Resumen de A weak version of the Mond conjecture

Roberto Giménez Conejero, Juan José Nuño Ballesteros

• We prove that a map germ f:(C,S) - (Cn +, 0) with isolated instability is stable if and only if ui(f)= 0, where ui (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 > 2, provided that (n,n +1) are nice dimensions in Mather’s sense (so ui (f) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the Ae-codimension of f is < ui(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.