On the fifth Whitney cone of a complex analytic curve

TL;DR

This paper introduces auxiliary multiplicities derived from the fifth Whitney cone of complex analytic curves, which characterize Lipschitz singularity types and refine bounds on the cone's components.

Contribution

It defines auxiliary multiplicities for the $C_5$-cone, linking them to Lipschitz classification and improving bounds on the number of cone components.

Findings

Auxiliary multiplicities characterize Lipschitz types of curve singularities.

Bounds for the number of $C_5$-cone components are improved.

The number of planes in the $C_5$-cone can vary in Lipschitz equisingular families.

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.