Specialization to the tangent cone and Whitney Equisingularity

TL;DR

This paper establishes that the absence of exceptional cones is crucial for Whitney's conditions to hold during the specialization to the tangent cone of an analytic singularity, advancing understanding of equisingularity.

Contribution

It proves a necessary and sufficient condition linking exceptional cones to Whitney's conditions in the specialization to tangent cones, generalizing previous hypersurface results.

Findings

Absence of exceptional cones ensures Whitney's conditions along the parameter axis.

The result generalizes L extsuperscript{e} and Teissier's hypersurface theorem to higher dimensions.

Provides a criterion for Whitney equisingularity in the context of tangent cone specialization.

Abstract

Let (X,0) be a reduced, equidimensional germ of analytic singularity with reduced tangent cone (C_{X,0},0). We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part \X^0 of the specialization to the tangent cone \phi: \X \to \C to satisfy Whitney's conditions along the parameter axis Y. This result is a first step in generalizing to higher dimensions L\^e and Teissier's result for hypersurfaces of \C^3 which establishes the Whitney equisingularity of X and its tangent cone under this conditions.