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

## TL;DR

This paper investigates a specific ruled quintic surface in projective 6-space, exploring its geometric properties and its connection to subplanes in finite projective geometry.

## Contribution

It introduces and analyzes a ruled quintic surface in PG(6,q), linking it to subplanes in PG(2,q^3) through the Bruck-Bose representation.

## Key findings

- Identifies the surface as a ruled quintic $ ext{V}^5_2$
- Studies its geometric properties
- Establishes its relationship with an $ extbf{F}_q$-subplane

## 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 $\mathcal V^5_2$, 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.