2016/04/08 by Javier Fernández de Bobadilla, J. Fernández de Bobadilla, J. J. Nuño-Ballesteros +6
Mathematics · #58K15 (Primary) #58K40 #58K65 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #math.AG #msc:58K15 #msc:58K40 #msc:58K65
paper · pdf · doi:10.48550/arxiv.1604.02422
19 pages
arxiv created 2016/04/08 · openalex publication_date 2016/04/08 · arxiv updated 2016/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Given a germ of holomorphic map f from \mathbb Cn to \mathbb Cn+1, we define a module M(f) whose dimension over \mathbb C is an upper bound for the \mathscr A-codimension of f, with equality if f is weighted homogeneous. We also define a relative version My(F) of the module, for unfoldings F of f. The main result is that if (n,n+1) are nice dimensions, then the dimension of M(f) over \mathbb C is an upper bound of the image Milnor number of f, with equality if and only if the relative module My(F) is Cohen-Macaulay for some stable unfolding F. In particular, if My(F) is Cohen-Macaulay, then we have Mond's conjecture for f. Furthermore, if f is quasi-homogeneous, then Mond's conjecture for f is equivalent to the fact that My(F) is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it suffices to prove it in a suitable family of examples.