Asymptotic syzygies grow exponentially:\\ A Remark on a paper of Ein-Lazarsfeld
Daniel Chun
Ziv Ran
%We compare Koszul cohomology of a variety and a subvariety %via certain filtrations on Koszul complexes. For all $2\leq q\leq \dim(X)$ and most relevant $p$ values, the dimension of the asymptotic Koszul cohomology group $K_{p,q(X, B;L_d)$ grows exponentially with $d$.
elaz-remark.tex
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\begin{document} 
\title{Asymptotic syzygies grow
 exponentially:\\
A Remark on a paper of Ein-Lazarsfeld}
\author{Daniel Chun}
\author {Ziv Ran}
{}
%\author{}
%\Large

%\thanks{\raggedright{

%Partially supported by NSA Grant MDA904-02-1-0094} }
%\thanks{arxiv.org/1308.2442}
\date {\today}% \enddate

%\affil University of California, Riverside\endaffil

\address {\nl UC Math Dept. \nl
Big Springs Road Surge Facility
\nl
Riverside CA 92521 US\nl 
ziv.ran @  ucr.edu\nl
\url{http://math.ucr.edu/~ziv/}
}

%\email {ziv.ran @ucr.edu}
% \subjclass[2010]{14J40, 32G07, 32J27, 53D17}
%\keywords{}

\begin{abstract}
%We compare Koszul cohomology of a variety and a subvariety
%via certain filtrations on Koszul complexes.
For all $2\leq q\leq \dim(X)$ and most relevant $p$ values, the dimension
of the asymptotic
Koszul cohomology group $K_{p,q}(X, B;L_d)$ grows exponentially with $d$.
\end{abstract}
\maketitle

%\section{Introduction}
While interest in the syzygies of a projective variety $X$ goes back to Hilbert,
it was Mark  L. Green's seminal paper \cite{green-koszul} that popularized the notion
that the syzygies of $X$ can be viewed as a type of cohomology,
called Koszul cohomology, attached to the pair $(X, L)$ where $L$ is a line
bundle on $X$. Since then, chiefly under
the leadership of Robert Lazarsfeld and Claire Voisin, numerous results
have been established on the computation or nonvanishing
 of syzygies in many cases.
 Recently
particular attention has been focused on the case
of \emph{asymptotic} Koszul cohomology groups
$K_{p,q}(X, B;L_d)$ where $L_d$ is sufficiently ample
 \cite{el}. Some of the results are stated in \cite{ein-lazarsfeld-yang}. \par
Larzarsfeld and Ein \cite{el} have shown, for a large range of $p$ and $q\geq 2$ that these groups
don't vanish. Our purpose here is to show that the construction of \cite{el} can be pushed to
yield lower bounds on the dimension of these
groups which are exponential in $d$. Our argument differs from \cite{el} mainly
 in the elimination of the induction, i.e. 'drilling down to the base case'.\par
 After this was written, Lazarsfeld communicated that he and Ein had been aware
 in a general way of
 similar bounds, based on an argument different from the one given here, which was never
 written down.  Nonetheless, the asymptotic behavior of the
 dimension of the $K_{p,q}(X, B; L_d)$ is stated
 an an open problem in \cite{el}. See \cite{el-sav} for further related results and problems.


%Our purpose here is to work out a local cohomology version of (asymptotic) Koszul 
%cohomology, with support in a subvariety $Y\subset X$.
%This takes the form of some filtrations on Koszul complexes on $X$,
%whose graded pieces are related either to $(Y, L|_Y)$ or to the projection
%of $X$ from $Y$ (or equivalently, the blowup of $X$ in $Y$
%and the sections of $L$ vanishing on $Y$).
%As a consequence we will prove a result on nonvanishing of certain asymptotic
%Koszul cohomology groups first obtained with a different proof by Ein and
%Lazarsfeld \cite{ein-lazarsfeld-inventiones}. Our proof
%is inspired by the argument in \cite{ein-lazarsfeld-inventiones}
%which also involves a (different) comparison with a different,
%trickier type of
%subvariety. Here we will use a simple complete intersection.
\section*{Notations and conventions}
Let $L$ be a line bundle and $B$ a coherent sheaf on a variety $X$. 
A \emph{linear system} on $X$, belonging to $L$, is by definition a 
vector space $V$ endowed with
a map, not necessarily  injective, $V\to H^0(X, L)$.
This induces  a map of coherent sheaves $V\otimes\O_X\to L$. 
 In our applications,
we will initially take $V$ complete, i.e. $V=H^0(X, L)$, but various
associated linear systems, e.g. restrictions on subvarieties, need not be complete.

