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 wit h no geometric conditions on the points. Whenever the conjecture is true, t he 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 a n easy proof of the conjecture in P-2, we prove it in P-3, by using some pr eliminary results which hold, more generally, in P-n. (C) 2001 Elsevier Sci ence B.V. All rights reserved.