Complements of discriminants of real singularities of type $X_{10}$

TL;DR

This paper provides a comprehensive list of the connected components of the complements of discriminant varieties for certain real singularities, highlighting novel examples and topological properties.

Contribution

It presents the first examples of non semi-quasihomogeneous plane function singularities and analyzes their topological features.

Findings

Connected components of discriminant complements for $X_{10}^3$ and $X_{10}^1$ are listed.

Discriminant complements of $X_9^{ imes}$ and $X_{10}^1$ have non-trivial 1-dimensional homology.

First examples of non semi-quasihomogeneous plane function singularities are provided.

Abstract

A conjecturally complete list of connected components of complements of discriminant varieties (aka wave fronts) of smooth function singularities of type $X_{10}^3$ and $X_{10}^1$ is presented; it are the first examples of not semi-quasihomogeneous plane function singularities. It is shown that the complements of discriminants of singularities of classes $X_9^{\pm}$ and $X_{10}^1$ have non-trivial 1-dimensional homology groups, in contrast to all simple singularity classes.