The image Milnor number and excellent unfoldings

TL;DR

This paper investigates the properties of the image Milnor number for germs with isolated instability, proving its conservation, confirming weak Mond's conjecture, and validating Houston's conjecture for corank 1 cases.

Contribution

It establishes the conservation of the image Milnor number, proves the weak Mond's conjecture, and confirms Houston's conjecture for corank 1 families.

Findings

The image Milnor number is conserved in families.

A germ with zero image Milnor number is stable.

Families with constant image Milnor number are excellent.

Abstract

We show three basic properties on the image Milnor number $\mu_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond's conjecture, which says that $\mu_I(f)=0$ if and only if $f$ is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(f_t)$ constant is excellent in Gaffney's sense. By technical reasons, in the two last properties we consider only the corank 1 case.