Given these data, the associated
\emph{Koszul complexes} are by definition the complexes
\eqspl{koszul}{
C_{p, c}(X, B, L, V):\ \  \bigwedge\limits^pV\otimes(B+(c+1)L)\to\bigwedge\limits^{p-1}V\otimes
(B+(c+2)L)\to...
}
where we use customary additive notation: $B+aL:=B\otimes L^{\otimes a}$.
We will often suppress $X, L$ when understood and denote this by $C_{p, c}(B, V)$
or $C_p(B,V)$ when $c=0$.
When $V$ is base-point free, the
complex \eqref{koszul} admits local homotopy operators, hence it is
locally acyclic, i.e. exact in strictly positive degrees.
We will denote the kernel of \eqref{koszul}, i.e. $\H^0(C_p(B,V))$, by $M_p(B, V)$.
In our applications, $L$ will have the form
\[L=L_d:=dA+P\]
with $A$ ample and $d$ as large as desired. Then the members of the
Koszul complex are themselves acyclic sheaves thanks to Serre vanishing.
Whenever the Koszul complex is locally 
acyclic and consists of acyclic sheaves, 
we have
\eqspl{}{
\HH^i(C_{p,c}(B, V))=H^{i}(M_{p,c}(B, V))=K_{p-i,c+1+i}(X, L, B, V), i>0.
}
It will be convenient to use the following
terminology: a nonnegative complex $C^\bullet$ of sheaves
is \emph{globally
acyclic} if it is locally acyclic and consists of acyclic sheaves. In this
case, the hypercohomology can be identified
with either the cohomology of the $\H^0$ sheaf, or with the
cohomology of the associated $H^0$ complex. Thus, we have
\eqspl{}{
\HH^i(C^\bullet)= H^i(\H^0(C^\bullet))=H^i(H^0(C^\bullet)).
}
If $C^\bullet$ is globally acyclic
and these groups are moreover trivial for all $i>0$, $C^\bullet $ is said
to be \emph{hyper-acyclic}.\par
We will often use $H^\bullet$ rather than $\HH^\bullet$ to denote hypercohomology of
a complex of sheaves, when there is no confusion.

\begin{comment}

\section{The local weight  and diagonal filtration}
Now suppose given a smooth subvariety $Y\subset X$ and set
\[V_0=\ker(V\to H^0(Y, L|_Y)), V_Y=V/V_0.\]
This induces a descending 
filtration $F^\bullet_Y$ called
the \emph{$Y$-weight filtration} on $\bigwedge\limits^pV$ with associated graded
\[\bigoplus \limits_{a=0}^p\bigwedge\limits^aV_0\otimes\bigwedge\limits^{p-a}
V_Y.\]
We denote the respective graded pieces by $V^{a, p-a}_Y$ or just $V^{a, p-a}$
when $Y$ is understood. Setting
\[r+1=\dim V, s=\dim V_Y, c_0=\dim(V_0),\]
we have that $V^{a, p-a}\neq 0$ precisely when $p-s\leq a\leq c_0$.
There is an associated $Y$-weight filtration, also denoted $F^\bullet_Y$
and called the \emph{vertical} filtration,
on the complex $C_{p, c}(B, V)$ with graded pieces $C_{a, c}(B, V_0)\otimes\bigwedge
\limits^{p-a}V_Y$. 
For visualization purposes only, it may be helpful to identify $V_Y$
non-canonically 
with a suitable subspace of $V$, which turns the 
the filtered complex $C_{p, c}(V, B)$
to a double complex with 'southward' differential
\[\delta_0:V^{a,b}\otimes (B+tL)\to V^{a-1, b}\otimes(B+(t+1)L)\]
and 'southwestward' differential
\[\delta_1:V^{a,b}\otimes (B+tL)\to V^{a, b-1}\otimes(B+(t+1)L).\]
However note that $V_0$ has of course a base locus,
namely $Y$ (at least). So the vertical complexes 
that are the graded pieces of the $F^\bullet_Y$ filtration
are not locally acyclic. \par
To deal with this circumstance, we will 
consider another filtration on $C_{p,c}(V, B)$ that we call the
\emph{diagonal filtration} and denote by $D^\bullet_Y$.
First let $\pi:\tilde X\to X$ be the blowup of $X$ in $Y$ with exceptional
divisor $\tilde Y$ and $\tilde L=\pi^*(L), \tilde B=\pi^*(B)$, and note that 
$V_0\subset H^0(\tilde X, \tilde L-\tilde Y)$,
i.e. $V_0$ is a linear system belonging to the line bundle $\tilde L-\tilde Y$.

