A characterisation of Baer subplanes

Abstract

Let ${{\mathcal {K}}}$ be a set of $q^2+2q+1$ points in $\text {PG}(4,q)$. We show that if every 3-space meets ${{\mathcal {K}}}$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then ${{\mathcal {K}}}$ is a ruled cubic surface. Moreover, ${{\mathcal {K}}}$ corresponds via the Bruck–Bose representation to a tangent Baer subplane of $\text {PG}(2,q^2)$. We use this to prove a characterisation in $\text {PG}(2,q^2)$ of a set of points ${{\mathcal {B}}}$ as a tangent Baer subplane by looking at the intersections of ${{\mathcal {B}}}$ with Baer-pencils.