Twisted cubic and orbits of lines in $\mathrm{PG}(3,q)$, II

TL;DR

This paper classifies line orbits under the twisted cubic stabilizer in projective space PG(3,q), identifying unique and multiple orbit classes, and determines subgroup structures for various line types.

Contribution

It provides a complete classification of line orbits in PG(3,q) under the twisted cubic stabilizer, including sizes, structures, and subgroup fixing lines.

Findings

Identified all line classes with a single orbit.

Proved most classes consist of two or three orbits.

Determined subgroup structures fixing each orbit.

Abstract

In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of $\mathrm{PG}(3,q)$ are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems which remain open for one type of lines are formulated and, for $5\le q\le37$ and $q=64$, a solution is provided.