Symbolic powers of ideals of generic points in \(\mathbb P^3\)
type: Item
instance of: scholarly article
title: Symbolic powers of ideals of generic points in \(\mathbb P^3\)
zbMATH Open document ID: 1259.14036
author: Marcin Dumnicki
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
review text: Let \(k\) be a field of characteristic zero and \(k[\mathbb{P}^N]=k[x_0,x_1,\ldots,x_N]\) be the ring of coordinates of the projective plane with the standard grading. Let \(I\subseteq k[\mathbb{P}^N]\) be an ideal of fat points in \(\mathbb{P}^N\) and \(M=(x_0,x_1,\ldots,x_N)\subseteq k[\mathbb{P}^N]\) the maximal homogeneous ideal. Harbourne and Huneke conjectured that \(I^{(Nr)}\subseteq M^{(N-1)r}I^r\), for all \(r\geq1\). The conjecture is known to be true for ideals of generic points in \(\mathbb{P}^2\). The author shows that is also true for any number of generic points in \(\mathbb{P}^3\) and for up to \(N+1\) generic points in \(\mathbb{P}^N\), for all \(N\geq2\). As a consequence, he also shows that the conjecture of Chudnovski holds for any number of generic points in \(\mathbb{P}^3\).
reviewed by: Anda-Georgina Olteanu
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 6094127
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.12.010
OpenAlex ID: W2963388287
cites work: Comparing powers and symbolic powers of ideals
cites work: The resurgence of ideals of points and the containment problem
cites work: Q3903262
cites work: An algorithm to bound the regularity and nonemptiness of linear systems in \(\mathbb P^n\)
cites work: Are symbolic powers highly evolved?
DOI: 10.1016/J.JPAA.2011.12.010
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description: scientific article; zbMATH DE number 6094127