## Abstract

Let L be a set of n lines in ℝ^{3} that is contained, when represented as points in the four-dimensional Plücker space of lines in ℝ^{3}, in an irreducible variety T of constant degree which is non-degenerate with respect to L (see below). We show: (1) If T is two-dimensional, the number of r-rich points (points incident to at least r lines of L) is O(n^{4/3+}^{ε}/r^{2}), for r ≥ 3 and for any ε > 0, and, if at most n^{1/}^{3} lines of L lie on any common regulus, there are at most O(n^{4/3+ε}) 2-rich points. For r larger than some sufficiently large constant, the number of r-rich points is also O(n/r). As an application, we deduce (with an ε-loss in the exponent) the bound obtained by Pach and de Zeeuw [16] on the number of distinct distances determined by n points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. (2) If T is two-dimensional, the number of incidences between L and a set of m points in ℝ^{3} is O(m + n). (3) If T is three-dimensional and nonlinear, the number of incidences between L and a set of m points in ℝ^{3} is O(m^{3/}^{5}n^{3/}^{5} + (m^{11/}^{15}n^{2/}^{5} + m^{1/}^{3}n^{2/}^{3})s^{1/}^{3} + m + n), provided that no plane contains more than s of the points. When s = O(min{n^{3/}^{5}/m^{2/}^{5}, m^{1/}^{2}}), the bound becomes O(m^{3/}^{5}n^{3/}^{5} + m + n). As an application, we prove that the number of incidences between m points and n lines in ℝ^{4} contained in a quadratic hypersurface (which does not contain a hyperplane) is O(m^{3/}^{5}n^{3/}^{5} +m+n). The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.

Original language | English |
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Title of host publication | 37th International Symposium on Computational Geometry, SoCG 2021 |

Editors | Kevin Buchin, Eric Colin de Verdiere |

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

ISBN (Electronic) | 9783959771849 |

DOIs | |

State | Published - 1 Jun 2021 |

Event | 37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States Duration: 7 Jun 2021 → 11 Jun 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 189 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 37th International Symposium on Computational Geometry, SoCG 2021 |
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Country/Territory | United States |

City | Virtual, Buffalo |

Period | 7/06/21 → 11/06/21 |

## Keywords

- Incidences
- Lines in space
- Polynomial partitioning
- Rich points