Degree of $\mathbb{K}$-uniruledness of the non-properness set of a polynomial map

Zbigniew Jelonek, Michał Lasoń

Published 2014-11-18Version 1

Let $X\subset\mathbb{C}^n$ be an affine variety covered by polynomial curves, and let $f:X\rightarrow\mathbb{C}^m$ be a generically finite polynomial map. The first author showed that the set $S_f$, of points at which $f$ is not proper, is covered by polynomial curves. We generalize this result to the field of real numbers. We prove that if $X\subset\mathbb{R}^n$ is a closed semialgebraic set covered by polynomial curves, and $f: X\rightarrow\mathbb{R}^m$ is a generically finite polynomial map, then the set $S_f$ is also covered by polynomial curves. Moreover, if $X$ is covered by curves of degree at most $d_1$, and the map $f$ has degree $d_2$, then the set $S_f$ is covered by polynomial curves of degree at most $2d_1d_2$. For the field of complex numbers we get a better bound by $d_1d_2$, which is best possible. Additionally, if $X=\mathbb{C}^n$ or $X=\mathbb{R}^n$, and a generically finite polynomial map $f$ has degree $d$, then the set $S_f$ is covered by polynomial curves of degree at most $d-1$. This bound is best possible.