Deforming Curves in Jacobians to Non-Jacobians II: Curves in C(e), 3⩽ e ⩽ g−3*

Abstract

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