Equations of $(1,d)$ -polarized abelian surfaces

Abstract

In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d ). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A via L ⊗n are all quadratic; a possible choice for a set of generators of IA are the Riemann theta relations. On the other hand, much less is known about embeddings via line bundles L of type (1, d ), that is line bundles L which are not powers of another line bundle on A. It is well-known that if d ≥ 5, and A is a general abelian surface, then L is very ample, while L can never be very ample for d < 5. However, even if d ≥ 5, L may not be very ample for special abelian surfaces. We will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface. A few special cases are well-documented in the literature: d = 4, in which case the general surface is a singular octic in P3, cf. [BLvS], and d = 5 in which case the abelian surface is described as the zero set of a section of the Horrocks-Mumford bundle [HM], whereas its homogeneous ideal is generated by 3 (Heisenberg invariant) quintics and 15 sextics (cf. [Ma]). Also, recent work by Manolache and Schreyer [MS] and by Ranestad [Ra] provides a description of the equations and syzygies in the case d = 7.