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Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

Part of: Singularities

Published online by Cambridge University Press:  11 January 2016

Christophe Eyral  and
Maria Aparecida Soares Ruas
Christophe Eyral
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland, [email protected]
Maria Aparecida Soares Ruas
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13566-590 São Carlos - SP, Brazil, [email protected]
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Abstract

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We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.

As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

MSC classification

Secondary: 32S15: Equisingularity (topological and analytic) 32S25: Surface and hypersurface singularities 32S05: Local singularities
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 29 - 50
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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