Milnor numbers in deformations of homogeneous singularities

TL;DR

This paper investigates how Milnor numbers, which measure the complexity of plane curve singularities, change under deformations of homogeneous singularities, providing a complete description for non-degenerate cases.

Contribution

It offers a complete characterization of Milnor numbers in deformations of homogeneous singularities for non-degenerate cases and partial results for the general case.

Findings

Complete description of Milnor numbers for non-degenerate deformations

Partial results on Milnor numbers in the general case

Insights into the behavior of singularities under deformation

Abstract

Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.