On a sharp bound for the regularity index of any set of fat points

We propose an upper bound for the regularity index of fat points of P-n with no geometric conditions on the points. Whenever the conjecture is true, the bound is sharp. It is, in fact, reached when there are points with high multiplicities either on a line or on some rational curve. Besides giving an easy proof of the conjecture in P-2, we prove it in P-3, by using some preliminary results which hold, more generally, in P-n. (C) 2001 Elsevier Science B.V. All rights reserved.