Set $L_Y=L\otimes\O_Y, B_Y=B\otimes\O_Y$.
\par
Now the diagonal filtration is defined as follows:
\eqspl{}{
D^a_YC_{p, c}(X, L, B, V)=\sum\limits_{a_1+a_2=a}
F^{a_1}_YC_{p,c}(X, L, \I_Y^{a_2}L, V), a=0,...,c_0.
}
(internal, non-direct sum).
Note that
\eqspl{}{
\gr^0_{D_Y}(C_{p,c}(B,V))=C_{p,c}(Y, L_Y, B_Y, V_Y)
}
and similarly,
\eqspl{}{
\gr^a_{D_Y}(C_{p,c})=\bigoplus\limits_{a_1+a_2=a} C_{p-a_1, c}(Y, L_Y, B_Y\otimes
\sym^{a_2}\check N_Y, V_Y)\otimes\bigwedge\limits^{a_1}V_0, a=0,...,c_0-1.
} where $\check N_Y=\I_Y/\I_Y^2$ is the conormal bundle,
so  $\sym^a\check N_Y=\I_Y^a/I_Y^{a+1}$.
Also, note that on the 'bottom piece' $D^{c_0}_Y$, the induced graded pieces
of the vertical filtration can be identified, without changing cohomology,
 with $C_{a_1, c}(\tilde X, \tilde L-\tilde Y, \tilde B+(a_1+1)\tilde Y, V_0)
\otimes\bigwedge^{p-a_1}V_Y, a_1=0,...,c_0$, i.e.
\eqspl{}{
\gr^{a_1}_{F_Y}(D^{c_0}_Y(C_{p,c}))=\pi_*(C_{a_1, c}(\tilde X, \tilde L-\tilde Y, \tilde B+(a_1+1+c-c_0)\tilde Y, V_0))
\otimes\bigwedge^{p-a_1}V_Y, a_1=0,...,c_0
} with higher derived images vanishing
by standard properties of smooth blowups.
\par
Here is a schematic of $D^c_YC_{p,c}(B, V)$, realized on $\tilde X$, assuming for simplicity that
$c=c_0$:
\tiny
\eqspl{}{
\begin{matrix}
V^{c, p-c}\otimes(\tilde B+(c+1)\tilde L)&&V^{c-1, p-c+1}\otimes(\tilde B+(c+1)\tilde L-\tilde Y)&
...&V^{1, p-1}\otimes(\tilde B+(c+1)\tilde L-(c-1)\tilde Y)&&V^{0,p}\otimes(\tilde B+c+1)\tilde L-c\tilde Y)\\
\downarrow&\swarrow&\downarrow&&\downarrow&\swarrow&\\
V^{c-1, p-c}\otimes(\tilde B+(c+2)\tilde L-2\tilde Y)&&V^{c-1, p-c+1}\otimes(\tilde B+(c+1)\tilde L-\tilde Y)&
...&V^{1, p-1}\otimes(\tilde B+(c+2)\tilde L-c\tilde Y)&&\\
\vdots&&\vdots&&\vdots&&\\
V^{c-1, p-c}\otimes(\tilde B+2c\tilde L-(c-1)\tilde Y)&&V^{0, p-c+1}\otimes(\tilde B+2c\tilde L-c\tilde Y)&&&&\\
\downarrow&\swarrow&&&&&&\\
V^{0, p-c}\otimes(\tilde B+(2c+1)\tilde L-c\tilde Y)&&&&&&
\end{matrix}
}
\normalsize
\ls
Now applying the standard spectral sequence of a filtered complex, we conclude
\begin{thm}
Notations as above, assume\par
(i) $\mathrm{Bs}(V_0)=Y$;\par
(ii) $H^i(Y, B_Y\otimes L_Y^{c+1}\otimes \sym^{a}\check N_Y)=0, i>0, a=0,...,c_0.$\par
Then there is a spectral sequence abutting to $K_{p-a-b, c+1+a+b}(X, L, B, V)$ with
\eqspl{}{
E_1^{a,b}=\begin{cases}
\bigoplus\limits_{a_2=0}^a K_{p-2a-b+a_2, c+1+a+b}
(Y, L_Y, B_Y\otimes\sym^{a_2}\check N_Y, V_Y)\otimes\bigwedge\limits^{a-a_2}V_0,
a=0,...,c_0-1;\\
K_{-b-c_0, c+1+a+b}(\tilde X, \tilde L-\tilde Y, \tilde B+(a+1+c-2c_0)\tilde L, V_0)
\otimes\bigwedge\limits^{p-a+c_0}V_Y, a=c_0,...,2c_0.
\end{cases}
}

