Subspaces intersecting each element of a regulus in one point, André-Bruck-Bose representation and clubs

Michel Lavrauw¹ ; Corrado Zanella¹

¹Università degli Studi di Padova

The electronic journal of combinatorics, Tome 23 (2016) no. 1

Résumé

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.

Détail

DOI : 10.37236/4662

Classification : 51E20
Mots-clés : club, linear set, subplane, André-Bruck-Bose representation, Segre variety