Augmentation of singularities: $\mu/\tau$-type conjectures and simplicity

Abstract

For function germs $g:(\mathbb C^n,0)\to (\mathbb C,0)$ it is well known that $1\leq\frac{\mu(g)}{\tau(g)}$ and it has recently been proved by Liu that $\frac{\mu(g)}{\tau(g)}\leq n$. We give an upper bound for the codimension of map-germs $f:(\mathbb C^n,0)\to (\mathbb C^p,0)$ given as augmentations of other map-germs with which we prove the analog to the first inequality (known as Mond's conjecture) for augmentations $h:(\mathbb C^n,0)\to (\mathbb C^{n+1},0)$. Furthermore, we show that the quotient given by the image Milnor number and the codimension of any augmentation in the pair of dimensions $(n,n+1)$ is less than $\frac{1}{4}(n+1)^2$ and prove the analog to the second inequality for map-germs with $n=1$ and augmentations with $n=2,3$. We then prove a characterization of when a map-germ is an augmentation, finding a counterexample for the characterization given by Houston. Next, we give sufficient conditions for when the augmentation is independent of the choice of stable unfolding by studying different notions of equivalence of unfoldings. Moreover, these results allow us to give sufficient conditions for the simplicity of an augmentation, providing context to locate the moduli for non-simple augmentations.