_{g-c,L}\boxtimes\omega_{C})(-D_{g-c,1}))=0.$

This can be deduced from

$H^{0}(C_{c}\times
C,N_{c,L(-\xi)}\boxtimes\omega_{C}(-\xi))(-D_{c,1}))=0~{}~{}\text{ for general
$\xi\in C_{g-2c}$}.$

When $c=0$, this is $H^{0}(C,\omega_{C}(-\xi))=0$ for general $\xi\in C_{g}$
but it is clear. Assume that $c\geq 1$ and $\dim W_{g-c}^{1}(C)=g-2c-2$.
Agostini (Agostini2, , Lemma 4.2) observed $\omega_{C}(-\xi)$ is $(c-1)$-very
ample under this assumption. The above vanishing is equivalent to

$H^{0}(C_{c},M_{c,\omega_{C}(-\xi)}\otimes N_{c,L(-\xi)})=0.$

As $M_{c,\omega_{C}(-\xi)}\otimes
N_{c,\omega_{C}(-\xi)}=\wedge^{c-1}M_{c,\omega_{C}(-\xi)}^{\vee}$, we have a
long exact sequence

$\begin{array}[]{l}\cdots\longrightarrow\wedge^{c-3}H^{0}(C,\omega_{C}(-\xi))\otimes
S^{2}E_{c,\omega_{C}(-\xi)}^{\vee}\longrightarrow\wedge^{c-2}H^{0}(C,\omega_{C}(-\xi))\otimes
E_{c,\omega_{C}(-\xi)}^{\vee}\\\
\longrightarrow\wedge^{c-1}H^{0}(C,\omega_{C}(-\xi))^{\vee}\otimes\mathscr{O}_{C}\longrightarrow
M_{c,\omega_{C}(-\xi)}\otimes N_{c,\omega_{C}(-\xi)}\longrightarrow
0.\end{array}$

Thus we reduce the problem to

$H^{i}(C_{c},S^{i}E_{c,\omega_{C}(-\xi)}^{\vee}\otimes
S_{c,L\otimes\omega_{C}^{-1}})=0~{}~{}\text{ for $0\leq i\leq c-1$}.$ (3)

When $c=1$, this is $H^{0}(C,L\otimes\omega_{C}^{-1})=0$, which holds since
$\deg L=2g+p$ but $L$ is $(p+1)$-very ample. When $c=2$, Agostini proved this
vanishing holds if and only if $g\neq p+4$ in (Agostini2, , Lemma 4.5). In the
exceptional case of $g=p+4$, some additional arguments are required to settle
Green–Lazarsfeld’s secant conjecture (see (Agostini2, , Section 5)). Recently,
Farkas Farkas2  has given a different account of the secant conjecture for the
case of $g=p+2+c$. It would be very nice if one can verify the vanishing (3)
for $c\geq 3$ when $g\neq p+2+c$.

##  3 Syzygies of secant varieties of smooth projective curves

In this section, we assume that $\operatorname{char}(\mathbf{k})=0$. Let $C$
be a smooth projective curve of genus $g\geq 0$, and $L$ be a line bundle on
$C$ with $\deg L\geq 2g+2k+1$ for an integer $k\geq 0$. The tautological
bundle $E_{k+1,L}$ on $C_{k+1}$ is globally generated with
$H^{0}(C_{k+1},E_{k+1,L})=H^{0}(C,L)$, so
$|\mathscr{O}_{\mathbf{P}(E_{k+1,L})}(1)|$ induces a morphism $\beta_{k}$ from
$B^{k}(L):=\mathbf{P}(E_{k+1,L})$ to $\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$. If
$\pi_{k}\colon B^{k}(L)\to C_{k+1}$ is the canonical projection, then
$\beta_{k}(\pi_{k}^{-1}(\xi))$ is the $k$-plane in $\mathbf{P}^{r}$ such that
$\beta_{k}(\pi_{k}^{-1}(\xi))\cap C=\xi$ for any $\xi\in C_{k+1}$. Thus
$\beta_{k}(B^{k}(L))$ is the $k$-th secant variety
$\Sigma_{k}=\Sigma_{k}(C,L)$ of $C\subseteq\mathbf{P}^{r}$. We may think that
$\beta_{k}\colon B^{k}(L)\to\Sigma_{k}$ is a resolution of singularities. Let
$Z_{k-1}:=\beta_{k}^{-1}(\Sigma_{k-1})$. Then
$Z_{k-1}=(k+1)H_{k}-\pi_{k}^{*}A_{k+1,L}$, where
$\mathscr{O}_{B^{k}(L)}(H_{k})=\mathscr{O}_{\mathbf{P}(E_{k+1,L})}(1)$ and
$A_{k+1,L}:=S_{k+1,L}(-2\delta_{k+1})$ (see (ENP1, , Proposition 3.15)).
Notice that $\omega_{B^{k}(L)}(Z_{k-1})=\pi_{k}^{*}S_{k+1,\omega_{C}}$.
Ein–Niu–Park ENP1  established the following theorem (cf. Theorem 1.3).

###### Theorem 3.1

Let $R:=R(\Sigma_{k};\mathscr{O}_{\Sigma_{k}}(1))$. We have the following:

  1. $(a)$

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

  2. $(b)$

$R^{i}\beta_{k,*}\omega_{B^{k}(L)}(Z_{k-1})=\begin{cases}\omega_{\Sigma_{k}}&\text{for
$i=0$}\\\ \omega_{\Sigma_{k-1}}&\text{for $i=1$}\\\ 0&\text{for $i\geq
2$}.\end{cases}$

  3. $(c)$

$\Sigma_{k}$ has normal Du Bois singularities.

  4. $(d)$

$\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay and
$H^{2k+1}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(\ell))=0$ for $\ell>0$. In
particular, $\operatorname{reg}(R)=2k+2$ unless $g=0$, in which case
$\operatorname{reg}(R)=k+1$.

  5. $(e)$

If $\deg L\geq 2g+2k+1+p$, then $\mathscr{O}_{\Sigma_{k}}(1)$ satisfies
$N_{k+2,p}$-property.

