Ruled quintic surfaces in $\mathrm{PG}(6,q)$

Abstract

We look at a scroll of $PG\left(6,q\right)$ that uses a projectivity to rule a conic and a twisted cubic. We show this scroll is a ruled quintic surface ${\mathcal{V}}_2^5$, and study its geometric properties. The motivation in studying this scroll lies in its relationship with an ${\mathbb{F}}_q$-subplane of $PG\left(2,q^3\right)$ via the Bruck–Bose representation.