A ruled quintic surface in $PG(6,q)$

Abstract

In this article we look at a scroll of $PG(6,q)$ 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 $PG(2,q^3)$ via the Bruck-Bose representation.