On the $PGL_2(q)$-orbits of lines of $PG(3,q)$ and binary quartic forms in characteristic three

Krishna Kaipa; Puspendu Pradhan

paper

On the $P G L_{2} (q)$ -orbits of lines of $P G (3, q)$ and binary quartic forms in characteristic three

arXiv:2508.11229

Abstract

We consider the problem of classifying the lines of the projective $3$ -space $P G (3, q)$ over a finite field $F_{q}$ into orbits of the group $P G L_{2} (q)$ of linear symmetries of the twisted cubic $C$ . The problem has been solved in literature in characteristic different from $3$ , and in this work, we solve the problem in characteristic $3$ . We reduce this problem to another problem, which is the classification of binary quartic forms into $P G L_{2} (q)$ -orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.

arXiv admin note: text overlap with arXiv:2312.07118

On the PGL2​(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three

Abstract

On the $P G L_{2} (q)$ -orbits of lines of $P G (3, q)$ and binary quartic forms in characteristic three