To prove the theorem, we proceed by the induction on $k$. As Ullery did in
Ullery , we apply the formal function theorem to prove the assertions
$(a),(b),(c)$, which are local questions. Here we focus on the normality of
$\Sigma_{k}$. We remark that the idea of proving the normality of a variety
using the formal function theorem was already presented in Ein’s old paper Ein
. Note that $\Sigma_{k}$ is normal if and only if the map
$\mathscr{O}_{\mathbf{P}^{r}}\to\beta_{k,*}\mathscr{O}_{B^{k}(L)}$ is
surjective at any point $x\in\Sigma_{m}\setminus\Sigma_{m-1}$ for $0\leq m\leq
k-1$. Notice that the unique $(m+1)$-secant plane containing $x\in\Sigma_{m}$
determines $\xi\in C_{m+1}$ uniquely. By the formal function theorem, we need
to confirm that the map

$\varprojlim\mathscr{O}_{\mathbf{P}^{r}}/\mathfrak{m}_{x}^{\ell}\longrightarrow\varprojlim
H^{0}(\mathscr{O}_{B^{k}(L)}/\mathscr{I}_{F_{x}}^{\ell})$

is surjective, where $\mathfrak{m}_{x}$ is the ideal sheaf of $x$ in
$\mathbf{P}^{r}$ and $\mathscr{I}_{F_{x}}$ is the ideal sheaf of
$F_{x}:=\beta_{k}^{-1}(x)=C_{k-m}$ in $B^{k}(L)$. To this end, consider the
commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{m}^{\ell}/\mathfrak{m}_{x}^{\ell+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\ell}}$$\textstyle{\mathscr{O}_{\mathbf{P}^{r}}/\mathfrak{m}_{x}^{\ell+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{\mathbf{P}^{r}}/\mathfrak{m}_{x}^{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathscr{I}_{F_{x}}^{\ell}/\mathscr{I}_{F_{x}}^{\ell+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathscr{O}_{B^{k}(L)}/(\mathscr{I}_{F_{x}}^{\ell+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathscr{O}_{B^{k}(L)}/\mathscr{I}_{F_{x}}^{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

and proceeding by induction on $\ell$, we reduce the problem to check that
$\alpha_{\ell}$ is surjective for all $\ell\geq 1$. Note that
$\mathfrak{m}_{x}^{\ell}/\mathfrak{m}_{x}^{\ell+1}=S^{\ell}T_{x}^{\vee}\mathbf{P}^{r}$
and
$\mathscr{I}_{F_{x}}^{\ell}/\mathscr{I}_{F_{x}}^{\ell+1}=S^{\ell}N_{F_{x}|B^{k}(L)}^{\vee}$
As $\alpha_{1}$ is an isomorphism ((ENP1, , Proposition 3.13)) and
$\alpha_{\ell}=\theta_{\ell}\circ S^{\ell}\alpha_{1}$, where

$\theta_{\ell}\colon S^{\ell}H^{0}(N_{F_{x}|B^{k}(L)}^{\vee})\longrightarrow
H^{0}(S^{\ell}N_{F_{x}|B^{k}(L)}^{\vee}),$

it is sufficient to show that $\theta_{\ell}$ is surjective for $\ell\geq 1$.
Based on Bertram’s work Bertram , it was shown in (ENP1, , Proposition 3.13)
that

$N_{F_{x}|B^{k}(L)}^{\vee}=\mathscr{O}_{F_{x}}^{\oplus 2m+1}\oplus
E_{k-m,L(-2\xi)}.$

Then the surjectivity of $\theta_{\ell}$ easily follows from the projective
normality of lower secant varieties
$\Sigma_{k-m-1}(C,L(-2\xi))\subseteq\mathbf{P}H^{0}(C,L(-2\xi))$ (notice that
$\deg L(-2\xi)\geq 2g+2(k-m-1)+1$). Therefore, $\Sigma_{k}$ has normal
singularities. The other assertions in $(a),(b),(c)$ can be similarly proven
except two cases. First, after establishing all the other assertions, we can
prove $\beta_{k,*}\omega_{B^{k}(L)}(Z_{k-1})=\omega_{\Sigma_{k}}$ in $(b)$
using (KSS, , Theorem 1.1) (see (CKP, , Proposition 3.6)). Second, we apply a
theorem of Kollár–Kovács (Kollar, , Corollary 6.28) and $(a)$ to show that
$\Sigma_{k}$ has Du Bois singularities as claimed in $(c)$. For this purpose,
we need to check that $Z_{k-1}$ has Du Bois singularities. In fact, one can
show that $(B^{k}(L),Z_{k-1})$ is a log canonical pair ((ENP1, , Proposition
3.15)) based on Bertram’s blow-up construction Bertram  (see also (ENP1, ,
Subsection 3.3)). This implies that $Z_{k-1}$ has Du Bois singularities by
(Kollar, , Corollary 6.32).

Now, we prove the remaining assertions $(d),(e)$ assuming $(a),(b),(c)$. By
Danila’s theorem (see ENP2 ),
$H^{0}(\mathbf{P}^{r},\mathscr{O}_{\mathbf{P}^{r}}(\ell))=H^{0}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(\ell))$
for $1\leq\ell\leq k+1$. Then one can easily see that

$H^{i}(\Sigma_{k},\wedge^{j}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{O}_{\Sigma_{k}}(k+1))=0~{}~{}\text{
for $i\geq 1$, $j\geq 0$, and $i\geq j-p$}$

implies that $(i)$ $H^{i}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(\ell))=0$ for
$i>0,\ell>0$, $(ii)$ $\Sigma_{k}\subseteq\mathbf{P}^{r}$ is projectively
normal, and $(iii)$ $\mathscr{O}_{\Sigma_{k}}(1)$ satisfies
$N_{k+2,p}$-property when $\deg L\geq 2g+2k+1+p$. Considering the short exact
sequence

$0\longrightarrow\wedge^{j}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}\longrightarrow\wedge^{j}M_{\mathscr{O}_{\Sigma_{k}}(1)}\longrightarrow\wedge^{j}M_{\mathscr{O}_{\Sigma_{k-1}}(1)}\longrightarrow
0$

and using the induction hypothesis, we transfer the problem to

$H^{i}(\Sigma_{k},\wedge^{j}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k}}(k+1))=0~{}~{}\text{
for $i\geq 1$, $j\geq 0$, and $i\geq j-p$}.$

By the Du Bois-type condition $(a)$, it is equivalent to

$H^{i}(B^{k}(L),\wedge^{j}M_{H_{k}}\otimes\pi_{k}^{*}A_{k+1,L})=0~{}~{}\text{
for $i\geq 1$, $j\geq 0$, and $i\geq j-p$},$

which can be identified with

$H^{i}(C_{k+1},\wedge^{j}M_{k+1,L}\otimes A_{k+1,L})=0~{}~{}\text{ for $i\geq
1$, $j\geq 0$, and $i\geq j-p$},$

since $\pi_{k,*}\wedge^{j}M_{H_{k}}=\wedge^{j}M_{k+1,L}$ and
$R^{i}\pi_{k,*}\wedge^{j}M_{H_{k}}=0$ for $i>0$ ((ENP1, , Lemma 5.1)). Let
$\Delta_{k+1}:=q_{k+1}^{*}\delta_{k+1}$ be the union of all pairwise diagonals
in $C^{k+1}$. As $\mathscr{O}_{C_{k+1}}(-\delta_{k+1})$ is a direct summand of
$q_{k+1,*}\mathscr{O}_{C^{k+1}}$ ((ENP1, , Lemma 3.5)), the desired vanishing
follows from

