Equisingularity of map germs from a surface to the plane

TL;DR

This paper investigates the invariants associated with map germs from a surface to the plane, establishing conditions for Whitney and Zariski equisingularity through relations between invariants and stability analysis.

Contribution

It introduces new relations between invariants of map germs and provides necessary and sufficient conditions for Whitney equisingularity in families of surface singularities.

Findings

Relations between invariants of map germs and stability conditions

Necessary and sufficient conditions for Whitney equisingularity

Equivalence of Zariski and Whitney equisingularity under constant cusp and fold counts

Abstract

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_\epsilon$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. As an application, we show that a family $(X_t,0)$ is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant on $t$.