Abstract
Let L be a set of n lines in ℝ³ that is contained, when represented as points in the fourdimensional Plücker space of lines in ℝ³, in an irreducible variety T of constant degree which is nondegenerate with respect to L (see below). We show:
(1) If T is twodimensional, the number of rrich 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}) 2rich points. For r larger than some sufficiently large constant, the number of rrich points is also O(n/r).
As an application, we deduce (with an εloss 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 twodimensional, the number of incidences between L and a set of m points in ℝ³ is O(m+n).
(3) If T is threedimensional 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.
BibTeX  Entry
@InProceedings{sharir_et_al:LIPIcs.SoCG.2021.56,
author = {Sharir, Micha and Solomon, Noam},
title = {{On Rich Points and Incidences with Restricted Sets of Lines in 3Space}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {56:156:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13855},
URN = {urn:nbn:de:0030drops138551},
doi = {10.4230/LIPIcs.SoCG.2021.56},
annote = {Keywords: Lines in space, Rich points, Polynomial partitioning, Incidences}
}
Keywords: 

Lines in space, Rich points, Polynomial partitioning, Incidences 
Collection: 

37th International Symposium on Computational Geometry (SoCG 2021) 
Issue Date: 

2021 
Date of publication: 

02.06.2021 