Equations of (1,d)-polarized Abelian Surfaces

Details Equations of (1,d)-polarized Abelian Surfaces Equations of (1,d)-polarized Abelian Surfaces

1996-09-04

arxiv.org/abs/alg-geom/9609001v2

Mathematics

Algebraic Geometry

50 pages with 1 figure, author-supplied PostScript file available at http://www.math.columbia.edu/~psorin/eprints/abel-eq.ps (

Scientific paper

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.

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