\end{thm}
\par ***********\par

\begin{comment}
define
a refinement $F^{\bullet, \bullet}_Y$ of the $Y$-weight filtration
by 'carving out' some diagonal complexes out of the $F^i$.
Let $\pi:\tilde X\to X$ be the blowup of $X$ in $Y$ with exceptional
divisor $\tilde Y$ and $\tilde L=\pi^*(L)$, and note that 
$V_0\subset H^0(\tilde X, \tilde L-\tilde Y)$,
i.e. $V_0$ is a linear system belonging to the line bundle $\tilde L-\tilde Y$.
Set $L_Y=L\otimes\O_Y, B_Y=B\otimes\O_Y$.
Now consider the restricted complex
$C_{p, c}(B, V)\otimes\O_Y$. It clearly admits as quotient
the 'diagonal' complex $C_{p, c}(Y, B_Y, L_Y, V_Y).$ 
Denote the kernel by
$F^{0, 1}_Y(C_{p, c}(B, V))$, or temporarily just $F^{0,1}$,
a subcomplex of $C_{p, c}(B,V)$. Next, set
\[F^{1,0}=F^{0,1}\cap F^1.\]
Thus,
\[ F^{0,1}/F^{1,0}=V^{0,p}\otimes (B+\tilde L-\tilde Y).\]
Then $F^{1,0}$ admits as quotient the 'diagonal complex 
$C_{p-1, c}(Y, B_Y, L_Y, V_Y)\otimes V_0$.
Let $F^{1,1}$ be the kernel. Then $F^{1,1}$ in turn admits the complex
$C_{p-1, c+1}(Y, B\otimes\check N_Y, L_Y, V_Y)$ as quotient,
where $\check N_Y=\I_{Y, X}/\I^2_{Y, X}$ is the conormal bundle.
Let $F^{1,2}$ be the kernel. Note that $F^{1,2}$ admits as quotient
$C_1(\tilde X, B, L-\tilde Y, V_0)\otimes\bigwedge\limits^{p-1}V_Y$,
and the kernel coincides with $F^{1,2}\cap F^2=:F^{2,0}$. Also,
$C_1(\tilde X, B, L-\tilde Y, V_0)$ is acyclic if $V_0$ is base-point free
as linear system belonging to $\tilde L-\tilde Y$.\par
Continuing in this way, we construct a double-indexed filtration $F^{\bullet, \bullet}$
such that
\eqspl{}{
F^{a,b}/F^{a, b+1}=C_{p-a, c+b}(Y, L_Y, B\otimes\sym^b(\check N_Y), V_Y)\otimes\bigwedge\limits^a V_0,b=0,...,a-1,
}
\eqspl{}{
F^{a,a}/F^{a+1, 0}=C_{a, c}(\tilde X, \tilde L-\tilde Y, \tilde B, V_0)\otimes \bigwedge\limits
^{p-a}V_Y.
}

