Subspaces intersecting each element of a regulus in one point, André-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 R of (n-1)-subspaces in PG(2n-1; q), with respect to the subgroup of PGL(2n; q) fixing 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 R of (n-1)-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in [2] on the André-Bruck-Bose representation of a q-subline in PG(2; qn). Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.