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)
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 \(2N-r\).
scientific article; zbMATH DE number 4081702