$H^{i}(C^{k+1},\wedge^{j}q_{k+1}^{*}M_{k+1,L}\otimes L^{\boxtimes
k+1}(-\Delta_{k+1}))=0~{}~{}\text{ for $i\geq 1$, $j\geq 0$, and $i\geq
j-p$}.$

Using Rathmann’s technique Rathmann1 , we can prove this vanishing (see (ENP1,
, Theorem 4.1)). It only remains to show that
$\Sigma_{k}\subseteq\mathbf{P}^{r}$ is arithmetically Cohen–Macaulay. It is
enough to verify that

$H^{i}(\Sigma_{k},\mathscr{O}_{\Sigma_{k}}(-\ell))=0~{}~{}\text{ for $1\leq
i\leq 2k$ and $\ell\geq 0$}.$ (4)

The proof of (4) in ENP1  is incomplete as was pointed out by Doyoung Choi (we
would like to thank him). By (Kollar, , Theorem 10.42), it was reduced to the
cases of $\ell=0,1$, and these cases are verified in the proof of (ENP1, ,
Theorem 5.8). Here (Kollar, , Theorem 10.42) states that if $X$ is a
projective pure dimensional scheme with Du Bois singularities and $A$ is an
ample line bundle on $X$, then $h^{i}(X,A^{-\ell})=h^{i}(X,A^{-1})$ for
$i<\dim X$ and $\ell\geq 1$. This statement contains a crucial typo:
$h^{i}(X,A^{-\ell})=h^{i}(X,A^{-1})$ should be replaced by
$h^{i}(X,A^{-\ell})\geq h^{i}(X,A^{-1})$. Thus (4) was only proven for
$\ell=0,1$. However, using the fact that

$H^{i}(C_{k+1},S_{k+1,\omega_{C}})=\begin{cases}S^{k+1}H^{0}(C,\omega_{C})&\text{for
$i=0$}\\\ S^{k}H^{0}(C,\omega_{C})&\text{for $i=1$}\\\ 0&\text{for $i\geq
2$}\end{cases}$

and the assertion $(b)$ (the former fact is also used to prove $(b)$), one can
establish (4) for $\ell=0$ and $\ell\gg 0$ by similar arguments in the proof
of (ENP2, , Theorem 5.8). Then the corrected version of (Kollar, , Theorem
10.42) implies that (4) holds for all $\ell\geq 0$.

As explained in the introduction, Theorem 3.1 determines the first $p+1$
columns of the Betti table of $R(\Sigma_{k};\mathscr{O}_{\Sigma_{k}}(1))$ when
$\deg L=2g+2k+1+p$. Assuming $\deg L\gg 0$, Choe–Kwak–Park CKP  determined the
remaining part of the Betti table using the _gonality sequence_
$(\gamma^{0}(C)+0,\gamma^{1}(C)+1,\gamma^{2}(C)+2,\ldots)$ of $C$, where

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

Note that $\gamma^{1}(C)+1=\operatorname{gon}(C)$. Let
$e:=\operatorname{codim}\Sigma_{k}=r-2k-1$. Recall Theorem 1.4: for each
$k+1\leq q\leq 2k+2$, if $e-g+1\leq p\leq e$, then

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

We give a sketch of the proof of this theorem. Using the Du Bois-type
condition $(a)$ and proceeding by induction on $q-k-1$, we reduce the
vanishing part to

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

where $p^{*}:=e-p$ and $q^{*}:=2k+2-q$ are introduced by duality theorem
((Green1, , Theorem 2.c.1)). By (CKP, , Lemma 3.3), it is equivalent to

$H^{q^{*}+1}(C_{p^{*}+q^{*}}\times C_{k+2},(N_{p^{*}+q^{*},L}\boxtimes
S_{k+2,\omega_{C}})(-D_{p^{*}+q^{*},k+2}))=0.$

Let $\operatorname{pr}_{1}\colon C_{p^{*}+q^{*}}\times C_{k+2}\to
C_{p^{*}+q^{*}}$ be the projection map. By the Leray spectral sequence for
$\operatorname{pr}_{1}$, it is enough to check that

$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 follows from Fujita–Serre vanishing since
$N_{p^{*}+q^{*},L}$ is sufficiently positive. When $i=0$, we consider 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 $\xi\in C_{p^{*}+q^{*}}$,
which is

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

Now, the gonality sequence condition $p\geq e-\gamma^{2k+2-q}(C)+1$ can be
restated as $\gamma^{q^{*}}(C)\geq p^{*}+1$, which gives
$h^{1}(C,\omega_{C}(-\xi))\leq q^{*}$ in view of the positivity of
$\omega_{C}$. Thus we obtain

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

which implies the desired cohomology vanishing for $i=0$.

For the nonvanishing part, it suffices to show that the map

$\begin{array}[]{l}H^{q-1}(\wedge^{p+q-1}M_{\mathscr{O}_{\Sigma_{k}}(1)}\otimes\mathscr{I}_{\Sigma_{k-1}|\Sigma_{k}}(1))\longrightarrow
H^{q}(\wedge^{p+q-1}M_{\mathscr{O}_{\Sigma_{k+1}}(1)}\otimes\mathscr{I}_{\Sigma_{k}|\Sigma_{k+1}}(1))\end{array}$

is nonzero. We can reduce to showing that the map

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

is nonzero. By the gonality sequence condition $p\leq e-\gamma^{2k+2-q}(C)$,
which can be rephrased as $\gamma^{q^{*}}(C)\leq p^{*}$, 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 above map over $\xi\in C_{p^{*}+q^{*}}$ is
$\operatorname{id}_{S^{k+1-q^{*}}H^{0}(C,\omega_{C}(-\xi))}\otimes\delta$,
where $\delta$ is the Koszul-like map

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

Since $\delta$ is clearly nonzero, it follows that the above map is nonzero.

In view of the effective gonality theorem (Theorem 1.2), it is a natural
problem to establish an effective lower bound for $\deg L$ such that the
conclusion of Theorem 1.4 holds. Some partial answers and related results are
discussed in (CKP, , Section 5).

