Asymptotic Np property of rational surfaces

Abstract

In this paper, we examine Green and Lazarsfeld's Np property for rational surfaces obtained by blowing up the projective plane P^2 at a subscheme of fat points Z. We show that these surfaces, embedded into projective spaces by linear system of plane curves of degree t containing Z, possess property Np for all t >> 0. This gives a positive answer to a conjecture of Geramita and Gimigliano. Our bound for the values t is better than the conjectural value, which is σ(Z) + p.