Disentangling mappings defined on ICIS

TL;DR

This paper investigates hypersurface germs as images of finite mappings on ICIS, extending Jacobian modules, and proves a case of the generalized Mond conjecture relating image Milnor number and codimension.

Contribution

It extends the Jacobian module definition for ICIS and proves the n=2 case of the generalized Mond conjecture relating invariants of hypersurface germs.

Findings

Extended Jacobian module controls the image Milnor number.

Proved the n=2 case of the generalized Mond conjecture.

Established equality conditions for weighted homogeneous cases.

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.