On finite polynomial mappings

TL;DR

This paper proves that for polynomial mappings from a smooth affine variety to complex space, a generic linear perturbation results in a finite mapping, and all such perturbations are topologically equivalent.

Contribution

It establishes that a generic linear perturbation of a polynomial map yields a finite mapping and that these mappings are topologically equivalent.

Findings

Existence of a Zariski open dense subset of linear maps making the sum finite.

All such perturbed mappings are topologically equivalent.

The result applies to smooth irreducible affine varieties of any dimension.

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.