Tropes, Torelli and theta characteristics

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.

References in corpus (2)

• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
• Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves