Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Abstract

Let Q ₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry $\mathcal {X}$ having as points all the elliptic ovoids of Q ₀ and as lines the maximal pencils of elliptic ovoids of Q ₀ pairwise tangent at the same point. We first prove that $\mathcal {X}$ is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q∖Q ₀, where Q is the classical (q,q ²)-generalized quadrangle admitting Q ₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of $\mathcal {X}$. We prove that any maximal clique in Γ is either a line of $\mathcal {X}$ or it consists of 6 or 4 points of $\mathcal {X}$ not contained in any line of $\mathcal {X}$, accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of $\mathcal {X}$ arise as subgeometries of Q defined over $\mathbb {F}_{2}$.