Regularity index of fat points in the projective plane

In this paper we determine a new upper bound for the regularity index of fat points of P^2, without requiring any gerometric condition on the points. This bound is intermediate between Segre's bound, that holds for points in general position, and the more general bound, that is attained when the points are collinear: in fact both of these bound can be recoverd as particular cases. Furthermore, our bound ccannot, in general, be sharpened: in fact, is attained if there are either many collinear points or collinear points with high multiplicities.