The jump of the Milnor number in the X_9 singularity class

TL;DR

This paper proves that for singularities in the X_9 class, the minimal non-zero change in the Milnor number during deformation is exactly 2, providing a precise understanding of their singularity behavior.

Contribution

It establishes that all X_9 singularities have a Milnor number jump of 2, clarifying their deformation properties and contributing to singularity theory.

Findings

Milnor number jump for X_9 singularities is 2

Provides a classification of deformation behavior in X_9 class

Enhances understanding of singularity invariants

Abstract

The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)).$ We prove that for the singularities in the $X_9$ singularity class their jumps are equal to 2.