Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

The flag geometry Γ=(P, L, I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting P equal to the set of all flags of Π, by putting L equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). 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). In two earlier papers we have shown that the dimension d of the projective space belongs to {6, 7, 8}, that the projective plane Π is Desarguesian, and we have classified the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace UL, M of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is automatically satisfied if d=6). In the present paper, we partly handle the case d=7; more precisely, we consider for d=7 the case where for all pairs (L, M) of opposite lines of Γ, the subspace UL, M has dimension 7 and where there exist four lines concurrent with L contained in a 4-dimensional subspace of PG(7, q).