Generalized quadrangles with an ovoid that is translation with respect to opposite flags

Abstract.

The article [6] contains the result that if a finite generalized quadrangle Γ of order s has an ovoid $\mathcal{O}$ that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then Γ is self-polar and $\mathcal{O}$ is the set of absolute points of a polarity. In particular, if Γ is the classical generalized quadrangle Q(4, q) then $\mathcal{O}$ is a Suzuki-Tits ovoid. In this article, we remove the need to assume that Γ is Q(4, q) in order to conclude that $\mathcal{O}$ is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that Γ is Q(4, q). At the same time, we also relax the requirement that Γ have order s.