We consider the orbits of the group \(G=PGL_2(q)\) on the points, lines and planes of the projective space PG(3, q) over a finite field \(\mathbb {F}_q\) 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 \(\mathcal {L}\) 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 \(\mathcal {L}\) were recently obtained by the authors. Let \(\mathcal {I}\) be the subset of \(\mathcal {L}\times PG(3,q)\) consisting of pairs (L, P) where L is a line incident with the point P. The decomposition of \(\mathcal {L}\times PG(3,q)\) into \(G \times G\) orbits yields a partition of \(\mathcal {I}\) . The problem that we solve in this work is to determine the sizes of the corresponding parts of \(\mathcal {I}\) .
