Quantitative properties of the non-properness set of a polynomial map

Abstract

Let $f$ be a generically finite polynomial map $f:{\mathbb{C}}^n\to {\mathbb{C}}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric curves of degree at most $d-1$. This bound is best possible. Moreover, we prove that if $X\subset {\mathbb{R}}^n$ is a closed algebraic set covered by parametric curves, and $f:X\to {\mathbb{R}}^m$ is a generically finite polynomial map, then the set $S_f$ of non-properness of $f$ is also covered by parametric curves. Moreover, if $X$ is covered by parametric curves of degree at most $d_1$, and the map $f$ has degree $d_2$, then the set $S_f$ is covered by parametric curves of degree at most $2d_1{d}_2$. As an application of this result we show a real version of the Białynicki-Birula theorem: Let $G$ be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety $X\subset {\mathbb{R}}^n$. Then the set $Fix(G)$ of fixed points has no isolated points.

final version, 13 pages

References in corpus (1)

• The set of fixed points of a unipotent group

Cited by in corpus (2)

• Some observations on the properness of identity plus linear powers: part 2
• Quantitative properties of the non-properness set of a polynomial map, a positive characteristic case