A weak version of Mond's conjecture

Abstract

We prove that a map germ $f:({\mathbb{C}}^n,S)\to ({\mathbb{C}}^{n+1},0)$ with isolated instability is stable if and only if ${\mu }_I(f)=0$, where ${\mu }_I(f)$ is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that $f$ has corank one. The proof here is also valid for corank $\ge 2$, provided that $(n,n+1)$ are nice dimensions in Mather's sense (so ${\mu }_I(f)$ is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the ${\mathcal{A}}_e$-codimension of $f$ is $\le {\mu }_I(f)$, with equality if $f$ is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of $f$ is a hypersurface.

18 pages, 5 figures