On embeddings of the flag geometries of projective planes in finite projective spaces
The incidence graph \(\Gamma\) of a projective plane of order \(s\) is a generalized hexagon of order \((s,1)\). A weak and full embedding of a generalized hexagon \({\mathcal H}\) of order \((s,t)\) in the projective space \(\text{PG}(d,s)\) is a map that sends points and lines of \(\mathcal H\) to points and lines respectively of the projective space, such that incidence is preserved and such that for any point \(p\) of \(\mathcal H\) the set of points not opposite \(p\) does not generate \(\text{PG}(d,s)\). The paper under review is one step towards the classification of weak and full embeddings of the incidence graphs of finite projective planes. The main result of the paper states that only the incidence graph of a Desarguesian projective plane allows such an embedding and the values of the dimension~\(d\) are restricted to \(6\), \(7\) and~\(8\). For \(d\in\{6,7\}\) some weak and full embeddings of \(\Gamma\) in \(\text{PG}(d,s)\) are given in rather explicit terms. These examples are derived from known embeddings of the dual classical generalized hexagons and the fact, that these hexagons contain incidence graphs of projective planes. The authors conjecture that all weak and full embeddings of the incidence graph of a finite projective plane arise in this fashion, that is, they are derived from embeddings of the dual classical hexagons over the same field as the projective plane.
scientific article; zbMATH DE number 1392725