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 $g:({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0)$ 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.

References in corpus (2)

• Topological Triviality of Families of Singular Surfaces
• Roots of Formal Power Series and New Theorems on Riordan Group Elements