Orbits and incidence matrices for points, planes and lines regarding the twisted cubic in PG(3,q), q = 2, 3, 4

TL;DR

This paper classifies the orbits of points, planes, and lines under the group fixing a twisted cubic in PG(3,q) for q=2,3,4, and determines the associated incidence matrices.

Contribution

It provides the first complete classification of these orbits and their incidence matrices for the specified finite fields.

Findings

Classified orbits of points, planes, and lines under G_q.

Determined incidence matrices between these orbits.

Solved open problems for q=2, 3, 4.

Abstract

In the three-dimensional projective space PG(3,q) over the finite field F_q with q elements, we consider the normal rational curve known as a twisted cubic and the projectivity group G_q that fixes it. For q = 2, 3, 4, we solve the open problems of classifying the orbits of points, planes, and lines under G_q and of determining the corresponding incidence matrices between points, planes, and lines partitioned into these orbits.