Incidence matrices for the class $\mathcal{O}_6$ of lines external to the twisted cubic in $\mathrm{PG}(3,q)$

TL;DR

This paper analyzes the incidence matrices related to the class of lines external to the twisted cubic in projective space, providing explicit structures for these matrices across all finite fields.

Contribution

It determines the incidence matrices for a significant subset of the class  of lines external to the twisted cubic in (3,q), advancing understanding of their geometric and algebraic structure.

Findings

Explicit incidence matrices for (3,q) for all q.

Partial classification of  orbits involving external lines.

Enhanced understanding of the geometric configuration of lines external to the twisted cubic.

Abstract

We consider the structures of the plane-line and point-line incidence matrices of the projective space $\mathrm{PG}(3,q)$ connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all $q$, even and odd, we determine the incidence matrices connected with a family of orbits of the class named $\mathcal{O}_6$. This class contains lines external to the twisted cubic. The considered family include an essential part of all $\mathcal{O}_6$ orbits, whose complete classification is an open problem.