An algebraic approach to the regularity index of fat points in \(P^ n\)
type: Item
instance of: scholarly article
title: An algebraic approach to the regularity index of fat points in \(P^ n\)
zbMATH Open document ID: 0821.14030
author: Trung Ngo Viet
published in: Kodai Mathematical Journal
publication date: 1995-10-03T00:00:00Z
review text: This paper is a survey of results about bounds for the regularity index of fat points in \(\mathbb{P}^ n\) [mainly by \textit{M. V. Catalisano}, \textit{G. Valla} and the author, Proc. Am. Math. Soc. 118, No. 3, 717-724 (1993; Zbl 0787.14030), and \textit{G. Valla} and the author, Math. Z. 219, No. 2, 187-201 (1995)]. The question is the following: Let \(X = \{P_ 1, \dots, P_ s\}\) be a set of points in \(\mathbb{P}^ n_ k\), where \(k = \overline k\) and \(\text{char} k = 0\), and let \(p_ i\) be the homogeneous ideal, in \(R = k[x_ 0, \dots, x_ n]\), associated to \(P_ i\); consider for \(m_ 1 \geq \cdots \geq m_ s \geq 0\), the scheme \(Z\) associated to the ideal \(I = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_ s^{m_ s}\); then, for every \(t \geq 0\), the vector space \(I_ t\) represents the space of hypersurfaces of \(\mathbb{P}^ n\) having a point of multiplicity at least \(m_ i\) at each \({\mathfrak p}_ i\). The multiplicity of \(Z\) is \(e(Z) = \sum_ i {m_ i + n - 1 \choose n}\), and its index of regularity \(r(Z)\) is the least integer \(g\) for which \(h_ A(t) = e(Z)\), where \(h_ A(t)\) is the Hilbert function of the coordinate ring \(A = R/I\) of \(Z\); in other words, \(r(Z)\) is the least integer \(t\) for which \(Z\) imposes \(e(Z)\) independent conditions on hypersurfaces of degree \(t\). Even for \(n = 2\), it is quite a hard problem to determine the value of \(r(Z)\) for a generic choice of \(X\) (it will only depend on the \(m_ i\)'s). A (sharp) bound for \(r(Z)\), for any choice of \(X\) in general position (i.e. no \(n + 1\) points on a hyperplane), can be found thanks to an approach described in the paper under review: consider the ideal \(J = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_{s - 1}^{m_{s - 1}}\), then \(r(Z) = \max \{m_ s - 1, r(R/J), r(R/J + {\mathfrak p}_ s^{m_ s})\}\). One can use this in order to work by induction (since \(r(Z)\) is known for small \(s)\), noting that \(r(R/(J + {\mathfrak p}_ s^{m_ s})…
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 729228
zbMATH Keywords: linear systems
zbMATH Keywords: rational normal curves
zbMATH Keywords: bounds for the regularity index of fat points
zbMATH Keywords: Hilbert function of the coordinate ring
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138040029
OpenAlex ID: W1997522963
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DOI: 10.2996/KMJ/1138040029
description: scientific article; zbMATH DE number 729228