A Jacobian module for disentanglements and applications to Mond’s conjecture

Abstract

Let $f:({\mathbb {C}}^n,S)\rightarrow ({\mathbb {C}}^{n+1},0)$ be a germ whose image is given by $g=0$. We define an ${\mathcal {O}}_{n+1}$-module M(g) with the property that ${\mathscr {A}}_e$-${\text {codim}}(f)\le {\text {dim}}_{\mathbb {C}}M(g)$, with equality if f is weighted homogeneous. We also define a relative version $M_y(G)$ for unfoldings F, in such a way that $M_y(G)$ specialises to M(g) when G specialises to g. The main result is that if $(n,n+1)$ are nice dimensions, then ${\text {dim}}_{\mathbb {C}}M(g)\ge \mu _I(f)$, with equality if and only if $M_y(G)$ is Cohen–Macaulay, for some stable unfolding F. Here, $\mu _I(f)$ denotes the image Milnor number of f, so that if $M_y(G)$ is Cohen–Macaulay, then Mond’s conjecture holds for f; furthermore, if f is weighted homogeneous, Mond’s conjecture for f is equivalent to the fact that $M_y(G)$ is Cohen–Macaulay. Finally, we observe that to prove Mond’s conjecture, it is enough to prove it in a suitable family of examples.