Subgeometries in the André/Bruck-Bose representation

Abstract

We consider the André/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.