On the conflict variety of an isolated hypersurface singularity
type: Item
instance of: scholarly article
title: On the conflict variety of an isolated hypersurface singularity
zbMATH Open document ID: 0767.32019
DOI: 10.1007/BF02571427
author: Klaus Wirthmüller
published in: Mathematische Zeitschrift
publication date: 1993-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/174440
review text: Let \(f:(\mathbb{C}^ n,0)\to(\mathbb{C},0)\) be a holomorphic germ with isolated singularity and \(F:(\mathbb{C}^ k\times\mathbb{C}^ n,0)\to(\mathbb{C}^ k\times\mathbb{C},0)\) its versal unfolding. The discriminant \(\Delta(F)\) of this unfolding is a branched cover of \((\mathbb{C}^ k,0)\). The branch locus of this cover \(D(F)\) contains as one irreducible component the bifurcation variety \(\text{Bif}(F)\). The union of all the other components is called the conflict variety \(\text{Con}(F)\). In the article the reduced hypersurfaces \(\text{Con}(F)\) and \(\text{Bif}(F)\) in \((\mathbb{C}^ k,0)\) are studied. For instance it is proved: \(\text{Con}(F)\) is irreducible except the cases of \(A_ 1\), \(A_ 2\) and \(D_ n\)- singularities. The singularity is uniquely determined (modulo stably right --- left equivalence) by \(\text{Con}(F)\cup\text{Bif}(F)\).
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 58K35
zbMATH DE Number: 108920
zbMATH Keywords: catastrophe theory
zbMATH Keywords: unfolding
zbMATH Keywords: discriminant
zbMATH Keywords: bifurcation variety
zbMATH Keywords: conflict variety
MaRDI profile type: Publication
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OpenAlex ID: W2328586228
description: scientific article; zbMATH DE number 108920