On the number of incidences between points and planes in three dimensions

We prove an incidence theorem for points and planes in the projective space P³ over any field 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 (m√n + mk),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 Ω[min(|S|^2/3,p)]. 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 Ω[min(|S|^½,p)] distinct distances to other points of S are attained.