Non-degenerate jumps of Milnor numbers of quasihomogeneous singularities
Volume 123 / 2019
Abstract
Let $f_0: (\mathbb {C}^n,0)\rightarrow (\mathbb {C},0)$ be a holomorphic function germ having an isolated critical point at $0\in \mathbb {C}^n$ and let $[f_0]$ be the singularity generated by $f_0$, i.e. the equivalence class of $f_0$ with respect to right-left holomorphic equivalence. The non-degenerate jump of Milnor number $\lambda ^{ {\rm nd}}(f_0)$ of $f_0$ is the minimal non-zero difference between the Milnor number of $f_0$ and the Milnor number of a generic element of $(f_t)$ among all holomorphic non-degenerate deformations $(f_t)$ of $f_0$. For the class $[f_0]$ we define $\lambda ^{ {\rm nd}}([f_0])$ as the minimum of $\lambda ^{ {\rm nd}}(g_0)$ over $g_0\in [f_0]$. We give a formula for $\lambda ^{ {\rm nd}}([f_0])$ when $f_0$ is quasihomogeneous in two variables.
