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

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.