Specialness and the Bose representation

TL;DR

This paper characterizes subconics of order q in projective geometry using the Bruck-Bose and Bose representations, highlighting the role of transversals of regular spreads in understanding varieties in PG(2,q^2).

Contribution

It provides a geometric explanation for the intrinsic role of transversals in characterizing varieties in PG(2,q^2) within different projective representations.

Findings

Characterization of subconics in Bruck-Bose representation.

Geometric explanation for transversals' role in varieties.

Insight into the structure of PG(2,q^2) representations.

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 $PG(2,q^2)$.