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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). (French) Zbl 0662.14001
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 J. Brun and André Hirschowitz on the stratification of \(Hilb^ N{\mathbb{P}}^ 2\) by “postulation”. The methods and techniques we use are those developed by A. Iarrobino, J. Briançon and 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 \(2N-r\).
Reviewer: J.Yameogo

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14N05 Projective techniques in algebraic geometry
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
14D99 Families, fibrations in algebraic geometry
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