On Segre's bound for fat points in Pn

For a scheme of fat points Z defined by the saturated ideal IZ, the regularity index computes the Castelnuovo-Mumford regularity of the Cohen-Macaulay ring R/IZ. For points in "general position" we improve the bound for the regularity index computed by Segre for P2 and generalised by Catalisano, Trung and Valla for Pn. Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for n + 3 arbitrary points in Pn. We propose a modification of Segre's conjecture for arbitrary points and we discuss some evidences.