### Subspaces Intersecting Each Element of a Regulus in One Point, André-Bruck-Bose Representation and Clubs

#### Abstract

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\mathcal{R}$ of $(n-1)$-subspaces in $\mathrm{PG}(2n-1,q)$, with respect to the subgroup of $\mathrm{PGL}(2n,q)$ fixing $\mathcal{R}$. 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 $\mathcal{R}$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results of Barwick and Jackson on the André -Bruck-Bose representation of a $q$-subline in $\mathrm{PG}(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.