Deforming curves in jacobians to non-jacobians II: curves in $C^{(e)}$, $3\le e\le g-3$

Abstract

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in symmetric powers $C^{(e)}$ of $C$ where $3\le e\le g-3$. More precisely, given a pencil $g_d^1$ of degree $d$ on $C$, let $X$ be the curve parametrizing divisors of degree $e$ in divisors of $g_d^1$ (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair $(C,g_d^1)$, we prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=2e$, dim$H^0(g_d^1)=e$ or $d=2e+1$, dim$H^0(g_d^1)=e+1$. The analogous result in the case $e=2$ without genericity assumptions was proved earlier.

ams-latex, 27 pages