A weak version of the Mond conjecture

Abstract

We prove that a map germ $$f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( C n , S ) → ( C n + 1 , 0 ) with isolated instability is stable if and only if $$\mu _I(f)=0$$ μ I ( f ) = 0 , where $$\mu _I(f)$$ μ 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 $$\ge 2$$ ≥ 2 , provided that $$(n,n+1)$$ ( n , n + 1 ) are nice dimensions in Mather’s sense (so $$\mu _I(f)$$ μ I ( f ) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $$\mathscr {A}_e$$ A e -codimension of f is $$\le \mu _I(f)$$ ≤ μ 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.