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

Given a germ $f:(\mathbb C^n,S)\to (\mathbb C^{n+1},0)$, we define an $\mathcal O_{n+1}$-module $M(f)$ with the property that $\mathcal A_e$-$\mbox{codim}(f)\le \dim_\mathbb C M(f)$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ for unfoldings $F=(u,f_u)$, in such a way that $M_y(F)$ specialises to $M(f)$ when $u=0$. The main result is that if $(n,n+1)$ are nice dimensions, then $\dim_\mathbb C M(f)\ge \mu_I(f)$, with equality if and only if $M_y(F)$ is Cohen-Macaulay, for some stable unfolding $F$. Here, $\mu_I(f)$ denotes the image Milnor number of $f$, so that if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$; furthermore, if $f$ is quasi-homogeneous Mond's conjecture for $f$ is equivalent to the fact that $M_y(F)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it is enough to prove it in a suitable family of examples.