Constructions of k-regular maps using finite local schemes

Jarosław Buczyński; Tadeusz Januszkiewicz; Joachim Jelisiejew; Mateusz Michałek

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arXiv · 1511.05707

Constructions of k-regular maps using finite local schemes

Jarosław Buczyński·Tadeusz Januszkiewicz·Joachim Jelisiejew·Mateusz Michałek

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Abstract

A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any $k$ points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m and k. Using the methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k<10, and we provide explicit examples for k<6. We also provide upper bounds for arbitrary m and k.

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BibTeX RIS

Jarosław Buczyński, Tadeusz Januszkiewicz, Joachim Jelisiejew, Mateusz Michałek. 2016-11-07. Constructions of k-regular maps using finite local schemes. https://arxiv.org/abs/1511.05707

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