SINGULAR LOCUS ON THE SPACE OF GENUS 2 CURVES WITH DECOMPOSABLE JACOBIANS.

Abstract

We study the singular locus on the algebraic surface Sn of genus 2 curves with a n,n-split Jacobian. Such surface was computed by Shaska in [15] for n=3, and Shaska at al. in [3] for n=5. We show that the singular locus for n=2 is exactly th locus of the curves of automorphism group D4 or D6. For n=3 we use a birational parametrization of the surface S3 discovered in [15] to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group D4 or D6. We further show that the birational parametrization used in S3 would work for all n≥7 if Sn is a rational surface.