Constant Milnor number implies constant multiplicity for quasihomogeneous singularities
type: Item
instance of: scholarly article
title: Constant Milnor number implies constant multiplicity for quasihomogeneous singularities
zbMATH Open document ID: 0594.32021
DOI: 10.1007/BF01172153
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155163
review text: The author studies that the multiplicity does not change for certain topologically trivial deformations of an isolated hypersurface singularity. His work applies to all \(\mu\)-constant first order deformations and to all \(\mu\)-constant deformations of a quasihomogeneous singularity, i.e., an isolated hypersurface singularity with \({\mathbb{C}}^*\)-action. His argument uses a valuation test of Lê and Saito. He can apply this to arbitrary isolated singularities. It is known that constant Milnor number implies constant multiplicity in some special cases. In the case of quasihomogeneous (semi-quasihomogeneous) singularities, the author gives a positive answer to Zariski's question whether for a hypersurface singularity the multiplicity is an invariant of the topological type. The statement of his result is the following: Let f be a quasi-homogeneous polynomial with isolated singularity and \(F: ({\mathbb{C}}^ n\times {\mathbb{C}},0)\to ({\mathbb{C}},0)\) a \(\mu\)-constant unfolding of f. Then, \(mult(f_ t)=mult(f)\) for small values of t, where \(f_ t(x)=F(x,t)\) for \((x,y)\in {\mathbb{C}}^ n\times {\mathbb{C}}\).
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14B07
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 3955400
zbMATH Keywords: multiplicity for isolated singularity
zbMATH Keywords: deformation of
zbMATH Keywords: singularities
zbMATH Keywords: quasihomogeneous singularity
zbMATH Keywords: isolated hypersurface singularity
zbMATH Keywords: Milnor number
zbMATH Keywords: \(\mu \) -constant unfolding
author: Gert-Martin Greuel
reviewed by: S. Ohyanagi
MaRDI profile type: Publication
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OpenAlex ID: W2001367859
description: scientific article; zbMATH DE number 3955400