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 $\mathcal{A}$-finite mappings $f:(X,S)\to ({\mathbb{C}}^{n+1},0)$ defined on an ICIS $(X,S)$ of dimension $n$. We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-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 ${\mu }_I(X,f)\ge codi{m}_{{\mathcal{A}}_e}(X,f)$, with equality if $(Y,0)$ is weighted homogeneous.

19 pages