  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.
