On rich points and incidences with restricted sets of lines in 3-space

Abstract

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(n4/3+ε/r2), for r ⩾ 3 and for any ε > 0, and, if at most n1/3 lines of L lie on any common regulus, there are at most O(n4/3+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 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 ( m3/5n3/5 + (m11/15n2/5 + m1/3n2/3)s1/3 + m + n ) , provided that no plane contains more than s of the points. When s = O(min{n3/5/m2/5, m1/2}), the bound becomes O(m3/5n3/5 + 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(m3/5n3/5 +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. 2012 ACM Subject Classification Theory of computation → Computational geometry