Morsifications of real plane curve singularities

TL;DR

This paper proves the existence of real morsifications for a broad class of real plane curve singularities and explores their connection to the singularity's topology, extending previous results.

Contribution

It extends the existence of real morsifications to singularities with complex conjugate branches and relates these to the topological classification of singularities.

Findings

Existence of real morsifications for singularities with complex conjugate branches.

Extension of Balke-Kaenders theorem to arbitrary real morsifications.

The A'Campo--Gusein-Zade diagram determines the topological type of the singularity.

Abstract

A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for real plane curve singularities having arbitrary real local branches and pairs of complex conjugate branches satisfying some conditions. This was known before only in the case of all local branches being real (A'Campo, Gusein-Zade). We also discuss a relation between real morsifications and the topology of singularities, extending to arbitrary real morsifications the Balke-Kaenders theorem, which states that the A'Campo--Gusein-Zade diagram associated to a morsification uniquely determines the topological type of a singularity.