Specialness and the Bose representation

Abstract

This article looks at subconics of order $q$ of $PG(2,q^2)$ and characterizes them in the Bruck-Bose representation in $PG(4,q)$. In common with other objects in the Bruck-Bose representation, the characterisation uses the transversals of the regular line spread $S$ associated with the Bruck-Bose representation.By working in the Bose representation of $PG(2,q^2)$ in $PG(5,q)$, we give a geometric explanation as to why the transversals of the regular spread $S$ are intrinsic to the characterisation of varieties of $PG(2,q^2)$.