Abstract
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.
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Z. Jelonek was supported by Polish National Science Centre Grant No. 2013/09/B/ST1/04162. M. Lasoń was supported by the Polish Ministry of Science and Higher Education Iuventus Plus Grant No. 0382/IP3/2013/72.
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Jelonek, Z., Lasoń, M. Quantitative properties of the non-properness set of a polynomial map. manuscripta math. 156, 383–397 (2018). https://doi.org/10.1007/s00229-017-0965-0
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DOI: https://doi.org/10.1007/s00229-017-0965-0