On the topology of the transversal slice of a quasi-homogeneous map germ

TL;DR

This paper analyzes the topological structure of generic hyperplane sections of quasi-homogeneous map germs from ^2 to ^3, providing conditions for quasi-homogeneity and Whitney equisingularity in unfoldings.

Contribution

It describes the embedded topological type of hyperplane sections of such map germs in terms of weights and degrees, and establishes criteria for quasi-homogeneity and Whitney equisingularity.

Findings

The topological type of hyperplane sections is characterized by weights and degrees.

A necessary condition for quasi-homogeneity involves the characteristic exponents of associated plane curves.

Unfoldings adding terms of the same degrees are Whitney equisingular.

Abstract

We consider a corank $1$, finitely determined, quasi-homogeneous map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $\gamma_f$, in terms of the weights and degrees of $f$. As a consequence, a necessary condition for a corank $1$ finitely determined map germ $g:(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of $f$ which adds only terms of the same degrees as the degrees of $f$ is Whitney equisingular.