A counterexample to the theorem of Beppo Levi in three dimensions
type: Item
instance of: scholarly article
title: A counterexample to the theorem of Beppo Levi in three dimensions
zbMATH Open document ID: 0688.14012
DOI: 10.1007/BF01393974
author: M. Spivakovsky
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143676
review text: The traditional method for embedded resolution of singularities of an algebraic variety is to blow up successively non-singular subvarieties contained in the locus S of points where the Hilbert-Samuel function is maximal. The aim is to find a finite sequence of blowing ups such that finally the Hilbert-Samuel function decreases strictly everywhere. A natural idea for finding an algorithm for resolution was to blow up \textit{maximal} non-singular subvarieties of S. This is known to work for surfaces (see ``theorem of B. Levi'' in the paper). Here the author shows by a counterexample (the hypersurface \(z^ 2+x^ 2y^ 2+wxy^ 3+wx^ 3y=0\) in \(k^ 4\), where char(k)\(\neq 2)\) that an analogous theorem is not true in higher dimensions.
reviewed by: Manfred Herrmann
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 4127372
zbMATH Keywords: embedded resolution of singularities
zbMATH Keywords: Hilbert-Samuel function
MaRDI profile type: Publication
cites work: Q5525147
cites work: On the characteristic functions of a local ring
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Reduction of the singularities of algebraic three dimensional varieties
Wikidata QID: Q124818216
OpenAlex ID: W2065687602
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description: scientific article; zbMATH DE number 4127372