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Géométrie des schémas de Hilbert ponctuels. (French) Zbl 0534.14002
Let \({\mathfrak O}_ r={\mathbb{C}}\{x_ 1,...,x_ r\}\), and let \(Hilb^ n{\mathfrak O}_ r\) denote the Hilbert scheme parametrizing ideals \(I\subseteq {\mathfrak O}_ r\) of finite colength \(=n\). Let \(Z=(Hilb^ n{\mathfrak O}_ 2)_{red}\), and for each positive integer \(\mu\), let \(Z_{\mu}\) denote the locally closed subset corresponding to ideals of order \(\mu\). The two most important results in the paper under review are: (1) \(Z_{\mu +1}\) is contained in the closure of \(Z_{\mu}\). - (2) The singular locus of Z is the closure of \(Z_ 2.\)
Statement (1) was conjectured by J. Briançon [Invent. Math. 41, 45-89 (1977; Zbl 0353.14004)], and gives an alternative proof of the main theorem of that paper: that Z is irreducible. - In the case \(r>2\), the same methods give a proof that most complete intersections cannot be deformed to an ideal of imbedding dimension 1. In the last part of the paper, returning to the case \(r=2\), the decomposition of Z into Hilbert- Samuel strata is studied. Many special cases and examples are given, but no general results on the incidence of the strata.
Reviewer: S.A.Strømme

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
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References:
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