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Specialization to the tangent cone and Whitney equisingularity

Arturo Giles Flores — 2013

Bulletin de la Société Mathématique de France

Let $(X, 0)$ be a reduced, equidimensional germ of an 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 $ϕ : 𝔛 \to ℂ$ to satisfy Whitney’s conditions along the parameter axis $Y$ . This result is a first step in generalizing to higher dimensions Lê and Teissier’s result for hypersurfaces of $ℂ^{3}$ which establishes the Whitney equisingularity of $X$ and its tangent cone under...

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