Whitney equisingularity in families of generically reduced curves

TL;DR

This paper investigates Whitney equisingularity in families of generically reduced curves, establishing equivalences with strong simultaneous resolution and invariance of key invariants, extending known results from reduced to generically reduced cases.

Contribution

It extends the characterization of Whitney equisingularity to flat deformations of generically reduced curves, linking it to strong resolution and invariants like Milnor number and multiplicity.

Findings

Whitney equisingularity is equivalent to strong simultaneous resolution.

Whitney equisingularity corresponds to invariance of Milnor number and multiplicity.

Topological triviality relates to Cohen-Macaulay property of a local ring.

Abstract

In this work we study equisingularity in a one-parameter flat family of generically reduced curves. We consider some equisingular criteria as topological triviality, Whitney equisingularity and strong simultaneous resolution. In this context, we prove that Whitney equisingularity is equivalent to strong simultaneous resolution and it is also equivalent to the constancy of the Milnor number and the multiplicity of the fibers. These results are extensions to the case of flat deformations of generically reduced curves, of known results on reduced curves. When the family $(X,0)$ is topologically trivial, we also characterize Whitney equisingularity through Cohen-Macaulay property of a certain local ring associated to the parameter space of the family.