Orbits of the class $\mathcal{O}_6$ of lines external with respect to the twisted cubic in $\mathrm{PG}(3,q)$

TL;DR

This paper studies the classification of line orbits in projective space related to the twisted cubic, focusing on the class , and provides a new approach to describe a significant subset of these orbits and their stabilizers.

Contribution

It introduces a novel method to identify and describe a large subset of  orbits in ,q, advancing the understanding of their structure and stabilizer groups.

Findings

Describes a family of  orbits for all q

Identifies stabilizer groups of these orbits

Provides partial classification of  orbits

Abstract

In the projective space $\mathrm{PG}(3,q)$, we consider 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. We propose an approach to obtain orbits of the class named $\mathcal{O}_6$, whose complete classification is an open problem. For all even and odd $q$ we describe a family of orbits of $\mathcal{O}_6$ and their stabilizer groups. The orbits of this family include an essential part of all $\mathcal{O}_6$ orbits.