On the fifth Whitney cone of a complex analytic curve

Abstract

From a procedure to calculate the $C_5$-cone of a reduced complex analytic curve $X\subset {\mathbb{C}}^n$ at a singular point $0\in X$, we extract a collection of integers that we call {\it auxiliary multiplicities} and we prove they characterize the Lipschitz type of complex curve singularities. We then use them to improve the known bounds for the number of irreducible components of the $C_5$-cone. We finish by giving an example showing that in a Lipschitz equisingular family of curves the number of planes in the $C_5$-cone may not be constant.