##  4 Syzygies of tangent developable surfaces and generic Green’s conjecture

For a moment, we assume that $\operatorname{char}(\mathbf{k})=0$. Let $C$ be a
smooth projective curve of genus $g\geq 3$. If $\operatorname{Cliff}(C)\geq 1$
(i.e., $C$ is nonhyperelliptic), then $\omega_{C}$ is very ample and M.
Noether’s theorem says that
$C\subseteq\mathbf{P}H^{0}(C,\omega_{C})=\mathbf{P}^{g-1}$ is projectively
normal. Petri’s theorem states that if $\operatorname{Cliff}(C)\geq 2$, then
$I_{C|\mathbf{P}^{g-1}}$ is generated by quadrics. In the 1980s, Green
(Green1, , Conjecture 5.1) formulated a very famous conjecture predicting that
$\omega_{C}$ satisfies $N_{(\operatorname{Cliff}(C)-1)}$-property. By the
duality theorem ((Green1, , Theorem 2.c.1)), this completely determines the
shape of the Betti table of $R(C;\omega_{C})$. This conjecture is still widely
open. However, in the 2000s, Voisin Voisin1 ; Voisin2  resolved Green’s
conjecture for general curves. For this purpose, it suffices to exhibit one
curve of each genus for which the assertion holds. If
$S\subseteq\mathbf{P}^{g}$ is a K3 surface of degree $2g-2$, then its general
hyperplane section is a canonical curve $C\subseteq\mathbf{P}^{g-1}$ and
$K_{p,2}(S;\mathscr{O}_{S}(1))=K_{p,2}(C;\omega_{C})$. Voisin used the
geometry of the Hilbert schemes of K3 surfaces for generic Green’s conjecture.
Recently, Kemeny Kemeny  gave a simpler proof of Voisin’s theorem for even
genus case and a streamlined version of her arguments for odd genus case.

Prior to Voisin’s work, in 1980s, O’Grady and Buchweitz–Schreyer independently
observed that one could use the tangent developable surface
$T\subseteq\mathbf{P}^{g}$ of a rational normal curve of degree $g$ to solve
Green’s conjecture for general curves of genus $g$. Note that
$T\subseteq\mathbf{P}^{g}$ is arithmetically Cohen–Macaulay and it is a
degeneration of a K3 surface. A general hyperplane section of $T$ in
$\mathbf{P}^{g}$ is a canonically embedded $g$-cuspidal rational curve
$\overline{C}\subseteq\mathbf{P}^{g-1}$ of degree $2g-2$, which is a
degeneration of a general canonical curve $C\subseteq\mathbf{P}^{g-1}$ with
$\operatorname{Cliff}(C)=\lfloor(g-1)/2\rfloor$. By the upper semicontinuity
of graded Betti numbers,
$K_{p,2}(\overline{C};\mathscr{O}_{\overline{C}}(1))=K_{p,2}(T;\mathscr{O}_{T}(1))=0$
implies $K_{p,2}(C;\omega_{C})=0$. For generic Green’s conjecture, it suffices
to prove that

$K_{p,2}(T;\mathscr{O}_{T}(1))=0~{}~{}\text{ for $0\leq
p\leq\lfloor(g-3)/2\rfloor$}.$

This was finally established by Aprodu–Farkas–Papadima–Raicu–Weyman AFPRW
when $\operatorname{char}(\mathbf{k})=0$ or $\geq(g+2)/2$ (see Theorem 1.5).
This important result gives not only an alternative proof of generic Green’s
conjecture but also an extension to positive characteristic. This circle of
ideas is surveyed in EL4 .

We explain the proof of Theorem 1.5 given in AFPRW . We have a short exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu_{*}\mathscr{O}_{\widetilde{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{\mathbf{P}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\nu\colon\widetilde{T}=\mathbf{P}^{1}\times\mathbf{P}^{1}\to T$ is a
resolution of singularities. It is easy to see that

$K_{p,2}(T;\mathscr{O}_{T}(1))=\operatorname{coker}\big{(}K_{p,1}(T,\nu_{*}\mathscr{O}_{\widetilde{T}};\mathscr{O}_{T}(1))\xrightarrow{~{}\gamma~{}}K_{p,1}(\mathbf{P}^{1},\omega_{\mathbf{P}^{1}};\mathscr{O}_{\mathbf{P}^{1}}(g))\big{)}.$

Let $U:=H^{0}(\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{1}}(1))$, $V:=D^{p+2}U$,
$W:=D^{2p+2}U$, and $q:=g-p-3$. Here $D^{m}U$ is the $m$-th divided power of
$U$ (see e.g., (AFPRW, , Section 3), (Park2, , Section 2)). The authors of
AFPRW  devoted considerable effort to show that $\gamma$ arises as the
composition

$S^{q}V\otimes
W\xrightarrow{\operatorname{id}_{S^{q}V}\otimes\Delta}S^{q}V\otimes\wedge^{2}V\xrightarrow{~{}\delta~{}}\ker(S^{q+1}V\otimes
V\xrightarrow{~{}\delta~{}}S^{q+2}V),$ (5)

where $\Delta$ is the co-Wahl map and $\delta$ is the Koszul differential. To
achieve this, they established an explicit characteristic-free Hermite
reciprocity for $\mathfrak{sl}_{2}$-representations, and they carried out
complicated algebraic computations. Now, $K_{p,2}(T,\mathscr{O}_{T}(1))$ is
the degree $q$ piece of the _Koszul module_ $W_{q}(V,W)$, which is defined to
be the homology of a complex

$S^{q}V\otimes W\xrightarrow{~{}\gamma~{}}S^{q+1}V\otimes
V\xrightarrow{~{}\delta~{}}S^{q+2}V.$

Then it is enough to prove that

$W_{q}(V,W)=0~{}~{}\text{ for $q\geq p$}.$ (6)

The vanishing result (6) was established in the previous work of the authors
of AFPRW  in characteristic zero and extended to positive characteristics in
AFPRW .

Park Park2  gave a geometric approach to proving Theorem 1.5 utilizing the
first secant variety $\Sigma:=\Sigma_{1}(\mathbf{P}^{1},L)$ of a rational
normal curve $C$ of degree $g$, where $L:=\mathscr{O}_{\mathbf{P}^{1}}(g)$. We
remark that Theorem 3.1 holds in positive characteristics when
$C=\mathbf{P}^{1}$. Note that $Q:=\sigma(D_{1,1})$ is a smooth conic in
$\mathbf{P}^{2}$, where
$\sigma:=\sigma_{1,1}\colon\mathbf{P}^{1}\times\mathbf{P}^{1}\to\mathbf{P}^{2}$
is the universal family over $(\mathbf{P}^{1})_{2}=\mathbf{P}^{2}$. Let
$\widetilde{T}:=\mathbf{P}(E|_{Q})$, where $E:=E_{2,L}$. Then $\widetilde{T}$
is linearly equivalent to $K_{B}+Z$, where $B:=B^{1}(L)$ and
$Z:=Z_{0}=\mathbf{P}^{1}\times\mathbf{P}^{1}$, and
$\widetilde{T}=\mathbf{P}^{1}\times\mathbf{P}^{1}$ when
$\operatorname{char}(\mathbf{k})=0$ or $\geq(g+2)/2$. Then
$\beta_{1}(\widetilde{T})=T$ is the tangent developable surface of
$v_{g}(\mathbf{P}^{1})\subseteq\mathbf{P}H^{0}(\mathbf{P}^{1},L)=\mathbf{P}^{g}$,
and $\beta_{1}|_{\widetilde{T}}$ coincides with the map $\nu$. One can easily
check that $T$ is a Weil divisor on $\Sigma$ linearly equivalent to
$-K_{\Sigma}$. Let $H:=H_{1}$ be the tautological divisor on
$B=\mathbf{P}(E)$. We can realize (5) as maps induced in cohomology of vector
bundles on $B$:

