Equinormalizable theory for plane curve singularities with embedded points and the theory of equisingularity

TL;DR

This paper establishes criteria for equinormalizability of families of generically reduced plane curve singularities, utilizing invariants like the δ-invariant and concepts of I-equisingularity, and explores their equivalence.

Contribution

It introduces new criteria based on δ-invariant and I-equisingularity for determining equinormalizability of plane curve singularities.

Findings

Criteria based on δ-invariant for equinormalizability.

Criteria based on I-equisingularity for families.

Equivalence of different equisingularity notions.

Abstract

In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $\delta$-invariant of a (non-reduced) curve singularity which is introduced by Br\"{u}cker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studied.