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Lower Bounds for Waldschmidt Constants of Generic Lines in \({\mathbb {P}}^3\) and a Chudnovsky-Type Theorem

  • Marcin Dumnicki
  • Mohammad Zaman Fashami
  • Justyna Szpond
  • Halszka Tutaj-GasińskaEmail author
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Abstract

The Waldschmidt constant \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\) of a radical ideal I in the coordinate ring of \({\mathbb {P}}^N\) measures (asymptotically) the degree of a hypersurface passing through the set defined by I in \({\mathbb {P}}^N\). Nagata’s approach to the 14th Hilbert Problem was based on computing such constant for the set of points in \({\mathbb {P}}^2\). Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore, the research focuses on looking for accurate bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\). In the paper, we deal with \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\), the Waldschmidt constant for s very general lines in \({\mathbb {P}}^3\). We prove that \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2s-1}\rfloor \) holds for all s, whereas the much stronger bound \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2.5 s}\rfloor \) holds for all s but \(s=4\), 7 and 10. We also provide an algorithm which gives even better bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\), very close to the known upper bounds, which are conjecturally equal to \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\) for s large enough.

Keywords

Asymptotic Hilbert function Chudnovsky conjecture containment problem symbolic powers Waldschmidt constants 

Mathematics Subject Classification

14N20 13F20 13P10 14C20 

Notes

Acknowledgements

This research has been carried out while the Zaman Fasham was visiting as a senior graduate student in the Department of Mathematics of the Pedagogical University of Cracow. Dumnicki and Tutaj-Gasińska were partially supported by National Science Centre, Poland, Grant 2014/15/B/ST1/02197, Szpond was partially supported by National Science Centre, Poland, Grant 2018/30/M/ST1/00148. We thank Tomasz Szemberg for helpful remarks.

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Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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

  • Marcin Dumnicki
    • 1
  • Mohammad Zaman Fashami
    • 2
  • Justyna Szpond
    • 3
  • Halszka Tutaj-Gasińska
    • 1
    Email author
  1. 1.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Faculty of MathematicsK. N. Toosi University of TechnologyTehranIran
  3. 3.Department of MathematicsPedagogical University of CracowKrakówPoland

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