$H^{1}(\widetilde{T},\wedge^{p+2}M_{H}|_{\widetilde{T}})\xrightarrow{~{}\alpha~{}}H^{2}(B,\wedge^{p+2}M_{H}\otimes\omega_{B}(Z))\xrightarrow{~{}\delta~{}}H^{2}(Z,\omega_{\mathbf{P}^{1}}\boxtimes\wedge^{p+2}M_{L}\otimes\omega_{\mathbf{P}^{1}}).$

It is easy to check that $\alpha=\operatorname{id}_{S^{q}V}\otimes\Delta$,
where $\Delta$ is the dual of the restriction map
$H^{0}(\mathbf{P}^{2},\mathscr{O}_{\mathbf{P}^{2}}(p+1))\rightarrow
H^{0}(Q,\mathscr{O}_{Q}(p+1))$. Put $M_{E}:=M_{2,L}$. The map $\delta$ is
naturally factored as

$\textstyle{H^{2}(\wedge^{p+2}M_{H}\otimes\omega_{B}(Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{S^{q}V}\otimes\iota}$$\textstyle{H^{2}(\sigma^{*}\wedge^{p+2}M_{E}\otimes(\omega_{\mathbf{P}^{1}}\boxtimes\omega_{\mathbf{P}^{1}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m\otimes\operatorname{id}_{V}}$$\textstyle{H^{2}(\omega_{\mathbf{P}^{1}}\boxtimes\wedge^{p+2}M_{L}\otimes\omega_{\mathbf{P}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{q}V\otimes\wedge^{2}V}$$\textstyle{S^{q}V\otimes
V\otimes V}$$\textstyle{S^{q+1}V\otimes V,}$

where $\iota$ is the canonical injection and $m$ is the multiplication map
identified with

$H^{1}(\mathbf{P}^{p+2}\times\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{p+2}}(q)\boxtimes\omega_{\mathbf{P}^{1}}(-p-2))\longrightarrow
H^{1}(\mathbf{P}^{p+2}\times\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{p+2}}(q+1)\boxtimes\omega_{\mathbf{P}^{1}}).$

This gives a quick verification of the descriptions of the maps in (5). Next,
regarding $V=H^{0}(\mathbf{P}^{p+2},\mathscr{O}_{\mathbf{P}^{p+2}}(1))$, we
can give a direct proof of (6) as follows. We put
$M_{V}:=M_{\mathscr{O}_{\mathbf{P}^{p+2}}(1)}$. Then (5) can be identified
with

$W\otimes
H^{0}(\mathbf{P}^{p+2},\mathscr{O}_{\mathbf{P}^{p+2}}(q))\longrightarrow\wedge^{2}V\otimes
H^{0}(\mathbf{P}^{p+2},\mathscr{O}_{\mathbf{P}^{p+2}}(q))\longrightarrow
H^{0}(\mathbf{P}^{p+2},M_{V}(q+1)).$

We can show that $W\otimes\mathscr{O}_{\mathbf{P}^{p+2}}\to M_{V}(1)$ is
surjective and its kernel is $(p+1)$-regular. This implies (6). For this last
part, we use the characteristic assumption $\operatorname{char}(\mathbf{k})=0$
or $\geq(g+2)/2$. It is worth noting that the characteristic assumption cannot
be improved. Indeed, if $2\leq\operatorname{char}(\mathbf{k})\leq(g+1)/2$,
then $K_{\lfloor(g-3)/2\rfloor,2}(T,\mathscr{O}_{T}(1))\neq 0$ (see (AFPRW, ,
Remark 5.17)).

Concerning the characteristic of the base field, Raicu–Sam (RS, , Theorem 1.5)
obtained a sharp result: Green’s conjecture holds for general curves of genus
$g$ when $\operatorname{char}(\mathbf{k})=0$ or
$\operatorname{char}(\mathbf{k})\geq\lfloor(g-1)/2\rfloor$. This confirms a
conjecture of Eisenbud–Schreyer. It has long been known that generic Green’s
conjecture would follow from the canonical ribbon conjecture. A _canonical
ribbon_ is a certain double structure on a rational normal curve, and it can
be realized as a hyperplane section of a _K3 carpet_
$X=X(a,b)\subseteq\mathbf{P}^{a+b+1}$ for integers $b\geq a\geq 1$, which is a
unique double structure on a rational normal surface scroll
$S(a,b)=\mathbf{P}(\mathscr{O}_{\mathbf{P}^{1}}(a)\oplus\mathscr{O}_{\mathbf{P}^{1}}(b))\subseteq\mathbf{P}^{a+b+1}$
such that $\omega_{X}=\mathscr{O}_{X}$ and $h^{1}(X,\mathscr{O}_{X})=0$. Then
$X$ is a degeneration of a K3 surface of degree $2(a+b)$, and its hyperplane
section (a canonical ribbon) is a degeneration of a general canonical curve
$C$ of genus $a+b+1$ with $\operatorname{Cliff}(C)=a$. By extending algebraic
arguments of AFPRW , Raicu–Sam (RS, , Theorem 1.1) proved that
$\mathscr{O}_{X}(1)$ satisfies $N_{(a-1)}$-property when
$\operatorname{char}(\mathbf{k})=0$ or $\operatorname{char}(\mathbf{k})\geq
a$. For Eisenbud–Schreyer’s conjecture, we only need to consider the case
$a=\lfloor(g-1)/2\rfloor$ and $b=\lfloor g/2\rfloor$. In this case, we take a
general double line $\ell\in|Q|=|\mathscr{O}_{\mathbf{P}^{2}}(2)|$. Then
$\beta_{1}(\mathbf{P}(E|_{\ell}))=X(\lfloor(g-1)/2\rfloor,\lfloor
g/2\rfloor)\subseteq\mathbf{P}^{g}$ is a K3 carpet. Along the same way as the
geometric proof of Theorem 1.5, we can obtain a simple geometric proof of
Raicu–Sam’s result (RS, , Theorem 1.1).

######  Theorem 4.1 ($\operatorname{char}(\mathbf{k})=0$ or
$\geq\lfloor(g-1)/2\rfloor$ and $\operatorname{char}(\mathbf{k})\neq 2$)

Consider a K3 carpet $X=X(\lfloor(g-1)/2\rfloor,\lfloor
g/2\rfloor)\subseteq\mathbf{P}^{g}$ with $g\geq 3$. Then $\mathscr{O}_{X}(1)$
satisfies $N_{\lfloor(g-3)/2\rfloor}$-property.

