Constructions of k-regular maps using finite local schemes

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BUCZYŃSKI, Jarosław, Tadeusz JANUSZKIEWICZ, Joachim JELISIEJEW, Mateusz MICHALEK, 2019. Constructions of k-regular maps using finite local schemes. In: Journal of the European Mathematical Society. European Mathematical Society (EMS). 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873

@article{Buczynski2019Const-52566, title={Constructions of k-regular maps using finite local schemes}, year={2019}, doi={10.4171/JEMS/873}, number={6}, volume={21}, issn={1435-9855}, journal={Journal of the European Mathematical Society}, pages={1775--1808}, author={Buczyński, Jarosław and Januszkiewicz, Tadeusz and Jelisiejew, Joachim and Michalek, Mateusz} }

Michalek, Mateusz eng 2021-01-26T12:39:47Z 2021-01-26T12:39:47Z Buczyński, Jarosław Jelisiejew, Joachim terms-of-use 2019 A continuous map R<sup>m</sup>→R<sup>N</sup> or C<sup>m</sup>→C<sup>N</sup> 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, but there are very few results on the existence of such maps for particular values m and k. Using 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≤9, and we provide explicit examples for k≤5. We also provide upper bounds for arbitrary m and k. Buczyński, Jarosław Januszkiewicz, Tadeusz Constructions of k-regular maps using finite local schemes Januszkiewicz, Tadeusz Jelisiejew, Joachim Michalek, Mateusz

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