Twisted cubic and plane-line incidence matrix in $\mathrm{PG}(3,q)$

TL;DR

This paper analyzes the incidence matrix between planes and lines in projective space PG(3,q), focusing on the structure under the twisted cubic's stabilizer group and deriving incidence counts for various line orbits.

Contribution

It introduces a detailed structural analysis of the plane-line incidence matrix in PG(3,q) relative to the twisted cubic's symmetry, including submatrix decompositions and incidence counts.

Findings

Derived incidence counts for lines in each plane and planes through each line.

Structured submatrices for unions of line orbits and their incidence properties.

Identified special cases with unique incidence patterns.

Abstract

We consider the structure of the plane-line incidence matrix of the projective space $\mathrm{PG}(3,q)$ with respect to the orbits of planes and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences between a union of line orbits and an orbit of planes are investigated. For the unions consisting of two or three line orbits, the original submatrices are split into new ones, in which the incidences are also considered. For each submatrix (apart from the ones corresponding to a special type of lines), the numbers of lines in every plane and planes through every line are obtained. This corresponds to the numbers of ones in columns and rows of the submatrices.