This gives a very quick proof of generic Green’s conjecture and
Eisenbud–Schreyer’s conjecture. So far, five different proofs of generic
Green’s conjecture are known: 1 Voisin’s first proof using K3 surfaces Voisin1
; Voisin2 , 2 Kemeny’s simplification of Voisin’s proof Kemeny  (Rathmann
Rathmann2  provided a different account of Kemeny’s proof for even genus
case), 3 Aprodu–Farkas–Papadima–Raicu–Weyman’s algebraic proof using tangent
developable surfaces of rational normal curves AFPRW , 4 Raicu–Sam’s algebraic
proof using K3 carpets RS , 5 Park’s geometric simplification of the main
results of AFPRW  and RS . We remark that $\leavevmode\hbox to8.69pt{\vbox
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positive characteristics. All proofs use K3 or K3-like surfaces. It would be
exceedingly interesting to find a direct proof through symmetric products of
curves. This may lead to the resolution of Green’s conjecture for all curves.

It is quite natural to study syzygies of tangent developable surfaces of
arbitrary smooth projective curves of genus $g\geq 1$. As a first step, Park
(Park2, , Theorem 1.3) showed the arithmetic normality of tangent developable
surfaces.

######  Theorem 4.2 ($\operatorname{char}(\mathbf{k})=0$)

Let $C$ be a smooth projective curve of genus $g\geq 1$, and $L$ be a line
bundle on $C$ with $\deg L\geq 4g+3$. Consider the tangent developable surface
$T$ of $C$ embedded in $\mathbf{P}H^{0}(C,L)=\mathbf{P}^{r}$. Then
$T\subseteq\mathbf{P}^{r}$ is arithmetically normal but not arithmetically
Cohen–Macaulay, and $H^{i}(T,\mathscr{O}_{T}(m))=0$ for $i>0,m>0$ but
$H^{1}(T,\mathscr{O}_{T})\neq 0,H^{2}(T,\mathscr{O}_{T})\neq 0$.

To prove the theorem, regarding $T$ as a Weil divisor on the first secant
variety of $C$ in $\mathbf{P}^{r}$, we transfer the problem to cohomology
vanishing of vector bundles on the second symmetric (or ordinary) product of
$C$. The most difficult part is to check the $2$-normality of
$T\subseteq\mathbf{P}^{r}$, which is turned out to be equivalent to the
cohomology vanishing $H^{1}(C\times C,(L\boxtimes L)(-3D_{1,1}))=0$. This was
previously established by Bertram–Ein–Lazarsfeld BEL  when $\deg L\geq 4g+3$.
The degree condition in Theorem 4.2 is optimal. If $\deg L=4g+2$, then
$T\subseteq\mathbf{P}^{r}$ is arithmetically normal if and only if $C$ is
neither elliptic nor hyperelliptic (see (Park2, , Remark 4.4)). Some relevant
conjectures and examples on syzygies of tangent developable surfaces are
presented in (Park2, , Section 4). For example, it was conjectured that if
$g=1$, then $K_{p,1}(T;\mathscr{O}_{T}(1))=0$ for $p\geq\lfloor(\deg
L-3)/2\rfloor$ and $K_{p,2}(T;\mathscr{O}_{T}(1))=0$ for $0\leq
p\leq\lfloor(\deg L-7)/3\rfloor$. In this case, $T$ is a degeneration of an
abelian surface, so this conjecture may have some applications to syzygies of
general abelian surfaces.

##  5 Asymptotic syzygies of algebraic varieties

In this section, we assume that $\operatorname{char}(\mathbf{k})\geq 0$. Let
$X$ be an $n$-dimensional smooth projective variety, $B$ be a line bundle on
$X$, and $L_{d}:=\mathscr{O}_{X}(dA+P)$ for an integer $d\gg 0$, where $A$ is
an ample divisor and $P$ is an arbitrary divisor on $X$. Put
$R_{d}:=R(X,B;L_{d})$ and $r_{d}:=h^{0}(X,L_{d})-1$. Assume that $d\gg 0$.
Then $r_{d}=\Theta(d^{n})$, and $H^{i}(X,B\otimes L^{m})=0$ for $i>0$ and
$m>0$. Thus $\operatorname{reg}(R_{d})\leq n+1$, so $K_{p,q}(X,B;L)=0$ for
$q\geq n+2$. We want to determine vanishing and nonvanishing of
$K_{p,q}(X,B;L_{d})$ for $0\leq q\leq n+1$. By (1) in the introduction, we
only need to focus on the cases of $1\leq q\leq n$. Motivated by the
nonvanishing results of EGHP ; OP , Ein–Lazarsfeld (EL2, , Theorem 4.1)
established the following:

######  Theorem 5.1 (Asymptotic Nonvanishing Theorem)

For each $1\leq q\leq n$, there are constants $C_{1},C_{2}>0$ such that if $d$
is sufficiently large, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\text{ for $C_{1}d^{q-1}\leq p\leq
r_{d}-C_{2}d^{n-1}$}.$

The influential paper EL2  opens the door to research on the asymptotic
behaviors of $K_{p,q}(X,B;L_{d})$ for $d$ increasing. It is natural to ask
whether $K_{p,q}(X,B;L_{d})=0$ for the values of $p$ outside the range in
Theorem 5.1. Based on Raicu’s result (Raicu, , Appendix), Park (Park1, ,
Theorem 1.1) proved the following theorem, which was previously conjectured by
Ein–Lazarsfeld (EL2, , Conjecture 7.1).

######  Theorem 5.2 (Asymptotic Vanishing Theorem)

For each $1\leq q\leq n$, there is a constant $C_{3}>0$ such that if $d$ is
sufficiently large, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\text{ for $p\leq C_{3}d^{q-1}$}.$

Park Park3  subsequently realized that the asymptotic vanishing theorem leads
to a quick proof of the more precise version of the asymptotic nonvanishing
theorem, that is Theorem 1.6: for each $1\leq q\leq n$, there are functions
$c_{q}(d)$ and $c_{q}^{\prime}(d)$ with

$c_{q}(d)=\Theta(d^{q-1})~{}~{}\text{ and
}~{}~{}c_{q}^{\prime}(d)=\begin{cases}\Theta(d^{n-q})&\text{if
$H^{q-1}(X,B)=0$ or $q=1$}\\\ q-1&\text{if $H^{q-1}(X,B)\neq 0$ and $q\geq
2$}\end{cases}$ (7)

such that if $d$ is sufficiently large, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\Longleftrightarrow~{}~{}c_{q}(d)\leq p\leq
r_{d}-c_{q}^{\prime}(d).$

It is natural to wonder whether $c_{q}(d)$ and $c_{q}^{\prime}(d)$ are
polynomials in $d$ when $d\gg 0$ (see (Park3, , Question 5.5)). It seems very
challenging even for $X=\mathbf{P}^{n}$ (cf. (EL3, , Conjecture 2.3)).
However, the answer is known for $q=1$ by Yang (DavidYang, , Theorem 1).
Furthermore, Ein–Lazarsfeld–Yang (ELY, , Theorem A) proved that if $B$ is
$p$-jet very ample, then $c_{1}(d)\geq p+1$, and Agostini (Agostini2, ,
Theorem A) proved that if $B$ is not $p$-very ample, then $c_{1}(d)\leq p+1$.
These results are already highly nontrivial and in fact a higher dimensional
generalization of the gonality conjecture. We refer to (Park3, , Section 5)
for more discussion on related open problems.

