Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 3

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 three 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) for d=6 and for d=7 in the case that there exists a line L of Γ and four distinct lines L1, L2, L3, L4 concurrent with Γ which generate a 4-dimensional space. In the present paper, we drop all these additional assumptions by completing the case d=7 and handling the case d=8. In particular, we find new examples for d=8 (contrary to our original conjecture (J. A. Thas and H. Van Maldeghem, Des. Codes Cryptogr.17 (1999), 97–104)). This means that we have now the complete classification of all fully and weakly embedded geometries Γ in PG(d, q), with Γ the flag geometry of a finite projective plane.