Deforming curves in jacobians to non-jacobians I: curves in C⁽²⁾

2001/03/29 by Elham Izadi, E. Izadi, Izadi, E.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14B10 #msc:14C25 #msc:14H40 #msc:14K12

paper · pdf · doi:10.48550/arxiv.math/0103204

amslatex, 25 pages

arxiv created 2005/06/21 · arxiv updated 2009/11/30

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 g¹_d of degree d on C, let X be the curve parametrizing pairs of points in divisors of g¹_d (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, dimH⁰ (g¹₅) = 3 and C has genus 4.

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