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Equisingularity of families of hypersurfaces and applications to mappings. (English) Zbl 1233.32017
A family of isolated singularities is called Whitney equisingular if the singular set of the variety defined by the family is a stratum in a Whitney stratification. Briancon, Speder and Teissier proved that a family is Whitney equisingular if and only if the \(\mu^\ast\)-sequence is constant in the family, here \(\mu^\ast\) is the sequence of the Milnor numbers of generic \(i\)-dimensional plane sections with the singularity. In the case of non-isolated singularities one assumes that the family can be stratified in a way that outside the parameter axis of the family one has a Whitney stratification and seeks conditions that give an equivalence between a collection of topological invariants and Whitney equisingularity of the parameter axis.
In their paper [J. Algebr. Geom. 8, No. 4, 695–736 (1999; Zbl 0971.13021)], T. Gaffney and R. Gassler define the sequence \(\chi^\ast\) of Euler characteristics of the Milnor fibres generalizing \(\mu^\ast\). They define another sequence \(m^\ast\), the relative polar multiplicities, and prove that Whitney equisingularity of a family implies that \((m^\ast, \chi^\ast)\) are constant in the family. The aim of the polar is to give further conditions to ensure the converse.

32S15 Equisingularity (topological and analytic)
32S30 Deformations of complex singularities; vanishing cycles
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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