Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

Abstract

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 an earlier paper, we have shown that the dimension d of the projective space belongs to {6, 7, 8}, and that the projective plane Π is Desarguesian. Furthermore, we have given examples for d=6, 7. In the present paper we show that for d=6, only these examples exist, and we also partly handle the case d=7. More precisely, we completely classify 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 of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is the case for all embeddings in PG(d, q), d∈{6, 7}). Together with Parts 2 and 3, this will provide the complete classification of all full and weak embeddings of Γ.