On the equi-normalizable deformations of singularities of complex plane curves

TL;DR

This paper investigates equi-normalizable deformations of complex plane curve singularities, introducing a dual graph invariant to analyze how singularities split while preserving the delta invariant, with detailed studies on specific singularity types.

Contribution

It introduces the dual graph as a new invariant for equi-normalizable deformations and provides bounds on classical invariants during such deformations.

Findings

Dual graph imposes restrictions on singularity collisions.

Bounds established on invariant variations in deformations.

Detailed analysis of ordinary multiple points and specific singularity types.

Abstract

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple points, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type $x^p+y^{pk}$ into the collections of $A_k$'s.