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

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\text{cR}$ of $(n-1)$-subspaces in $\text{PG}(2n-1,q)$, with respect to the subgroup of $\text{PGL}(2n,q)$ fixing $\text{cR}$. 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 $\text{cR}$ 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 $\text{PG}(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.