Q1113953
type: Item
instance of: scholarly article
title: Sur l'alignement dans les schémas de Hilbert ponctuels du plan. La famille des N-uplets de \({\mathbb{P}}^ 2\) contenant au moins r points sur une droite. (Alignment in the punctual Hilbert schemes of the plane. The family of N-tuples containing at least r points on a line)
zbMATH Open document ID: 0662.14001
DOI: 10.1007/BF01455071
author: Joachim Yaméogo
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164615
review text: Let \(Hilb^ N{\mathbb{P}}^ 2\) be the Hilbert scheme parametrizing the closed finite subschemes Z of length N in the projective plane. In this paper we are interested in the stratification of these schemes Z by the number of points they have on a line. The subject arises from a study made by \textit{J. Brun} and \textit{André Hirschowitz} on the stratification of \(Hilb^ N{\mathbb{P}}^ 2\) by ``postulation''. The methods and techniques we use are those developed by \textit{A. Iarrobino}, \textit{J. Briançon} and \textit{M. Granger}, who studied the geometry of \(Hilb^ N{\mathbb{C}}\{x,y\}\). There are essentially three steps which could be summarized as follows: First, consider the subschemes Z which are supported on one point and contain a fixed subscheme S of length r: we obtain a scheme of dimension \(N-r\) and list its irreducible components. Fix a line L and consider those Z which have at least r points on L: we prove that the corresonding scheme is irreducible of dimension \(2N-r\). Fix only N and r (r at least 2): we prove that the corresponding Hilbert scheme is irreducible of dimension \(2N-r\).
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 32S60
Mathematics Subject Classification ID: 14D99
zbMATH DE Number: 4081702
zbMATH Keywords: vertical-escalier
zbMATH Keywords: Zariski tangent space
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: stratification
zbMATH Keywords: postulation
MaRDI profile type: Publication
cites work: Q3701517
cites work: Description de Hilb sup(n) C{X,Y}
cites work: Le problème de Brill-Noether pour les idéaux de ${P}\sp 2$
cites work: Algebraic Families on an Algebraic Surface
cites work: Q5732721
OpenAlex ID: W182027630
label: Sur l'alignement dans les schémas de Hilbert ponctuels du plan. La famille des N-uplets de \({\mathbb{P}}^ 2\) contenant au moins r points sur une droite. (Alignment in the punctual Hilbert schemes of the plane. The family of N-tuples containing at least r points on a line)
description: scientific article; zbMATH DE number 4081702