### Abstract

Kelly's theorem states that a set of n points affinely spanning C^{3} must determine at least one ordinary complex line (a line passing through 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 a theorem giving stronger bounds that take advantage of the existence of lines with four and 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.

Original language | English (US) |
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Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |

Editors | Matthew J. Katz, Boris Aronov |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 151-1515 |

Number of pages | 1365 |

ISBN (Electronic) | 9783959770385 |

DOIs | |

State | Published - Jun 1 2017 |

Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: Jul 4 2017 → Jul 7 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 77 |

ISSN (Print) | 1868-8969 |

### Other

Other | 33rd International Symposium on Computational Geometry, SoCG 2017 |
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Country | Australia |

City | Brisbane |

Period | 7/4/17 → 7/7/17 |

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Combinatorial geometry
- Designs
- Incidences
- Polynomial method

## Cite this

*33rd International Symposium on Computational Geometry, SoCG 2017*(pp. 151-1515). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 77). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2017.15