Classification of embeddings of the flag geometries of projective planes in finite projective spaces, part 1

摘要

The flag geometry Gamma = (P, L, I) of a finite projective plane Pi of order s is the generalized hexagon of order is (s, l) obtained from Pi by putting P equal to the set of all flags of Pi, by putting L equal to the set of all points and lines of Pi, and where I is the natural incidence relation (inverse containment), i.e., Gamma is the dual of the double of Pi in the sense of H. Van Maldeghem (1998, "Generalized Polygons," Birkhauser Verlag, Basel). Then we say that Gamma is fully and weakly embedded in the finite projective space PG(d, q) if Gamma is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q if the set of points of Gamma generates PG(d, q), and if the set of points of Gamma not opposite any given point of Gamma 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 Pi 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 Gamma(Gamma as above) in the case that there are two opposite lines L, M of Gamma with the property that the subspace of PG(d, q) generated by all lines of Gamma meeting either L or M has dimension 6 (which is the case for all embeddings in PG(d, q) d epsilon {6, 7}). Together with Parts 2 and 3, this will provide the complete classification of all full and weak embeddings of Gamma. (C) 2000 Academic Press. [References: 6]