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.