A weak version of the Mond conjecture

• Giménez Conejero, R. ^[2] ; Nuño-Ballesteros, J. J. ^[1]
  1. [1] Universitat de València

     Universitat de València

     Valencia, España

     
  2. [2] Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, 1053, Hungary
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 3, 2024, págs. 753-770
• Idioma: inglés
• DOI: 10.1007/s13348-023-00411-x
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• Resumen

  • We prove that a map germ f:(\mathbb {C}^n,S)\rightarrow (\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 \mathscr {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.

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