Constructions of k-regular maps using finite local schemes

Kurzanzeige
dc.contributor.authorBuczyński, Jarosław
dc.contributor.authorJanuszkiewicz, Tadeusz
dc.contributor.authorJelisiejew, Joachim
dc.contributor.authorMichalek, Mateusz
dc.date.accessioned2021-01-26T12:39:47Z
dc.date.available2021-01-26T12:39:47Z
dc.date.issued2019eng
dc.description.abstractA continuous map Rm→RN or Cm→CN 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.eng
dc.description.versionpublishedeng
dc.identifier.doi10.4171/JEMS/873eng
dc.identifier.urihttps://kops.uni-konstanz.de/handle/123456789/52566
dc.language.isoengeng
dc.rightsterms-of-use
dc.rights.urihttps://rightsstatements.org/page/InC/1.0/
dc.subjectk-regular embeddings, secants, punctual Hilbert scheme, finite Gorenstein schemeseng
dc.subject.ddc510eng
dc.titleConstructions of k-regular maps using finite local schemeseng
dc.typeJOURNAL_ARTICLEeng
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kops.citation.bibtex
@article{Buczynski2019Const-52566,
  year={2019},
  doi={10.4171/JEMS/873},
  title={Constructions of k-regular maps using finite local schemes},
  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}
}
kops.citation.iso690BUCZYŃSKI, 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). 2019, 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873deu
kops.citation.iso690BUCZYŃSKI, 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). 2019, 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873eng
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    <dcterms:abstract xml:lang="eng">A continuous map R&lt;sup&gt;m&lt;/sup&gt;→R&lt;sup&gt;N&lt;/sup&gt; or C&lt;sup&gt;m&lt;/sup&gt;→C&lt;sup&gt;N&lt;/sup&gt; 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.</dcterms:abstract>
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kops.sourcefieldJournal of the European Mathematical Society. European Mathematical Society (EMS). 2019, <b>21</b>(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873deu
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kops.sourcefield.plainJournal of the European Mathematical Society. European Mathematical Society (EMS). 2019, 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873eng
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source.periodicalTitleJournal of the European Mathematical Societyeng
source.publisherEuropean Mathematical Society (EMS)eng

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