On m-ovoids of dual twisted triality hexagons

TL;DR

This paper investigates the existence of m-ovoids in dual twisted triality hexagons, showing that such structures generally lack m-ovoids except in a specific case with s=3 and m=2.

Contribution

It proves that dual twisted triality hexagons do not have m-ovoids for all nontrivial m, except for one special case, extending previous results on extremal generalized hexagons.

Findings

No m-ovoids in dual twisted triality hexagons for all nontrivial m, except when s=3 and m=2.

Dual twisted triality hexagons are unique among extremal generalized hexagons in this property.

The case s=3, m=2 is an isolated exception where an m-ovoid exists.

Abstract

A generalised hexagon of order $(s,t)$ is said to be \emph{extremal} if $t$ meets the Haemers-Roos bound, that is, $t=s^3$. The \emph{dual twisted triality hexagons} associated to the exceptional Lie type groups $\,^3D_4(s)$ have these parameters, and are the only known such examples. It was shown in the work of De Bruyn and Vanhove that an extremal generalised hexagon has no 1-ovoids. In this note, we prove that a dual twisted triality hexagon has no $m$-ovoids for every possible (nontrivial) value of $m$, except for the isolated case where $s=3$ and $m=2$.