**************************
\section{Complete intersections}
Now set
\[Y=H_1\cap ...\cap H_c\]
where $H_i\in |L_d|$ are general, $1<c<n$, and let $V=H^0(X, L_d)$. 
Thus \[s=\dim V_Y=h^0(L_d)-c, K_Y=K_X|_Y+cL_d.\] 
Also, $\O_Y$ admits on $X$ the standard
Koszul resolution with members $\binom{c}{i}(-iL), L=L_d$, which
implies that
\eqspl{}{
H^i(Y, (c+k)L)=0, i,k>0.
}\par 
Let $\phi_0:\tilde X\to\P^{c-1}$ be the natural (flat, surjective)
map, which corresponds to the linear system 
$V_0$ pulled back on $\tilde X$, and whose fibres are codimension-$(c-1)$
complete intersections containing $Y$.
If $H^1(\O_X)=0$ then $V_Y=H^0(Y, L_d\otimes\O_Y)$.
Note that in this case the vertical complexes 
$C_{a,c}(\tilde X, \tilde L-\tilde Y, \tilde B, V_0)$
are essentially just pullbacks of the standard Koszul complex on $\P^{c-1}$, 
truncated if $a<c$. When $a\geq c$, these complexes are just exact.
When $a<c$, the complex can be identified with
$\phi_0^*(\Omega^a_{\P^{c-1}}(a+1))\otimes(cL)$,
and it is easy to see that this complex is hyper-acyclic (e.g. by resolving
by the 'tail end' of the pulled-back Koszul complex from $\P^{c-1}$).
\par
We consider the Koszul complex $C_{p,c}(V, B)$ with its diagonal filtration
$D^{ \bullet}_Y$ and note at the outset, by a standard
and elementary computation based on the Koszul
resolution of $\O_Y$,  that all the {diagonal} complexes appearing
 as graded pieces of the filtration
  are globally acyclic, i.e. are locally acyclic and have
acyclic members. Hence these graded pieces have hypercohomology
that coincides on the one hand with the cohomology of the $H^0$
complex (i.e. a suitable Koszul cohomology) and on the 
other with the cohomology of the kernel sheaf, i.e.
$\H^0$.\par

Then we have an exact sequence of complexes
\[\exseq{D^{1}_Y(C_{p, c}(V, B))}{C_{p, c}(V, B)}{C_{p, c}(Y, L_Y, V_Y, B_Y)}.\]
This leads to an exact cohomology sequence
%\small
\eqspl{long}{H^{n-2c-1}(X, C_{p, c}(V, B))\to H^{n-2c-1}(Y, C_{p, c}(Y, L_Y, V_Y, B_Y))\to 
H^{n-2c} (X, D^1_Y(C_{p, c}(V, B)))
}
\normalsize
where the first and second terms can be identified respectively
with the appropriate Koszul cohomology groups. As for the third term,
note that by global acyclicity each graded piece of the diagonal filtration
has $H^{n-2c}=0$ (it being a $K_{*, \dim(Y)+1}$ on $Y$). 
On the other hand, each graded piece of the
vertical filtration on $D^c_Y(C_{p,c}(V, B))$ is globally hyper-acyclic
as we have seen. In particular we have
\eqspl{}{
 H^{n-2c}
(D^1_Y(C_p(V, B)))=0.
}
Thus, the first map in \eqref{long} is surjective. In particular, whenever $p$ is in the range
$[s-n-h^0(Y, K_X|_Y+cL_Y-B_Y), s-n-1]$,
we have by \cite{ein-lazarsfeld-inventiones}, Prop. 5.1 that the middle group above is nonzero. Consequently,
\eqspl{}{
H^{n-2c-1}(X, C_p(V, B))\neq 0.
} Thus,
\begin{cor}
Assumptions as above, we have $K_{p, m }(X, L, B, V)\neq 0, m>n/2$.
\end{cor}
\vfill\eject
\end{comment}
\section*{Exponenetial Ein-Lazarsfeld}
Let $Y\subset X$ be a smooth subvariety of dimension $m=n-c$ and consider a subspace $V_0\subset
V^+_0:=V\cap H^0(\I_Y(L))$ whose zero-locus is $m$-dimensional.
In our application, we will take $V=H^0(X, L)$.
Let $V_Y=V/V_0$, which admits a natural map to $L_Y:=L.\O_Y$. 
Suppose $p\geq v_Y:=\dim(V_Y)$.
Then the complex $C_{v_Y}(Y, B_Y, L_Y, V_Y)$ is exact,
hence so is $C_{v_Y}(Y, B_Y, L_Y, V_Y)\otimes \wedge^{p-v_Y}V_0$.
Now there is a
a natural quotient map 
\[\wedge^iV\to\wedge^{i-v_Y}V_0\otimes
\wedge^{v_Y}V_Y\]
(where the target is taken to be $\wedge^{v_Y}V_Y$ if $i=v_Y$ and zero if
$i<v_Y$). This induces a
 natural termwise surjective
map  
\[C_p(X, B, L, V)\to C_{v_Y}(Y, B_Y, L_Y, V_Y)\otimes \wedge^{p-v_Y}V_0 .\] 
The kernel of the latter map, denoted
$C^0_p(X, B, L, V)$, is hence a subcomplex 
quasi-isomorphic to $C_p(X, B, L, V)$. We call it the \emph{adapted} Koszul complex
(with respect to $Y$). Note that there is a natural map
\eqspl{x}{C^0_p(X, B, L, V)\to\wedge^{v_Y}V_Y\otimes\wedge^{p-v_Y}V_0\otimes\I_{Y/X}(B+L).
}
In the sequel we will use for $Y$ the subvariety used by Ein-Lazarsfeld \cite{el}, 
which in turn
is based on a construction of Eisenbud et al. \cite{eisenbud-restricting}. 
For a very ample divisor
$H$ fixed independently of $d$,
this $Y$
is of type 
\eqsp{H_1\cap...\cap H_{c-2}\cap D_1\cap D_2,
H_i\in |H|, D_1\in|H+B-K_X|, D_2\in|L-(c-1)H|
.}

