Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces

Abstract

The flag geometry\(\Gamma = (\mathcal{P},\mathcal{L},I)\) of a finite projective plane II of orders is the generalized hexagon of order (s, 1) obtained from II by putting\(\mathcal{P}\) equal to the set of all flags of II, by putting\(\mathcal{L}\) equal to the set of all points and lines of II and where I is the natural incidence relation (inverse containment), that is, Γ is the dual of the double of II in the sense of [8]. Then we say that Γ is fully (and weakly) embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q) (and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q)). We have classified all such embeddings in [3, 4, 5, 6]. In the present paper, we weaken the hypotheses in some special cases, and we give an alternative formulation of the classification.