A note on 1-parameter stable unfoldings

TL;DR

This paper characterizes conditions for when a map-germ admits a 1-parameter stable unfolding and shows that many finitely determined map-germs of multiplicity 4 do not admit such unfoldings.

Contribution

It provides two new characterizations for the existence of 1-parameter stable unfoldings and demonstrates the existence of infinitely many map-germs that lack such unfoldings.

Findings

Two characterizations of 1-parameter stable unfoldings

Existence of infinitely many finitely determined map-germs of multiplicity 4 without such unfoldings

Insights into the structure of map-germs in singularity theory

Abstract

We give two characterisations of when a map-germ admits a 1-parameter stable unfolding, one related to the $\mathscr K_e$-codimension and another related to the normal form of a versal unfolding. We then prove that there are infinitely many finitely determined map-germs of multiplicity 4 from $\mathbb K^3$ to $\mathbb K^3$ which do not admit a 1-parameter stable unfolding.