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

TL;DR

This paper classifies the orbits of lines in projective 3-space under the action of PGL_2(q) in characteristic three, linking it to the classification of binary quartic forms, and describes the incidence structures of these orbits.

Contribution

It provides the first classification of PGL_2(q)-orbits of lines in PG(3,q) specifically in characteristic three, extending previous results to this case.

Findings

Classification of binary quartic forms into PGL_2(q)-orbits in characteristic three

Complete orbit classification of lines in PG(3,q) under PGL_2(q) in characteristic three

Determination of point-line and line-plane incidence structures for these orbits

Abstract

We consider the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $\mathbb{F}_q$ into orbits of the group $PGL_2(q)$ 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 $PGL_2(q)$-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.