The ideal resolution for generic 3-fat points in \(\mathbb P^2\).
type: Item
instance of: scholarly article
title: The ideal resolution for generic 3-fat points in \(\mathbb P^2\).
zbMATH Open document ID: 1042.13009
published in: Journal of Pure and Applied Algebra
publication date: 2004-02-02T00:00:00Z
review text: Let \(P_1,\dots,P_n\) be generic points in \(\mathbb P^2\), denote by \(Y=Y(n)\) the 0-dimensional scheme of \(m\)-fat points with support on the \(P_i\)'s, i.e. the scheme defined by the ideal \(I_Y= {\mathfrak p}_1^m\cap \dots \cap {\mathfrak p}_n^m\) , where \({\mathfrak p}_i\) is the homogeneous ideal of the point \(P_i\) in \(k[X_0, X_1,X_2]\) (\(k\) an algebraically closed field of characteristic zero). The paper deals with the problem of determining the degrees of a minimal set of generators for the ideal \(I_Y\), when \(m=3\) [for the case \(m=2\) see \textit{M. Idà}, J. Algebra 216, No. 2, 741--753 (1999; Zbl 0943.13008)]. Since \(Y\) is 0-dimensional of codimension 2, this is the same as determining the graded Betti numbers of a minimal free resolution of \(I_Y\). The authors prove that if \(n\neq 2,3,5\) the natural maps \(\sigma_k:H^0({\mathcal I}_Y(k)\otimes H^0({\mathcal O}_{\mathbb P^2} (1)) \rightarrow H^0({\mathcal I}_Y(k+1))\) (where \({\mathcal I}_Y\) is the ideal sheaf associated to \(I_Y\)) are of maximal rank for each \(k\). As \(Y\) has maximal Hilbert function, this means that the ideal \(I_Y\) has the same ''good resolution'' as the ideal of a scheme of \(6n\) generic (reduced) points.
reviewed by: Caterina Cumino
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 2035896
zbMATH Keywords: fat points
zbMATH Keywords: minimal set of generators
zbMATH Keywords: minimal free resolution
zbMATH Keywords: Hilbert function
author: Alessandro Gimigliano
author: Monica Idà
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.07.002
OpenAlex ID: W2099952672
cites work: An asymptotic vanishing theorem for generic unions of multiple points
cites work: Generators for the homogeneous ideal of s general points in \({\mathbb{P}}_ 3\)
cites work: Linear systems of plane curves with a composite number of base points of equal multiplicity
cites work: Fat points on a conic
cites work: On the resolution of points in generic position
cites work: Linear systems of plane curves with base points of equal multiplicity
cites work: Gale duality and free resolutions of ideals of points
cites work: Q4699515
cites work: Resolutions of fat points ideals involving eight general point of \(\mathbb{P}^2\)
cites work: Resolutions of ideals of quasiuniform fat point subschemes of 𝐏²
cites work: The ideal of forms vanishing at a finite set of points in \({\mathbb{P}}^ n\)
cites work: Q5602063
cites work: Q3751693
cites work: The ideal generation problem for fat points
cites work: Q4143433
cites work: La methode d'Horace pour l'interpolation à plusieurs variables
cites work: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques.
cites work: The minimal resolution of the ideal of a general arrangement of a big number of points in \(\mathbb{P}^ n\)
cites work: On the homogeneous ideal of the generic union of lines in 3.
cites work: The minimal free resolution for the first infinitesimal neighborhoods of \(n\) general points in the plane
cites work: The minimal resolution conjecture
cites work: Systems of plane curves with prescribed singularities: The case of multiplicities less than or equal to four
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: The minimal free resolution of the homogeneous ideal of \(s\) general points in \(\mathbb{P}^ 4\)
DOI: 10.1016/J.JPAA.2003.07.002
Recommended article: On the minimal free resolution for fat point schemes of multiplicity at most 3 in \(\mathbb P ^2\)
Recommended article: Q4882187
Recommended article: Q4534652
Recommended article: The minimal free resolution for the first infinitesimal neighborhoods of \(n\) general points in the plane
Recommended article: The role of the cotangent bundle in resolving ideals of fat points in the plane
Recommended article: On the resolution of ideals of fat points
Recommended article: Q5487006
Recommended article: Free resolutions of fat point ideals on \(\mathbb{P}^2\)
Recommended article: Q4784965
Recommended article: Resolutions of ideals of fat points with support in a hyperplane
description: scientific article; zbMATH DE number 2035896