| The purpose of this talk is to study algebraic properties of embeddings of abelian surfaces. Following Green's seminal work on the relation between Koszul cohomology and positivity of line bundles, Green and Lazarsfeld introduced 'property N_p' for a natural number p, a sequence of increasingly restrictive conditions on the higher syzygies associated to a very ample line bundle. First we explain how ensuring property N_p for an ample line bundle L on an abelian surface can be reduced to a very general problem in projective geometry: the construction of singular divisors with given numerical behaviour and singularities, then we show how to use infinitesimal Newton-Okounkov bodies to deal with certain instances of the construction problem. This is an account of joint work with Victor Lozovanu. |