Upper bounds for the regularity index of fat points
type: Item
instance of: scholarly article
title: Upper bounds for the regularity index of fat points
zbMATH Open document ID: 0835.14020
author: Trung Ngo Viet
author: G. Valla
published in: Journal of Algebra
publication date: 1996-04-15T00:00:00Z
review text: Let \(P_1, \ldots, P_s\) be \(s\) points in \(\mathbb{P}^n\). Given positive integers \(m_1 \geq m_2 \geq \cdots \geq m_s\), we can consider the cycle \(Z = \sum m_i P_i\). That is, take the subscheme \(Z\) of \(\mathbb{P}^n\) defined by the ideal sheaf \(I = \prod {\mathcal I}^{m_j}_j\) where \({\mathcal I}_j\) is the sheaf of ideals defining \(P_j\). The smallest integer \(r(Z)\) such that \(H^1 (\mathbb{P}^n, I(r(Z))) = 0\) is called the regularity index of \(Z\). It is of considerable interest to find good upper bounds for \(r(Z)\) in terms of the \(m_i\)'s and a lot of work has been done in this direction. The study of these cycles is classical, but it has bloomed in the recent times. If the points were arbitrary, then the following bound is known: \(r(Z) \leq (\sum m_i)-1\) and equality if and only if the points lie on a line. The interested reader can find the references in the paper. If one assumes that the points are in general position, then the bound can be improved. Let \(\delta (Z) = [{1 \over n} (\sum^n_{i = 1} m_i + n - 2)]\), where as usual, \([x]\) stands for the integral part of \(x\). Then \(r(Z) \leq \max \{m_1 + m_2 - 1, \delta (Z)\}\). This, the authors call the Segre bound, since Segre proved it for \(n = 2\). This bound is attained if the points lie on a rational normal curve. In this paper, the authors prove that under some mild hypotheses, the converse also holds. i.e. if this bound is attained, then the points lie on a rational normal curve. They also give a better bound if the points are assumed to be in uniform position.
reviewed by: N. Mohan Kumar
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 802375
zbMATH Keywords: zero dimensional subscheme
zbMATH Keywords: points in general position
zbMATH Keywords: regularity index
zbMATH Keywords: Segre bound
MaRDI profile type: Publication
full work available at URL: jabr.1995.1239
OpenAlex ID: W2032601770
DOI: 10.1006/JABR.1995.1239
description: scientific article; zbMATH DE number 802375