Incidence of lines, points and planes in PG(3, q) with respect to the twisted cubic

Abstract

We consider the orbits of the group on the points, lines and planes of the projective space PG(3, q) over a finite field of characteristic different from 2 and 3. The points of PG(3, q) can be identified with projective space of binary cubic forms, and the set of lines of PG(3, q) can be thought of as pencils of cubic forms. The action of G on PG(1, q) naturally induces an action of G on binary cubic forms f(X, Y). The points of PG(3, q) decompose into five G orbits. The G orbits on were recently obtained by the authors. Let be the subset of consisting of pairs (L, P) where L is a line incident with the point P. The decomposition of into orbits yields a partition of . The problem that we solve in this work is to determine the sizes of the corresponding parts of .