The biLipschitz Geometry of Complex Curves: An Algebraic Approach

Abstract

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On the fifth Whitney cone of a complex analytic curve

“…In particular, we have the equality between Milnor numbers, µ(π(X), 0) = µ( X, 0), which is a contradiction since µ( X, 0) < µ(π(X), 0) by [1, Prop. IV.2] or [5,Prop. 8.4.6].…”

“…Remark 4.12 Since all C 5 -generic projections of a germ of curve have the same topological type (see [5,Prop. 8.4.6] or [1, Prop.…”

Nash Modification on Toric Curves

“…where ⊗ denotes the analytic tensor product which is the operation on the analytic algebras that corresponds to the fibre product of analytic spaces (for more details see [2] and [11]). Definition 1.1.…”

“…A linear projection π : (C n , 0) → (C m , 0) with kernel D is called C 5 -general (or generic) with respect to (X , 0) if it is transversal to the cone C 5 (X , 0), meaning D C 5 (X , 0) = {0}. When π is generic, it induces a homeomorphism between (X , 0) and its image (π(X ), 0), and these two germs have the same multiplicity, for a detailed explanation see [11,Section 8.4].…”

“…It was first studied by Pham and Teissier in [18] and its associated germ (X s , 0) is important in the study of biLipschitz equisingularity. The case of curves is pretty well understood and described in [4], [7], [10], [11], [17], [18]. However not much is known in higher dimensions.…”