Generalized hexagons and Singer geometries

Abstract

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