Generalized hexagons and Singer geometries

In this paper, we consider a set L of lines of PG(5, q) with the properties that (1) every plane contains 0, 1 or q + 1 elements of L, (2) every solid contains no more than q(2) + q + 1 and no less than q + 1 elements of L, and (3) every point of PG(5, q) is on q + 1 members of L, and we show that, whenever (4) q not equal 2 (respectively, q = 2) and the lines of L through some point are contained in a solid (respectively, a plane), then L is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon H(q) in PG(5, q), with q even. We present examples of such sets L not satisfying (4) based on a Singer cycle in PG(5, q), for all q.

Jaar van publicatie:2008