On Segre's bound for fat points in PnPn

For a scheme of fat points Z   defined by the saturated ideal IZIZ, the regularity index computes the Castelnuovo–Mumford regularity of the Cohen–Macaulay ring R/IZR/IZ. For points in “general position” we improve the bound for the regularity index computed by Segre for P2P2 and generalised by Catalisano, Trung and Valla for PnPn. Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for n+3n+3 arbitrary points in PnPn. We propose a modification of Segre's conjecture for arbitrary points and we discuss some evidences.