$Y$ has dimension $m=q-2$ and
satisfies 
\eqspl{vanishing}{
h^{q-1}(\I_{Y/X}(B+L))=1; h^{q-1+i}(\I_{Y/X}(B+L+kH))=0, k>i\geq 0. 
}
Now let $\bar X$ be a smooth subvariety of $X$ transverse to (or disjoint from) $Y$ and let $\bar L, \bar B$ etc
denote restrictions on $\bar X$. Then for any divisor $B'$  we have an analogous map
\eqspl{xbar}{C^0_p(\bar X, \bar B', \bar L, V)\to\wedge^{v_Y}V_Y
\otimes\wedge^{p-v_Y}V_0\otimes\I_{\bar Y/\bar X}(\bar B'+\bar L).
}
Let $V_K, \bar V$ denote respectively the kernel and image of
the restriction map $V\to H^0(\bar L)$.
Then we have an exact diagram
\eqspl{v}{
\begin{matrix}
&0&&0&&0&\\
&\downarrow&&\downarrow&&\downarrow&\\
0\to&V_0\cap V_K&\to&V_K&\to&V_{K,Y}&\to 0\\
&\downarrow&&\downarrow&&\downarrow&\\
0\to&V_0&\to&V&\to&V_Y&\to 0\\
&\downarrow&&\downarrow&&\downarrow&\\
0\to&\bar V_0&\to&\bar V&\to&\bar V_Y&\to 0\\
&\downarrow&&\downarrow&&\downarrow&\\
&0&&0&&0&.
\end{matrix}
} 
\par
Now take $\bar X$ general of the form
\[\bar X=H'_1\cap...\cap H'_{m+1}, H'_i\in|H|\]
where $H$ is a fixed very ample divisor as above. Note that
a Koszul resolution of $\O_{\bar X}$ on $X$ remains exact
upon tensoring with $\I_Y$ and its locally free twists, and yields a torsion-free (not locally free)
Koszul resolution of $\O_{\bar X}$, since $\bar Y=Y\cap\bar X=\emptyset$ by
transversality. 
Thus we have exact
\small
\eqspl{koszul-torsionfree}{
\I_{Y/X}(B+L)\to(m+1)\I_{Y/X}(B+L+H)\to...\to\I_{Y/X}(B+L+(m+1)H)
\to\O_{\bar X}(B+L+(m+1)H).
}
\normalsize
Then setting $B'=B+(m+1)H$, consider an analogous 
torsion-free Koszul resolution of
$\wedge^{p-v_Y}V_Y\otimes\I_{\bar Y/\bar X}(\bar B'+\bar L)=
\wedge^{p-v_Y}V_Y\otimes \O_{\bar X}(\bar B+\bar L)$ , 
together with a compatible one for 
$C^0_p(\bar X, \bar B', \bar L, V)=C_p(\bar X, \bar B', \bar L, V)$.
This takes the form of a morphism of complexes
\tiny
\eqspl{map}{
\begin{matrix}
C^0_p(X, B, L, V)&\to...\to&(m+1)C^0_p(X, B+L+(m+1)H)&\to&C_p(\bar X, \bar B', \bar L, V)\\
\downarrow&&\downarrow&&\downarrow\\
\wedge^{v_Y}V_Y\otimes\wedge^{p-v_Y}V_0\I_{Y/X}(B+L)&\to...\to& (m+1)\wedge^{v_Y}V_Y\wedge^{p-v_Y}V_0\otimes\I_{Y/X}
(B+L+(m+1)H)&\to&\wedge^{v_Y}V_Y\wedge^{p-v_Y}V_0\otimes \O_{\bar X}(\bar B'+\bar L)
\end{matrix}
}
\normalsize
Pick a natural number $\bar p\leq v_Y$ independent of $d$, e.g. $\bar p=1$,
 and choose $V_0 $
so that $\dim(\bar V_Y)=\bar p$. 
For example, we can choose $V_0$ generated by $n$ general elements of 
$V_0^+$ (which cut out $Y$), plus the inverse image in $V_0^+$
of a general codimension-$(\bar p+n)$ subspace of the image of
$V_0^+\to\bar V$ (cf. \eqref{v}).
Note that in that case
$\dim(\bar V_0)$ is asymptotically $d^m$ for large d.
%hence $\dim(V_0\cap V_K)$ is asymptotically $d^n$. 
Then
the rightmost column of \eqref{map}
fits in a diagram, in which the upper horizontal arrows represent
a direct summand inclusion (with a suitable shift):
\eqspl{xbar1}{
\begin{matrix}
C_{\bar p}(\bar X, \bar B', \bar L, \bar V)\otimes\wedge^{p-\bar p}V_K&
\leftrightarrows& 
C_p(\bar X, \bar B', \bar L, V)\\
\downarrow&&\downarrow\\
\wedge^{v_Y}V_Y\otimes \wedge^{p-v_Y}(V_K\cap V_0)\otimes\O_{\bar X}
(\bar B'+\bar L)&\hookrightarrow&\wedge^{v_Y}V_Y\otimes\wedge^{p-v_Y}V_0
\otimes\O_{\bar X}
(\bar B'+\bar L)\\
\parallel&&\parallel\\
\wedge^{\bar p}\bar V_Y\otimes\wedge^{v_Y-\bar p}V_{Y, K}\otimes
\wedge^{p-v_Y}(V_K\cap V_0)\otimes\O_{\bar X}
(\bar B'+\bar L)&\hookrightarrow& 
\wedge^{\bar p}\bar V_Y\otimes\wedge^{v_Y-\bar p}V_{Y, K}\otimes
\wedge^{p-v_Y}V_0\otimes\O_{\bar X}
(\bar B'+\bar L)
\end{matrix}
}

