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

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 [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(m^3/5n^3/5 + (m^11/15n^2/5 + m^1/3n^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^3/5n^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^3/5n^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.