A weak version of the Mond conjecture

Gimenez Conejero, Roberto and Nuno-Ballesteros, J. J. (2024) A weak version of the Mond conjecture. COLLECTANEA MATHEMATICA. ISSN 0010-0757

Preview

Text s13348-023-00411-x.pdf - Published Version Download (767kB) | Preview

Abstract

We prove that a map germ f : (Cn , S) → (Cn+1, 0) with isolated instability is stable if and only if μI ( f ) = 0, where μI ( 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 μI ( 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 ≤ μI ( 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.

Actions (login required)

Edit Item