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

Abstract: 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 $\mathcal I$.