Moduli of abelian surfaces, symmetric theta structures and theta characteristics

TL;DR

This paper investigates the birational geometry of moduli spaces of abelian surfaces with additional structures, focusing on how polarizations, theta characteristics, and theta-null maps influence their geometric properties.

Contribution

It elucidates the relationship between polarization parities, symmetric theta structures, and theta characteristics in the context of moduli spaces of abelian surfaces.

Findings

Determines how polarization parities affect the relation between level and theta structures.

Shows when theta characteristics are necessary for defining Theta-null maps.

Analyzes the birational properties and Kodaira dimension of these moduli spaces.

Abstract

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geometry and the Kodaira dimension of these moduli spaces.