Equations of (1,d)-polarized Abelian Surfaces

Abstract

We study the equations of abelian surfaces embedded in P^{n-1} with a line bundle of polarization of type (1,n). For n>9, we show that the ideal of a general abelian surface with this polarization is generated by quadrics, and if the embedding is Heisenberg invariant, we give a very specific description of these quadrics. These quadrics are produced using a generalization of the Moore matrices which yields a uniform description of these ideals. We prove that these equations generate the ideals using a degeneration argument, degenerating abelian surfaces to schemes obtained from Stanley-Reisner ideals of certain triangulations of the torus. Furthermore, we obtain information on how to compute the moduli spaces of abelian surfaces of type (1,n) with canonical level structure. In a sequel to this paper, we will study the structure of the equations for n<=9, and also give detailed geometric descriptions for the moduli spaces of abelian surfaces of low degree.