% admits as direct summand the arrow
%\[ C_{\bar p}(\bar X, \bar B', \bar L, \bar V)\otimes\wedge^{p-\bar p}V_K
%\to\wedge^{\bar p}\bar V_Y\otimes\wedge^{v_Y-\bar p}V_{K,Y}\otimes \wedge^{p-v_Y}
%(V_K\cap V_0)\otimes\O_{\bar X}(\bar B'+\bar L).\]
By \cite{el}, Prop. 4.6,  the upper 
left arrow in \eqref{xbar1} induces a surjection on $H^0$ if $d$
is large enough. Now considering
the $E_1$ spectral sequence associated to the resolution \eqref{map}, we see from the 
vanishings of \cite{el} (cf. \eqref{vanishing})
that $H^{q-1}(\I_{Y/X}(B+L))$,
which is 1-dimensional, injects
into $H^0(\O_{\bar X}(\bar B'+\bar L))$. Therefore the image of the
left vertical map in \eqref{map} on $H^{q-1}$ must contain
$\wedge^{v_Y}V_Y\otimes\wedge^{p-v_Y}(V_0\cap V_K)H^{q-1}(\I_{Y/X}(B+L))$.
Consequently,
\eqspl{}{
\dim K_{p,q}(X, B, L)\geq \binom{\dim(V_0\cap V_K)}{p-v_Y}.
}
%\eqref
Choosing $p>v_Y\sim d^{q-1}$, hence $\dim(V_0)\sim d^n$, 
we can arrange things so that $\dim(V_0\cap V_K)\sim d^n$ while
$p-v_Y\sim d$, hence the above lower bound is exponential in $d$.
On the other hand by the very definition of Koszul cohomology,
its dimension is at most exponential in $d$. Hence we conclude:
%its diemension grows at most exponentially in $d$. Therefore we conclude
%and adjusting $V_0, V_K$ appropriately,
%we conclude
\begin{thm}
For $n=\dim(X)\geq 2$, 
$q\geq 2$ and $p$ asymptotically between $d^{q-1}$ and $d^n$, 
the dimension of $K_{p,q}(X, B; L_d)$ grows exponentially with $d$;
in particular,
\[\lim\limits_{d\to\infty}\dim K_{p,q}(X, B, L_d)=\infty.\]
\end{thm}
%In fact, with appropriate choices (e.g. $\dim(V_0\cap V_K)\sim d^n, n>1,  p-v_Y\sim d$), 
%we can arrange that $\dim K_{p,q}(X, B, L_d)$ grows 
%at least like $\binom{d^n}{d}$, i.e. exponentially with $d$. 
This sheds some light on Problem
7.3 in \cite{el}.
\vfill\eject
\bibliographystyle{plain}
\bibliography{../mybib}
\end{document}


