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

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.