Deforming curves in jacobians to non-jacobians I: curves in $C^{(2)}$

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 the second symmetric power $C^{(2)}$ of $C$. More precisely, given a pencil $g_d^1$ of degree $d$ on $C$, let $X$ be the curve parametrizing pairs of points in divisors of $g_d^1$ (see the paper for the precise scheme-theoretical definition). We prove that if $X$ deforms infinitesimally out of the jacobian locus with $JC$ then either $d=4$ or $d=5$, dim$H^0(g_5^1)=3$ and $C$ has genus 4.

amslatex, 25 pages