%*************\par
\begin{comment}
Now, just for amusement's sake, if the
third group were zero and $p$ is in the range $[s-n-h^0(_Y-B), s-n-1]$, 
then the second group
is nonzero  by a (relatively elementary) 
result of Green and others (\cite{green-koszul},\S 3) (see
\cite{ein-lazarsfeld-yang}, Remark 1.8) 
(note that  $K_Y=K_X+cL_d$), 
and hence the first goup is nonzero.
But actually, I claim that the third group is nonzero. Consequently if $p$ is in the range
$[s-n, s]$,
then the second group is zero by the same result of Green and others,
hence the fourth group is nonzero. Either way, we get
some nonvanising Koszul cohomologies $K_{p,q}$ for all $q>1$. \par
To prove the claim, consider the exact sequence
\eqspl{}{
\exseq{F^{1,0}(C_p(V, B))}{F^{0,1}(C_p(V, B))}{\bigwedge\limits^pV_Y\otimes(B+\tilde L-\tilde Y)}.
}
It is easy to see that
\[H^{n-c+1}(B+\tilde L-\tilde Y)\neq 0, H^{n-c+2}(B+t(\tilde L-\tilde Y)=0, \forall t>0.\]
The vanishing assertion, plus obvious dimension considerations,
prove vanishing of  $H^{n-c+2}(F^{1,0}(C_p(V, B)))$, 
hence surjectivity of 
$H^{n-c+1}(F^{0,1}(C_p(V, B)))\to 
H^{n-c+1}(\bigwedge\limits^pV_Y\otimes(B+\tilde L-\tilde Y))$;
then the nonvanishing assertion proves the nonvanishing of 
$H^{n-c+1}(F^{0,1}(C_p(V, B)))$.
\end{comment}
\vfill\eject
\bibliographystyle{plain}
\bibliography{../mybib}
\end{document}

myheader3.tex
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elaz-remark.bbl
\begin{thebibliography}{1}

\bibitem{el-sav}
L~Ein and R~Lazarsfeld.
\newblock Syzygies of algebraic varieties.

\bibitem{el}
L.~Ein and R.~Lazarsfeld.
\newblock Asymptotic syszygies of algebraic varieties.
\newblock {\em Inventiones math.}, 190:603--646, 2012.

\bibitem{ein-lazarsfeld-yang}
L.~Ein, R.~Lazarsfeld, and D.~Yang.
\newblock A vanishing theorem for weight one syzygies.
\newblock \url{Arxiv.org/1512.06384v1}.

\bibitem{eisenbud-restricting}
D.~Eisenbud, M.~Green, K.~Hulek, and S.~Popsecu.
\newblock Restricting linear syzygies: algebra and geometry.
\newblock {\em Compos. Math.}, 141:1460--1478, 2005.

\bibitem{green-koszul}
M.~L. Green.
\newblock Koszul cohomology and the geometry of projective varieties.
\newblock {\em J. Differential Geometry}, 19:125--171, 1984.

\end{thebibliography}
