Hyper-reguli in PG(5,q)

TL;DR

This paper proves that in projective space PG(5,q), an André hyper-regulus has exactly two switching sets and characterizes the planes intersecting these sets, providing a counting-based proof.

Contribution

It establishes the precise number of switching sets for André hyper-reguli in PG(5,q) using a simple counting argument, clarifying their geometric structure.

Findings

Exactly two switching sets for André hyper-reguli in PG(5,q)

Number of planes meeting each hyper-regulus in a point is 2(q^2+q+1)

Counting argument confirms the structure of these hyper-reguli

Abstract

A simple counting argument is used to show that for all $q$, an Andr\'e hyper-regulus $\mathbb X$ in $PG(5,q)$ has exactly two switching sets. Moreover, there are exactly $2(q^2+q+1)$ planes in $PG(5,q)$ that meet every plane of $\mathbb X$ in a point, namely the planes in the switching sets.