Conics in Baer subplanes

Abstract

This article studies conics and subconics of $PG\left(2,q^2\right)$ and their representation in the André/Bruck–Bose setting in $PG\left(4,q\right)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG\left(2,q^2\right)$ corresponds in $PG\left(4,q\right)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3- and 4-dimensional normal rational curve in $PG\left(4,q\right)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG\left(2,q^2\right)$.