A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set L of q 5 + q 4 + q 3 + q 2 + q + 1 lines of PG ( 6 , q ) with the properties that (1) every point of PG ( 6 , q ) is incident with either 0 or q + 1 elements of L , (2) every plane of PG ( 6 , q ) is incident with either 0, 1 or q + 1 elements of L , (3) every solid of PG ( 6 , q ) is incident with either 0, 1, q + 1 or 2 q + 1 elements of L , and (4) every hyperplane of PG ( 6 , q ) is incident with at most q 3 + 3 q 2 + 3 q members of L , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon H ( q ) in PG ( 6 , q ) .