Jacobians with a vanishing theta-null in genus 4

TL;DR

This paper proves that in genus 4, a principally polarized abelian variety with a vanishing theta-null and degenerate Hessian at that point must be a Jacobian, confirming a conjecture and exploring higher genus implications.

Contribution

It proves a conjecture linking vanishing theta-nulls and Jacobians in genus 4, and discusses potential generalizations and geometric interpretations.

Findings

Confirmed the conjecture for genus 4 abelian varieties.

Connected the degeneracy of the Hessian to Jacobian characterization.

Explored higher genus generalizations and geometric interpretations.

Abstract

In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.