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On Rich Points and Incidences with Restricted Sets of Lines in 3-Space Sharir, Micha Solomon, Noam Lines in space Rich points Polynomial partitioning Incidences Let L be a set of n lines in ℝ³ that is contained, when represented as points in the four-dimensional Plücker space of lines in ℝ³, 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}/r²), 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 [J. Pach and F. de Zeeuw, 2017] 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 ℝ³ 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 ℝ³ 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 ℝ⁴ 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Micha Sharir and Noam Solomon 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.56 urn:nbn:de:0030-drops-138551 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.56 eng https://creativecommons.org/licenses/by/4.0/legalcode