On Segre's bound for fat points in ${\mathbb{P}}^n$

Abstract

For a scheme of fat points $Z$ defined by the saturated ideal ${\mathcal{I}}_Z$, the regularity index computes the Castelnuovo-Mumford regularity of the Cohen-Macaulay ring $R/{\mathcal{I}}_Z.$ For points in " general position" we improve the bound for the regularity index computed by Segre for ${\mathbb{P}}^2$ and generalised by Catalisano, Trung and Valla for ${\mathbb{P}}^n$. Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for $n+3$ arbitrary points in ${\mathbb{P}}^n$. We propose a modification of Segre's conjecture for arbitrary points and we discuss some evidences.

Revised version: statement of Theorem 3.6 corrected, Section 4.1 with a proof of Conjecture 4.1 for n=3 added. Accepted for publication in JPAA. 19 pages

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