manuscripta mathematica

, Volume 156, Issue 3–4, pp 383–397 | Cite as

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

  • Zbigniew JelonekEmail author
  • Michał Lasoń
Open Access


Let f be a generically finite polynomial map \(f: \mathbb {C}^n\rightarrow \mathbb {C}^m\) 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 \(f: X\rightarrow \mathbb {R}^m\) 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 \(2d_1d_2\). 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.

Mathematics Subject Classification

14R25 14P10 14R99 


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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Mathematics of the Polish Academy of SciencesWarszawaPoland

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