Specialization to the tangent cone and Whitney Equisingularity

Abstract

Let (X, 0) be a reduced, equidimensional germ of analytic singularity with reduced tangent cone (CX,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 → 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 this conditions.