Abstract
Kelly’s theorem states that a set of n points affinely spanning \({\mathbb {C}}^3\) must determine at least one ordinary complex line (a line incident to exactly two of the points). Our main theorem shows that such sets determine at least 3n / 2 ordinary lines, unless the configuration has \(n-1\) points in a plane and one point outside the plane (in which case there are at least \(n-1\) ordinary lines). In addition, when at most n / 2 points are contained in any plane, we prove stronger bounds that take advantage of the existence of lines with four or more points (in the spirit of Melchior’s and Hirzebruch’s inequalities). Furthermore, when the points span four or more dimensions, with at most n / 2 points contained in any three-dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.
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Notes
We note that while the Fermat configuration as stated lives in the projective plane, it can be made affine by any projective transformation that moves a line with no points to the line at infinity.
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Acknowledgements
Zeev Dvir: Research was supported by NSF CAREER award DMS-1451191 and NSF Grant CCF-1523816. Shubhangi Saraf: Research was supported in part by NSF Grants CCF-1350572 and CCF-1540634. Charles Wolf: Research was supported in part by NSF Grant CCF-1350572.
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Basit, A., Dvir, Z., Saraf, S. et al. On the Number of Ordinary Lines Determined by Sets in Complex Space. Discrete Comput Geom 61, 778–808 (2019). https://doi.org/10.1007/s00454-018-0039-4
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DOI: https://doi.org/10.1007/s00454-018-0039-4