Ample Divisors on Abelian Surfaces

Abstract

Let A be an abelian variety of dimension g, and let L be an ample line bundle on A. The algebraic equivalence class of L is determined by c^L) e H(A, Z). The latter can be viewed as the space of alternating bilinear forms on the free abelian group H^A, Z) of rank 2g. The ampleness of L implies that c^L) is (weakly) non-degenerate and consequently it determines positive integers <51} ...,Sg with Sx \S2 | . . . \Sg. Any line bundle in the algebraic equivalence class of L is a translate of L, so that the question of whether L is very ample on a fixed abelian variety A is a property of c^L). However, this is not so if A is allowed to vary. The classical theorem of Lefschetz asserts that if Si ^ 3, the line bundle L is always very ample. It is a necessary condition as well, if we require that whenever c^L) is of type (<5), L is very ample. For, it is obvious that if A = Ax x A2 and L = p*Li ® p*L2, where Lx is a line bundle of degree at most 2 on an elliptic curve Al} then L is not very ample. We wish to investigate the question of whether, given (S) = (<5l5 ...,Sg), there exist some abelian variety A and a very ample line bundle L on A of type {S\.