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Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities. (English) Zbl 1408.32029
Summary: 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:
 32S15 Equisingularity (topological and analytic) 32S25 Complex surface and hypersurface singularities 32S05 Local complex singularities
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##### References:
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