Minimal Whitney stratification and Euler obstruction of discriminants

Abstract

In this work we show how to obtain the minimal Whitney stratification of the discriminant of finitely determined map germs from $(\mathbb {C}^m,0)$ to $(\mathbb {C}^p,0)$, of corank one if $n<p$, and only with $A_k$ singularities, when $m = n+p$ with $n \ge 0$. We apply the theory developed by Gaffney which shows how to compute a Whitney stratification of discriminants of any finitely determined holomorphic map germ in the nice dimensions of Mather, or in its boundary. For the pairs cited above we show that both stratifications coincide. We also compute the local Euler obstruction at 0 in a class of discriminants of finitely determined map germs from $\mathbb {C}^{n+p}$ to $\mathbb {C}^p$ with $n\ge 0$ and only with $A_k$ singularities.