Subspaces intersecting each element of a regulus in one point, Andr\'e-Bruck-Bose representation and clubs

TL;DR

This paper investigates the properties of subspaces intersecting reguli in projective geometry, with applications to finite geometry structures like spreads and linear sets, improving existing classifications and representations.

Contribution

It provides new conditions for subspace extendability to Desarguesian spreads and enhances the understanding of the Andre9-Bruck-Bose representation of sublines.

Findings

Necessary conditions for subspace extendability to spreads.

Improved results on Andre9-Bruck-Bose representation.

Applications to classification of linear sets and clubs.

Abstract

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\cR$ of $(n-1)$-subspaces in $\PG(2n-1,q)$, with respect to the subgroup of $\PGL(2n,q)$ fixing $\cR$. Such results have consequences on several aspects of finite geometry. First of all, a necessary condition for an $(n-1)$-subspace $U$ and a regulus $\cR$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in \cite{BaJa12} on the Andr\'e-Bruck-Bose representation of a $q$-subline in $\PG(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.