Equisingularity in one parameter families of generically reduced curves

TL;DR

This paper investigates conditions for equisingularity in one-parameter families of generically reduced curves, establishing key equivalences and exploring stability issues under certain modifications.

Contribution

It proves the equivalence between Whitney regularity and Zariski's discriminant criterion, and analyzes stability of equisingularity conditions under blow-up and Nash modification.

Findings

Whitney regularity is equivalent to Zariski's discriminant criterion.

Topological triviality implies smoothness of the normalized surface.

Whitney regularity and equisaturation are not stable under certain modifications.

Abstract

We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.