Flat lax and weak lax embeddings of finite generalized hexagons (Q1272778)

A finite generalized hexagon \({\mathcal H} = (P,B,I)\) is laxly embedded in \(\text{PG}(d,q)\) if \(P\) is a set of points of \(\text{PG}(d,q)\) generating \(\text{PG}(d,q)\), if every line \(L \in B\) is contained in some line of \(\text{PG}(d,q)\), and if no two distinct lines of \(\mathcal H\) are contained in the same line of \(\text{PG}(d,q)\). A lax embedding is called flat if for every point \(x \in P\) the set of points collinear with \(x\) in \(\mathcal H\) is contained in some plane of \(\text{PG}(d,q)\), and it is called weak if for every point \(x \in P\) the set of points not opposite \(x\) in \(\mathcal H\) is contained in some hyperplane of \(\text{PG}(d,q)\). The authors prove that every lax embedding of a hexagon which is both flat and weak is a natural embedding of some classical hexagon, defined over a subfield of \(\text{GF}(q)\). They also obtain partial results for lax embeddings which satisfy only one of the two additional properties.