Following Park1 ; Park3 , we give an outline of the proof of Theorem 1.6.
First, we prove the asymptotic vanishing theorem for products of projective
spaces (see (Park1, , Subsection 3.2)). Let
$Y:=\mathbf{P}^{n_{1}}\times\cdots\times\mathbf{P}^{n_{k-1}}$ and
$X:=Y\times\mathbf{P}^{n_{k}}$. Set

$\begin{array}[]{l}B_{Y}:=\mathscr{O}_{\mathbf{P}^{n_{1}}}(b_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbf{P}^{n_{k-1}}}(b_{k-1}),~{}L_{Y}:=\mathscr{O}_{\mathbf{P}^{n_{1}}}(d_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbf{P}^{n_{k-1}}}(d_{k-1});\\\
B:=B_{Y}\boxtimes\mathscr{O}_{\mathbf{P}^{n_{k}}}(b_{k}),~{}L:=L_{Y}\boxtimes\mathscr{O}_{\mathbf{P}^{n_{k}}}(d_{k}).\end{array}$

For simplicity, we put $n:=n_{k}$. Fix $2\leq q\leq n$. Assuming
$d:=\min\\{d_{1},\ldots,d_{k}\\}\gg 0$ and $p\leq(1/n_{1}!\cdots
n_{k}!)(d^{q-1}+bd^{q-2})=\Theta(d^{q-1})$, we show that

$K_{p,q}(X,B;L)=H^{q-1}\big{(}Y\times\mathbf{P}^{n},\wedge^{p+q-1}M_{L}\otimes((L_{Y}\otimes
B_{Y})\boxtimes\mathscr{O}_{\mathbf{P}^{n}}(d_{k}+b_{k}))\big{)}=0.$

Regarding $\mathbf{P}^{n}=(\mathbf{P}^{1})_{n}$, let
$\sigma:=\operatorname{id}_{Y}\times\sigma_{n-1,1}$, where
$\sigma_{n-1,1}\colon\mathbf{P}^{n-1}\times\mathbf{P}^{1}\to\mathbf{P}^{n}$ is
the universal family. It suffices to show that

$H^{q-1}\big{(}Y\times\mathbf{P}^{n-1}\times\mathbf{P}^{1},\wedge^{p+q-1}\sigma^{*}M_{L}\otimes((L_{Y}\otimes
B_{Y})\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k}+b_{k})\boxtimes\mathscr{O}_{\mathbf{P}^{1}}(d_{k}+b_{k}+n-1))\big{)}=0.$

To this end, using the geometry of symmetric products of $\mathbf{P}^{1}$,
Park (Park1, , Subsection 3.1) constructed a short exact sequence

$0\longrightarrow\bigoplus\mathscr{O}_{Y\times\mathbf{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbf{P}^{1}}(-n)\longrightarrow\sigma^{*}M_{L}\longrightarrow
M_{L_{Y}\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k})}\boxtimes\mathscr{O}_{\mathbf{P}^{1}}(d_{k})\longrightarrow
0.$

Considering the filtration for $\wedge^{p+q-1}\sigma^{*}M_{L}$, we can reduce
the problem to

$H^{q-1}\big{(}Y\times\mathbf{P}^{n-1}\times\mathbf{P}^{1},(\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k})}\otimes((L_{Y}\otimes
B_{Y})\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k}+b_{k})))\boxtimes\mathscr{O}_{\mathbf{P}^{1}}(a_{i})\big{)}=0$

for $i\leq p+q-1$, where $a_{i}:=i(d_{k}+n)+d_{k}+b_{k}+2n-qn-1-pn$. By the
Künneth formula, it is equivalent to

$\begin{array}[]{l}(i)~{}H^{q-1-j}\big{(}Y\times\mathbf{P}^{n-1},\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k})}\otimes((L_{Y}\otimes
B_{Y})\boxtimes\mathscr{O}_{\mathbf{P}^{n-1}}(d_{k}+b_{k}))\big{)}=0~{}\text{
or }\\\
(ii)~{}H^{j}(\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{1}}(a_{i}))=0~{}~{}\text{
for $0\leq i\leq p+q-1$ and $j=0,1$}.\end{array}$

Assume $3\leq q\leq n$. We proceed by induction on $n_{1}+\cdots+n_{k}$. When
$j=0$, $(i)$ holds for $i\leq p+q-1$ by induction. When $j=1$, $(i)$ holds for
$i\leq(1/n_{1}!\cdots
n_{k-1}!(n_{k}-1)!)(d^{q-2}+bd^{q-3})+q-2=\Theta(d^{q-2})$ by induction, and
$(ii)$ holds for $i\geq(1/n_{1}!\cdots
n_{k-1}!(n_{k}-1)!)(d^{q-2}+bd^{q-3})+q-1$ since $a_{i}\geq-1$. When $q=2$,
the argument is slightly different but easier (see (Park1, , Subsection 3.2)).

Next, we deduce Theorem 5.2 for general case from the same assertion for
products of three projective spaces proven in the previous paragraph. This
reduction argument was done by Raicu (Raicu, , Appendix) (see also (Park1, ,
Subsection 3.3)) shortly after Ein–Lazarsfeld’s asymptotic vanishing
conjecture (EL2, , Conjecture 7.1) was raised. We can choose suitable positive
integers $a_{1},a_{2},a_{3}$ depending only on $X,A,P$ and positive integers
$d_{1},d_{2},d_{3}$ grow linearly in $d$ in such a way that if

$\begin{array}[]{l}A_{1}:=\mathscr{O}_{X}(a_{1}A),~{}A_{2}:=\mathscr{O}_{X}(a_{2}A+P),~{}A_{3}:=\mathscr{O}_{X}(a_{3}A-P),\\\
\mathbf{P}^{n_{1}}:=\mathbf{P}H^{0}(X,A_{1}),~{}\mathbf{P}^{n_{2}}:=\mathbf{P}H^{0}(X,A_{2}),~{}\mathbf{P}^{n_{3}}:=\mathbf{P}H^{0}(X,A_{3}),\\\
Y:=\mathbf{P}^{n_{1}}\times\mathbf{P}^{n_{2}}\times\mathbf{P}^{n_{3}},~{}L_{Y}:=\mathscr{O}_{\mathbf{P}^{n_{1}}}(d_{1})\boxtimes\mathscr{O}_{\mathbf{P}^{n_{2}}}(d_{2})\boxtimes\mathscr{O}_{\mathbf{P}^{n_{3}}}(d_{3}),\end{array}$

then $L_{d}=A_{1}^{d_{1}}\otimes A_{2}^{d_{2}}\otimes A_{3}^{d_{3}}$ and there
is a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{P}^{r}:=\mathbf{P}H^{0}(X,L_{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$linear
subspace$\textstyle{\mathbf{P}H^{0}(Y,L_{Y})=:\mathbf{P}^{N}.}$

