Maximal families of superabundant point systems in \(\mathbb{P}^ 2\)
type: Item
instance of: scholarly article
title: Maximal families of superabundant point systems in \(\mathbb{P}^ 2\)
zbMATH Open document ID: 0779.14018
DOI: 10.1007/BF01445236
author: Marc-Antoine Coppo
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164895
review text: The author addresses the problem of classifying point systems (or \(N\)- uples of points) in the projective plane. More precisely, if \(Z\subset\mathbb{P}^ 2\) is a finite subscheme of degree \(N\) and \(d\) is a positive integer, consider the linear system \(I_ Z(d)\) of plane curves of degree \(d\) containing \(Z\). The subscheme \(Z\) is called superabundant (for the linear system of plane curves of degree \(d)\) if the dimension of \(I_ Z(d)\) is strictly greater than the expected dimension \({d+2\choose 2}-1-N\). The set of superabundant subschemes \(Z\) is a subscheme \(W_ N[d]\) of the Hilbert scheme \(\text{Hilb}^ N\mathbb{P}^ 2\). The author's main result is to give the dimension of \(W_ N[d]\) and determine when it is irreducible, in terms of \(d\) and \(N\).
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 59944
zbMATH Keywords: classifying point systems
zbMATH Keywords: superabundant subschemes
zbMATH Keywords: Hilbert scheme
reviewed by: Ragni Piene
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59944