Subgeometries in the Andr\'e/Bruck-Bose representation

TL;DR

This paper studies the geometric representation of substructures in a finite projective plane, extending previous results and correcting earlier findings in the context of Andre9/Bruck-Bose representation.

Contribution

It characterizes the representation of various sublines and subplanes in the Andre9/Bruck-Bose model, extending and correcting prior work for general parameters.

Findings

Characterization of _{q^k}-sublines tangent to or contained in the line at infinity.

Description of _q- sublines external to the line at infinity.

Analysis of _q- subplanes tangent and _{q^k}-subplanes secant to the line at infinity.

Abstract

We consider the Andr\'e/Bruck-Bose representation of the projective plane $\mathrm{PG}(2,q^n)$ in $\mathrm{PG}(2n,q)$. We investigate the representation of $\mathbb{F}_{q^k}$-sublines and $\mathbb{F}_{q^k}$-subplanes of $\mathrm{PG}(2,q^n)$, extending the results for $n=3$ of \cite{BarJack2} and correcting the general result of \cite{BarJack1}. We characterise the representation of $\mathbb{F}_{q^k}$-sublines tangent to or contained in the line at infinity, $\mathbb{F}_q$-sublines external to the line at infinity, $\mathbb{F}_q$-subplanes tangent to and $\mathbb{F}_{q^k}$-subplanes secant to the line at infinity.