On the Number of Incidences Between Points and Planes in Three Dimensions

Abstract

We prove an incidence theorem for points and planes in the projective space ℙ³ over any Field $\mathbb{F}$, whose characteristic p ≠ 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.

This yields a bound on the number of incidences between m points and n planes in ℙ³, with m≥n as

$$O\left( {m\sqrt n + mk} \right)$$

, where k is the maximum number of collinear planes, provided that n = O(p²) if p > 0. Examples show that this bound cannot be improved without additional assumptions.

This gives one a vehicle to establish geometric incidence estimates when p >0. For a non-collinear point set S⊆F² and a non-degenerate symmetric or skew-symmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is $\Omega \left[ {\min \left( {{{\left| S \right|}^{\frac{2}{3}}},p} \right)} \right]$. This is also the best known bound over ℝ, where it follows from the Szemerédi-Trotter theorem. Also, a set S ⊆ F³, not supported in a single semi-isotropic plane contains a point, from which $\Omega \left[ {\min \left( {{{\left| S \right|}^{\frac{1}{2}}},p} \right)} \right]$ distinct distances to other points of S are attained.