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.