  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 \sigma(Z) + p.
