Tropes, Torelli and theta characteristics

TL;DR

This paper provides an effective Torelli theorem for genus 3 curves over fields with characteristic not 2, showing how odd theta characteristics determine a specific del Pezzo surface and reconstructing the curve from it.

Contribution

It introduces an explicit geometric method to recover genus 3 curves from their odd theta characteristics via a unique del Pezzo surface, extending Torelli's theorem.

Findings

Configuration of odd theta characteristics determines a degree two del Pezzo surface.

Curve is recovered as normalization of a curve on the surface.

Explicit description of the quadratic twist in Torelli's theorem.

Abstract

Our main result is an effective version of the Torelli theorem in genus $3$ and any characteristic not $2$: the configuration of the odd theta characteristics of a curve $C$ of genus $3$ determines a del Pezzo surface $S$ of degree two and that $C$ is recovered as the normalization of a certain curve on $S$. This surface is not the one usually associated to $C$; we describe the distinction. We also give an explicit geometrical description of the quadratic twist (observed by Serre) that arises in the statement of the Torelli theorem.