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

TL;DR

This paper introduces a Jacobian module for holomorphic map germs, establishing bounds for invariants like the $ ext{A}$-codimension and image Milnor number, and links Cohen-Macaulay properties of modules to Mond's conjecture.

Contribution

It defines a new Jacobian module for map germs and connects its properties to Mond's conjecture, providing a new approach to prove it.

Findings

The module $M(f)$ bounds the $ ext{A}$-codimension.

The module $M_y(F)$ bounds the image Milnor number.

Cohen-Macaulayness of $M_y(F)$ implies Mond's conjecture.

Abstract

Given a germ of holomorphic map $f$ from $\mathbb C^n$ to $\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\mathbb C$ is an upper bound for the $\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particular, if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$. Furthermore, if $f$ is quasi-homogeneous, then 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 suffices to prove it in a suitable family of examples.