Abstract. Let (S, L) be a polarized abelian surface of Picard rank one and let φ be the function which takes each ample line bundle L ′ to the least integer k such that L ′ is k-very ample but not (k + 1)-very ample. We use Bridgeland's stability conditions and Fourier-Mukai techniques to give a closed formula for φ(L n ) as a function of n showing that it is linear in n for n > 1. As a byproduct, we calculate the walls in the Bridgeland stability space for certain Chern characters.

A topological space X is an S-paracompact if there exists a bijective function f from X onto a paracompact space Y such that for every separable subspace A of X the restriction map f | A from A onto f (A) is a homeomorphism. Moreover, if Y is Hausdorff, then X is called S 2-paracompact. We investigate these two properties.