We may regard $B$ as a coherent sheaf on $Y,\mathbf{P}^{r},\mathbf{P}^{N}$.
Since

$\begin{array}[]{l}\text{(syzygies of $B$ on
$\mathbf{P}^{N}$)}=\text{(syzygies of $B$ on
$\mathbf{P}^{r}$)}\otimes\text{(Koszul complex of linear forms)},\end{array}$

it follows that

$\min\\{p\mid K_{p,q}(X,B;L_{d})\neq 0\\}=\min\\{p\mid K_{p,q}(Y,B;L_{Y})\neq
0\\}.$

By considering the minimal free resolution of $B$ on $Y$ with
$\mathbf{Z}^{3}$-grading, we can deduce Theorem 5.2 (the case of $q=1$ is
trivial) from the result proven in the previous paragraph. Actually this
argument shows that Theorem 5.2 holds when $X$ is a possibly singular
projective variety and $B$ is an arbitrary coherent sheaf on $X$.

Finally, we prove Theorem 1.6 using the asymptotic vanishing theorem (Theorem
5.2). As in EL2 , the argument proceeds by induction on $n=\dim X$. Let $H$ be
a suitably positive very ample line bundle on $X$, and choose a general member
$\overline{X}\in|H|$. Put

$\overline{L}_{d}:=L_{d}|_{\overline{X}},~{}~{}\overline{B}:=B|_{\overline{X}},~{}~{}\overline{H}:=H|_{\overline{X}},~{}~{}V_{d}^{\prime}:=H^{0}(X,L_{d}\otimes
H^{-1}).$

We have noncanonical splitting

$\begin{array}[]{l}K_{p,q}(X,\overline{B};L_{d})=\bigoplus_{j=0}^{p}\wedge^{j}V_{d}^{\prime}\otimes
K_{p-j,q}(\overline{X},\overline{B};\overline{L}_{d});\\\
K_{p,q}(X,\overline{B}\otimes\overline{H};L_{d})=\bigoplus_{j=0}^{p}\wedge^{j}V_{d}^{\prime}\otimes
K_{p-j,q}(\overline{X},\overline{B}\otimes\overline{H};\overline{L}_{d}).\end{array}$

There are exact sequences

$\begin{array}[]{l}K_{p+1,q-1}(X,B\otimes H;L_{d})\rightarrow
K_{p+1,q-1}(X,\overline{B}\otimes\overline{H};L_{d})\xrightarrow{\theta_{p,q}}K_{p,q}(X,B;L_{d})\rightarrow
K_{p,q}(X,B\otimes H;L_{d});\\\\[3.0pt] K_{p,q}(X,B\otimes
H^{-1};L_{d})\rightarrow
K_{p,q}(X,B;L_{d})\xrightarrow{\theta_{p,q}^{\prime}}K_{p,q}(X,\overline{B};L_{d})\rightarrow
K_{p-1,q+1}(X,B\otimes H^{-1};L_{d}).\end{array}$

When $q=1$ or $2$, the map $\theta_{p,q}$ should be slightly modified (see
(Park3, , Section 3)). By Theorem 5.2 and induction on $n$, we can easily get
the estimation of $c_{q}(d)$ and $c_{q}^{\prime}(d)$ in (7). Furthermore,
$\theta_{c_{q}(d),q}$ and $\theta_{r_{d}-c_{q}^{\prime}(d),q}$ are nonzero
maps. Thus there are

$\begin{array}[]{l}\text{$\alpha\in
K_{c_{q}(d)+1-j_{0},q-1}(\overline{X},\overline{B}\otimes\overline{H};\overline{L}_{d})$
for some $0\leq j_{0}\leq c_{q}(d)+1$ lifted to
$K_{c_{q}(d),q}(X,B;L_{d})$;}\\\ \text{ $\beta\in
K_{r_{d}-c_{q}^{\prime}(d)-j_{0}^{\prime},q}(\overline{X},\overline{B};\overline{L}_{d})$
for some $r_{d}^{\prime}-c_{q}^{\prime}(d)\leq j_{0}^{\prime}\leq
r_{d}^{\prime}$ lifted to
$K_{r_{d}-c_{q}^{\prime}(d),q}(X,B;L_{d})$.}\end{array}$

Then we can lift $\alpha$ and $\beta$ to $K_{p,q}(X,B;L_{d})$ for
$c_{q}(d)\leq p\leq r_{d}-\Theta(d^{n-1})$ and $\Theta(d^{n-1})\leq p\leq
r_{d}-c_{q}^{\prime}(d)$, respectively (see (Park3, , Theorem 3.1) for more
details on lifting syzygies from hypersurfaces). This gives the desired
nonvanishing for $K_{p,q}(X,B;L_{d})$ as stated in Theorem 1.6.

We close this paper by recalling open problems on the quantitative behavior of
the nonzero graded Betti numbers $\kappa_{p,q}(X,B;L_{d})$ as $d\to\infty$.
Ein–Erman–Lazarsfeld (EEL, , Conjecture B) (resp. Park (Park3, , Conjecture
5.7)) conjectured that the Betti numbers $\kappa_{p,q}(X,B;L_{d})$ are
normally distributed (resp. form a unimodal sequence) for each $1\leq q\leq
n$. It is also expected that the Betti numbers $\kappa_{p,0}(X,B;L_{d})$ are
increasing and the Betti numbers $\kappa_{p,n+1}(X,B;L_{d})$ are decreasing
(see (Park3, , Remark 5.9)). These have been verified when $n=1$ (see (EEL, ,
Proposition A) and (Park3, , Proposition 5.8)) but they are widely open when
$n\geq 2$. It seems that these problems are already very challenging even for
$X=\mathbf{P}^{n}$. However, from the perspective of Boij–Söderberg theory BS
; ES , Ein–Erman–Lazarsfeld (EEL, , Corollary D) proved that each row of the
“random” Betti table is normally distributed.

###### Acknowledgements.

The author would like to thank Daniele Agostini, Junho Choe, Lawrence Ein,
Sijong Kwak, Wenbo Niu for the many interesting discussions we have had on
syzygies of algebraic varieties. The author was partially supported by the
National Research Foundation (NRF) funded by the Korea government (MSIT)
(NRF-2021R1C1C1005479 and NRF-2022M3C1C8094326).

Competing InterestsNone

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