The Bose representation of PG(2,q^3) in PG(8,q)

Abstract

This article looks at the Bose representation of $PG(2,q^3)$ as a 2-spread of $PG(8,q)$. It is shown that an ${\mathbb{F}}_q$-subline of $PG(2,q^3)$ corresponds to a 2-regulus, and an ${\mathbb{F}}_q$-subplane corresponds to a Segre variety $S_{2;2}$. Moreover, the extension of these varieties to $PG(8,q^3)$ and $PG(8,q^6)$ is determined. These are used to determine the structure of an ${\mathbb{F}}_q$-conic of $PG(2,q^3)$ in the Bose representation in $PG(8,q)$.