Disentangling mappings defined on ICIS

Abstract

We study germs of hypersurfaces $(Y,0)\subset (\mathbb C^{n+1},0)$ that can be described as the image of $\mathscr A$-finite mappings $f:(X,S)\rightarrow (\mathbb C^{n+1},0)$ defined on an ICIS $(X,S)$ of dimension $n$. We extend the definition of the Jacobian module given by Fern\'andez de Bobadilla, Nu\~no-Ballesteros and Pe\~nafort-Sanchis when $X=\mathbb C^n$, which controls the image Milnor number $\mu_I(X,f)$. We apply these results to prove the case $n=2$ of the generalised Mond conjecture, which states that $\mu_I(X,f)\geq codim_{\mathscr A_e} (X,f)$, with equality if $(Y,0)$ is weighted homogeneous.