Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$

Abstract

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \mathbb{C}^3 \to \mathbb{C}^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j) \leq 2$ for $1 \leq i < j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U \subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ is $\mathcal{A}$-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping $(f_1,f_2,f_3):\mathbb{C}^3 \to \mathbb{C}^3$, where $\deg f_i=d_i$.