Moduli of abelian surfaces, symmetric theta structures and theta characteristics

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.