Hyperelliptic genus 4 curves on abelian surfaces

TL;DR

This paper investigates genus 4 hyperelliptic curves on abelian surfaces, revealing their unique existence on general (1, 3)-polarized surfaces and analyzing their Jacobian structures.

Contribution

It establishes the uniqueness of genus 4 hyperelliptic curves on general (1, 3)-polarized abelian surfaces and explores their Jacobian subvarieties.

Findings

Exactly one such curve exists up to translation.

Their Jacobians contain a surface and its dual as complementary subvarieties.

Provides structural insights into hyperelliptic curves on abelian surfaces.

Abstract

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1, 3)-polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.