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.
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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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Dumnicki, M., Zaman Fashami, M., Szpond, J. et al. Lower Bounds for Waldschmidt Constants of Generic Lines in \({\mathbb {P}}^3\) and a Chudnovsky-Type Theorem. Mediterr. J. Math. 16, 53 (2019). https://doi.org/10.1007/s00009-019-1328-8
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DOI: https://doi.org/10.1007/s00009-019-1328-8
Keywords
- Asymptotic Hilbert function
- Chudnovsky conjecture
- containment problem
- symbolic powers
- Waldschmidt constants