Disentangling mappings defined on ICIS

We study germs of hypersurfaces \((Y,0)\subset (\mathbb {C}^{n+1},0)\) $(Y,0)\subset ({\mathbb{C}}^{n+1},0)$ that can be described as the image of \({\mathscr {A}}\) $\mathcal{A}$ -finite mappings \(f:(X,S)\rightarrow (\mathbb {C}^{n+1},0)\) $f:(X,S)\to ({\mathbb{C}}^{n+1},0)$ 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 \(X=\mathbb {C}^n\) $X={\mathbb{C}}^n$ , which controls the image Milnor number \(\mu _I(X,f)\) ${\mu }_I(X,f)$ . We apply these results to prove the case \(n=2\) $n=2$ of the generalised Mond conjecture, which states that \({\mu _I(X,f)\ge \text{codim}_{\mathscr {A}_e}(X,f)}\) ${\mu }_I(X,f)\ge {\text{codim}}_{{\mathcal{A}}_e}(X,f)$ , with equality if (Y, 0) is weighted homogeneous.