Torelli's theorem for high degree symmetric products of curves
Najmuddin Fakhruddin
We show that two smooth projective curves $C_1$ and $C_2$ of genus $g$ which have isomorphic symmetric products are isomorphic unless $g=2$. This extends a theorem of Martens.
math0208180.tex
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\title{Torelli's theorem for high degree symmetric products of curves}
\author{Najmuddin Fakhruddin}
\address{School of Mathematics\\ Tata Institute of Fundamental Research\\ Homi Bhabha Road\\ Mumbai 400005\\ India}
\email{naf@math.tifr.res.in}

\begin{document}

\begin{abstract}
We show that two smooth projective curves $C_1$ and $C_2$ of genus $g$
which have 
isomorphic symmetric products are isomorphic unless $g=2$.
This extends a theorem of Martens.
\end{abstract}

%\today
\maketitle



Let $k$ be an algebraically closed field and $C_1$ and $C_2$ two smooth projective
curves of genus $g > 1$ over $k$ . It is a consequence of Torelli's theorem
that if $ Sym^{g-1}C_1 \cong Sym^{g-1} C_2$, then $C_1 \cong C_2$. The same holds for
the $d$'th symmetric products, for $1 \leq d < g-1$ as a consequence of a
theorem of Martens \cite{martens}. In this note we shall show that with one exception
the same result continues to hold for all $d \geq 1$, i.e.~ we have the following

\begin{thrm}
Let $C_1$ and $C_2$ be smooth projective curves of genus $g \geq 2$ over an algebraically
closed field $k$. If $Sym^d C_1 \cong Sym^d C_2$ for some $d \geq 1$, then $C_1 \cong C_2$
unless  $g=d=2$.
\end{thrm}

It is well known that there exist non-isomorphic curves of genus $2$ over 
$\mathbf{C}$ with isomorphic Jacobians. Since the second symmetric power 
of a genus $2$ curve is isomorphic to the blow up of the Jacobian in a point,
it follows that our result is the best possible.

\begin{proof}[Proof of Theorem]
Let $C_i$, $i=1,2$ be two curves of genus $g>1$
with $Sym^d C_1 \cong Sym^d C_2$ for some $d \geq 1$.
Since the Albanese variety of $Sym^d C_i$, $d \geq 1$, is isomorphic to the Jacobian $J(C_i)$,
it follows that $J(C_1) \cong J(C_2)$.
If  $d \leq g-1$, the theorem follows immediately from \cite{martens}, since the
image of $Sym^d C_i$ in $J(C_i)$ (after choosing a basepoint) is $W_d (C_i)$. Note that
in this case it sufices to have a birational  isomorphism from 
 $Sym^d C_1 $ to $ Sym^d C_2$.

Suppose $g \leq d \leq 2g -3$. Then the Albanese map from $Sym^d C_i$ to $J(C_i)$
is surjective with general fibre of dimension $d -g$. 
Interpreting the fibres as complete linear systems of degree $d$ on $C_i$,
it follows by Serre duality
that the subvariety of $J(C_i)$ over which the fibres are of dimension $> d-g$
is isomorphic to $W_{2g - 2 -d}(C_i)$. Therefore if $Sym^d C_1 \cong
Sym^d C_2$, then $W_{2g -2 -d}(C_1) \cong W_{2g -2 -d}(C_2)$, so 
Martens' theorem implies that $C_1 \cong C_2$.

Now suppose that $d > 2g-2$ and $g>2$. By choosing some isomorphism we identify 
$J(C_1)$ and  $J(C_2)$ with a fixed abellian variety $A$.
If $\phi:Sym^d C_1 \to Sym^d C_2$
is our given isomorphism, from the universal property of the Albanese
morphism we obtain a commutative diagram
\[
\xymatrix{
Sym^d C_1 \ar[r]^{\phi} \ar[d]_{\pi_1} & Sym^d C_2 \ar[d] _{\pi_2} \\
A \ar[r]^{f} & A 
}
\]
where  
the $\pi_i$'s are the Albanese morphisms corresponding to some basepoints and
$f$ is an automorphism of $A$ (not necessarily preserving the origin).
By replacing $C_2$ with $f^{-1}(C_2)$ we may then assume that $f$ is the identity.

Since $d > 2g -2 $, the maps $\pi_i$, $i=1,2$  make $Sym^d C_i$ into projective
bundles over $A$. By a theorem of Schwarzenberger \cite{schwarzenberger},
$Sym^d C_i \cong Proj(E_i)$, where $E_i$ is a vector bundle on $A$
of rank $d -g + 1$
with $c_j(E_i) = [W_{g-j}(C_i)]$, $i=1,2$, $0 \leq j \leq g-1$, in the group of cycles
on $A$ modulo numerical equivalence. Since $\phi$ is an isomorphism of 
projective bundles, it follows that there exists a line bundle $L$ on $A$
such that $E_1 \cong E_2 \otimes L$.

