On Embeddings of the Flag Geometries of Projective Planesin Finite Projective Spaces

Abstract

The flag geometry \Gamma=(\cP,\cL,\op) of a finite projective plane \Pi of order s is the generalized hexagon of order (s,1) obtained from \Pi by putting \cP equal to the set of all flags of\Pi , by putting \cL equal to the set of all points and lines of \Pi and where \op is the natural incidence relation (inverse containment), i.e.,\Gamma is the dual of the double of \Pi in the sense of Van Maldeghem Mal:98. 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). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to \{6,7,8\}.