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 a"(TM)(3) over any Field , whose characteristic p not equal 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 P-3, with m >= n as O (m root n vertical bar mk), where k is the maximum number of collinear planes, provided that n = O(p (2)) 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 SaS dagger F-2 and a non-degenerate symmetric or skew-symmetric bilinear form omega, the number of distinct values of omega on pairs of points of S is . This is also the best known bound over a"e, where it follows from the Szemer,di-Trotter theorem. Also, a set S aS dagger F-3, not supported in a single semi-isotropic plane contains a point, from which distinct distances to other points of S are attained.