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

## TL;DR

This paper explores the Bose representation of PG(2,q^3) within PG(8,q), revealing how sublines, subplanes, and conics correspond to geometric varieties like 2-reguli and Segre varieties, and analyzing their extensions.

## Contribution

It establishes the correspondence between substructures in PG(2,q^3) and geometric varieties in PG(8,q), and determines their extensions to larger projective spaces.

## Key findings

- Subline corresponds to a 2-regulus.
- Subplane corresponds to a Segre variety S_{2;2}.
- Extension of varieties to PG(8,q^3) and PG(8,q^6) analyzed.

## 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 $PG(8,q)$.