On the collisions of singular points of complex algebraic plane curves

TL;DR

This paper investigates how two singular points on complex algebraic plane curves merge during generic degenerations, classifying possible outcomes and establishing bounds on resulting singularity invariants.

Contribution

It introduces a precise notion of generic degeneration, proposes a classification method for linear singularity types, and provides bounds and conditions for singularity invariants during merging.

Findings

Established a strict upper bound on the multiplicity of merged singularities.

Proposed a classification method for generic degenerations of linear singularity types.

Provided a sufficient condition for constant delta collision.

Abstract

We study the "generic" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is proposed in the case of linear singularity types. We discuss possible bounds on the singularity invariants of the resulting type in terms of the initial types. In particular the strict upper bound on the resulting multiplicity is proved and a sufficient condition for $\delta=const$ collision is given.