Hyper-reguli in PG(5,q)

Abstract

A simple counting argument is used to show that for all $q$, an André 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.