On Segre's bound for fat points in ℙⁿ

2015/04/20 by Edoardo Ballico, Olivia Dumitrescu, Ballico, Edoardo +3
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Point processes and geometric inequalities #math.AC #math.AG #msc:13D40 #msc:14C20

paper · pdf · doi:10.48550/arxiv.1504.05151

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

arxiv created 2015/10/25 · arxiv updated 2015/10/27

Abstract

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

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