Let $\theta_i = [W_{g-1}(C_i)]$, so by  
Poincar\'e's formula  $[W_{g-j}(C_i)] = \theta_i^j/j!$, $i=1,2$, $i \leq j \leq g-1$.
 The lemma
below implies that $\theta_i^{g-1} = \theta_2^{g-1}$ in the group of cycles
modulo numerical equivalence on $A$.
Since $\theta_i^g = g!$, this implies that $\theta_1 \cdot [C_2] = g$.
By Matsusaka's criterion \cite{matsusaka},
 it follows that $W_{g-1}(C_1)$ is a theta
divisor for $C_2$, which by Torelli's theorem implies that $C_1 \cong C_2$.

If $d = 2g-2$ and $g>2$, then we can still apply the previous argument.
In this case we also have that $Sym^d(C_i) \cong Proj(E_i)$, $i =1,2$ but $E_i$ is
a coherent sheaf which is not locally free.
However on the complement of some point of $A$ it does become
locally free and the previous formula for the Chern classes remains valid.

The above argument clearly does not suffice if $g=2$. To
handle this case   we shall use some
properties of
Picard bundles for which we refer the reader to \cite{mukai}.
Suppose that $d>2$ and $C_i$, $i=1,2$ are two \emph{non-isomorphic} curves of genus
$2$ with $Sym^d C_1 \cong Sym^d C_2$. Using the same argument (and notation)
as the $g>2$ case, it follows that
there exist embeddings of $C_i$, $i=1,2$, in $A$ and a line bundle $L$ on $A$
such that $E_1 \cong E_2 \otimes L$ and $L^{\otimes d-1} \cong \mathcal{O}(C_1 - C_2)$
(we identify $C_i$, $i=1,2$ with their images).


For $i \geq 1$, let $G_i$ denote the $i$-th Picard sheaf associated to $C_2$,
so that $Proj(G_i) \cong Sym^i(C_2)$. ($G_i$ is the sheaf denoted by $F_{2-i}$ in
\cite{mukai} and $G_d \cong E_2$). 
There is an exact sequence (\cite[p.172]{mukai}):
\begin{equation}
0 \to \mathcal{O}_A \to G_i \to G_{i-1} \to 0 \label{eq:G}
\end{equation}
for all $i>1$. We will use this exact sequence and induction on $i$ to 
compute the cohomology of sheaves of the form $E_1 \otimes P \cong E_2 \otimes L \otimes P$,
where $P \in Pic^0(A)$.

Consider first the cohomology of $G_1$,
 which is the pushforward of a line bundle of degree $1$ on a 
translate of $C_2$. Since we have assumed that $C_1 \ncong C_2$, it follows
that $C_1 \cdot C_2 > 2$. Since $C_1^2 = C_2^2 = 2$,
$deg(L|_{C_2}) =  (C_1 - C_2) \cdot C_2/(d-1) > 0$.
 By Riemann-Roch it follows that $h^j(A,G_1 \otimes L \otimes P)$, $j = 1,2$
is independent of
$P$, except possibly for one  $P$ if $deg(L|_{C_2}) = 1$, and  $h^2(A,G_1 \otimes L \otimes P)=0$
since $G_1$ is supported on a curve.

 Now $C_1 \cdot C_2 > 2$ also
implies that $c_1(L)^2 < 0$. By the index theorem, it follows that
$h^0(A,L \otimes P) = h^2(A,L \otimes P) = 0$ and $h^1(A, L \otimes P)$ is independent
of $P$.
 Therefore by  tensoring the exact sequence \eqref{eq:G} with
$L \otimes P$ and considering the long exact sequence of cohomology,
we obtain an exact sequence
\begin{multline}
0 \to H^0(A, G_i \otimes L \otimes P) \to H^0(A, G_{i-1} \otimes L \otimes P) \to
H^1(A, L \otimes P) \\
\to H^1(A, G_i \otimes L \otimes P) \to H^1(A, G_{i-1} \otimes L \otimes P)
\to 0 \label{eq:G2}
\end{multline}
and isomorphisms $H^2(A, G_i \otimes L \otimes P) \to H^2(A, G_{i-1} \otimes L \otimes P)$
for all $i>1$. 
By induction, it follows that $H^2(A, G_i \otimes L \otimes P) = 0$ for all $i>0$.
Since the Euler characteristic of $G_i \otimes L \otimes P$ is 
independent of $P$,  the above exact sequence \eqref{eq:G2} along with
induction shows that for all $i>0$ and $j=0,1,2$,
$h^j(G_i \otimes L \otimes P)$ is independent of $P$, except for possibly
one $P$. In particular, this holds for $i = d$ hence $h^j(A, E_1 \otimes P)$
is independent of $P$ except again for possibly one $P$. We obtain a contradiction
by using the computation of the cohomology of Picard sheaves in 
Proposition 4.4 of \cite{mukai}: This implies that $h^1(A,E_1 \otimes P)$
is one or zero depending on whether the point in $A$ corresponding to
$P$ does or does not lie on a certain curve (which is a translate of $C_1$).



