Lipschitz Stratification of Complex Hypersurfaces in Codimension 2

TL;DR

This paper proves that the Zariski canonical stratification of complex hypersurfaces in three dimensions is locally bi-Lipschitz trivial along codimension two strata, using Lipschitz vector fields to achieve trivializations.

Contribution

It establishes Lipschitz triviality of Zariski stratifications for complex surface singularities in orm, extending bi-Lipschitz triviality results to non-isolated singularities.

Findings

Zariski stratification is bi-Lipschitz trivial along codimension two strata.

Lipschitz vector fields can be integrated to produce trivializations.

The stratification is given by singular set and polar curves.

Abstract

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated, singularities in $\mathbb{C}^3$. We show that a natural stratification of such a family given by the singular set and the generic family of polar curves provides a Lipschitz stratification in the sense of Mostowski. In particular, such families are bi-Lipschitz trivial by trivializations obtained by integrating Lipschitz vector fields.