On the homogeneous ideal of collinear punctual subschemes of \(\mathbb{P}^2\)
type: Item
instance of: scholarly article
title: On the homogeneous ideal of collinear punctual subschemes of \(\mathbb{P}^2\)
zbMATH Open document ID: 0929.14031
published in: Collectanea Mathematica
publication date: 1997-09-15T00:00:00Z
full work available at URL: https://eudml.org/doc/41095
review text: Fix positive integers \(s,n_i\), \(1\leq i\leq s\), a finite number of lines, \(D_1,\dots, D_s\), of \(\mathbb{C}\mathbb{P}^2\), points \(P_1,\dots,P_s\) with \(P_i\in D_i\), for all \(i\) and let \(Z(i)\) be the length \(n_i\) subscheme of \(D_i\) with support \(P_i\). Set \(Z:= \bigcup_{1\leq i\leq s}Z(i)\). Assume \(D_i\) and \(P_i\) general. Here we show (under mild assumptions on the integers \(n_i\)) that the homogeneous ideal of \(Z\) has the expected number of generators in each degree and hence we compute the minimal free resolution of \(Z\). The proof depends heavily on the proofs in a paper by \textit{A. Eastwood} [Manuscr. Math. 67, No.~3, 227-249 (1990; Zbl 0722.41010)], in which the corresponding problem for the postulation was solved using degenerations of collinear schemes to suitable 0-dimensional subschemes of \(\mathbb{C}\mathbb{P}^2\) and an inductive technique called the Horace method [cf. \textit{A. Hirschowitz}, Manuscr. Math. 50, 337-388 (1985; Zbl 0571.14002)].
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 1060667
zbMATH Keywords: collinear punctual subschemes
zbMATH Keywords: number of generators
zbMATH Keywords: minimal free resolution
zbMATH Keywords: postulation
zbMATH Keywords: Horace method
author: E. Ballico
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1060667