% The definition of
% the Fourier-Mukai transform and the base change theorem for cohomology
% imply that the Fourier-Mukai transform of $E_1$ cannot be a sheaf supported
% on a curve.
% But by the duality property of the Fourier-Mukai transform, the transform of $E_1$
% must be a coherent sheaf on $\hat{A} \cong A$ which is isomorphic to the pushforward
% of a line bundle on  (a translate of) $C_1$, hence we obtain a contradiction.






\end{proof}


\begin{lem}
Let $X$ be an algebraic variety of dimension $g \geq 3$
 and let $E_i$, $i=1,2$
be vector bundles on $X$ of rank $r$.
Suppose $c_1(E_i) = \theta_i$,
$c_j(E_i) = \theta^j_i/j!$ for $i=1,2$ and $ j =2,3$ ($j=2$ if $g = 3$),
and  $E_1 \cong  E_2 \otimes L$
for some line bundle $L$ on $X$. Then $\theta_1^j = \theta_2^j$ for
all $j > 1$ ($j=2$  if $g=3$).
\end{lem}
\begin{proof}
Since $E_1 \cong E_2 \otimes L$, $c_1(E_1) = c_1(E_2) + r c_1(L)$, hence 
$c_1(L) = (\theta_1 - \theta_2)/r$. 

For a vector bundle $E$ of rank $r$ and a line bundle $L$
on any variety, we have the
following formula for the Chern polynomial (\cite{fulton-it}, page 55):
\[
c_t(E \otimes L) = \sum_{k=0}^r t^k c_t(L)^{r-k}c_i(E) .
\]
Letting $E = E_1$, $E \otimes L = E_2$, and expanding out the terms of degree
$2$ and $3$, one easily sees that $\theta_1^j = \theta_2^j$ for $j = 2$
and also for $j=3$ if $g>3$.
(Note that this only requires knowledge of $c_j(E_i)$ for $j =1,2,3$).
Since any integer $n>1$ can be written as $n = 2a + 3b$ with $a,b \in \mathbf{N}$,
the lemma follows.
\end{proof}

We do not know whether the theorem holds if $k$ is not algebraically closed.

\begin{rem}
S.~Ramanan has suggested it may also be possible to prove  the
theorem when $g>2$ 
by computing the Chern classes of the push forward of the tangent bundle
of $Sym^d C$ to $J(C)$.
\end{rem}
\begin{ack}
We thank A.~Collino for some helpful correspondence, in particular for 
informing us about the paper \cite{martens}, and S.~Ramanan for the above 
remark.
\end{ack}

\begin{thebibliography}{1}

\bibitem{fulton-it}
{\sc W.~Fulton}, {\em Intersection theory}, Springer-Verlag, Berlin,
  second~ed., 1998.

\bibitem{martens}
{\sc H.~H. Martens}, {\em An extended {T}orelli theorem}, Amer. J. Math., 87
  (1965), pp.~257--261.

\bibitem{matsusaka}
{\sc T.~Matsusaka}, {\em On a characterization of a {J}acobian variety}, Memo.
  Coll. Sci. Univ. Kyoto. Ser. A. Math., 32 (1959), pp.~1--19.

\bibitem{mukai}
{\sc S.~Mukai}, {\em Duality between ${D}({X})$\ and ${D}(\hat {X})$\ with its
  application to {P}icard sheaves}, Nagoya Math. J., 81 (1981), pp.~153--175.

\bibitem{schwarzenberger}
{\sc R.~L.~E. Schwarzenberger}, {\em Jacobians and symmetric products},
  Illinois J. Math., 7 (1963), pp.~257--268.

\end{thebibliography}


\end{document}










math0208180.extracted_bib
\begin{thebibliography}{1}

\bibitem{fulton-it}
{\sc W.~Fulton}, {\em Intersection theory}, Springer-Verlag, Berlin,
  second~ed., 1998.

\bibitem{martens}
{\sc H.~H. Martens}, {\em An extended {T}orelli theorem}, Amer. J. Math., 87
  (1965), pp.~257--261.

\bibitem{matsusaka}
{\sc T.~Matsusaka}, {\em On a characterization of a {J}acobian variety}, Memo.
  Coll. Sci. Univ. Kyoto. Ser. A. Math., 32 (1959), pp.~1--19.

\bibitem{mukai}
{\sc S.~Mukai}, {\em Duality between ${D}({X})$\ and ${D}(\hat {X})$\ with its
  application to {P}icard sheaves}, Nagoya Math. J., 81 (1981), pp.~153--175.

\bibitem{schwarzenberger}
{\sc R.~L.~E. Schwarzenberger}, {\em Jacobians and symmetric products},
  Illinois J. Math., 7 (1963), pp.~257--268.

\end{thebibliography}