On finite polynomial mappings

Abstract

Let $X\subset {\mathbb{C}}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F:X\to {\mathbb{C}}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear mappings $\mathcal{L}({\mathbb{C}}^n,{\mathbb{C}}^m)$ such that for every $L\in U$ the mapping $F+L$ is a finite mapping. Moreover, we can choose $U$ in this way, that all mappings $F+L;L\in U$ are topologically equivalent.