Abstract

In this paper we continue our study begun in “Generalized hexagons and Singer geometries” (2008), aiming at characterizing the embedding of the split Cayley hexagons $\mathsf{H}\left(q\right)$, $q$ even, in $PG\left(5,q\right)$ by intersection numbers with respect to their lines. We prove that, for $q\ne 3$, every pseudo-hexagon (i.e. a set $\mathcal{ℒ}$ of lines of $PG\left(5,q\right)$ 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 $PG\left(5,q\right)$ is on $q+1$ members of $\mathcal{ℒ}$) which is 1-polarized at some point $x$ (i.e., the lines of $\mathcal{ℒ}$ through $x$ do not span $PG\left(5,q\right)$) is either the line set of the standard embedding of $\mathsf{H}\left(q\right)$ in $PG\left(5,q\right)$, or $q=2$ (in the latter case all pseudo-hexagons are classified in the paper cited).

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Mathematical Subject Classification 2000

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