# On the Schottky problem for genus five Jacobians with a vanishing theta null

## TL;DR

This paper solves the weak Schottky problem for genus five Jacobians with a vanishing theta null by characterizing when a principally polarized abelian variety is a Jacobian based on the rank of its tangent cone.

## Contribution

It provides a criterion involving the rank of the tangent cone at a vanishing theta null to identify genus five Jacobians, answering a question posed by Grushevsky and Salvati Manni.

## Key findings

- Principal polarization with a low-rank tangent cone implies Jacobian structure.
- Degeneration and ramification loci analysis are key methods used.
- The result characterizes Jacobians among abelian varieties with specific theta null properties.

## Abstract

We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.