, Volume 116, Issue 1, pp 87-109

Deforming Curves in Jacobians to Non-Jacobians I: Curves in C (2)

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Abstract

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic 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 \(\mathbb{C}^{(2)}\) 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° \((g_{5}^{1}) = 3\) and C has genus 4

This material is based upon work partially supported by the National Security Agency under Grant No. MDA904-98-1-0014 and the National Science Foundation under Grant No. DMS-0071795. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Security Agency (NSA)