Generalized quadrangles of order \((q,q^ 2)\), \(q\) even, containing \(W(q)\) as a subquadrangle (Q1900076)

Let \({\mathcal S}\) be a generalized quadrangle of order \((q, q^2)\), with \(q\) even. Suppose that \({\mathcal S}\) contains a subquadrangle \({\mathcal S}'\) of order \(q\) isomorphic to the symplectic quadrangle \(W(q)\). It is well- known that every point \(x\) outside \({\mathcal S}'\) is collinear to the points of an ovoid of \({\mathcal S}'\). If by varying \(x\), all such ovoids are classical ovoids, i.e., ovoids with the 2-transitive group \(L_2 (q^2)\) acting on it, then \textit{J. A. Thas} and \textit{S. E. Payne} [Geom. Dedicata 52, No. 3, 227-253 (1994; Zbl 0804.51007)] show that \({\mathcal S}\) is isomorphic to \(Q (5,q)\) (a quadric of projective Witt-index 1 in 5- dimensional projective space over the field with \(q\) elements). The main result of the paper under review is to weaken the previous assumption as follows: suppose every ovoid obtained by a point not in \({\mathcal S}'\) is either classical, or semi-classical (i.e., the Suzuki group \(Sz (q)\) acts on it as a 2-transitive automorphism group, and the ovoid is called a Suzuki-Tits ovoid), then the conclusion above is still true, i.e., \({\mathcal S}\) is isomorphic to \(Q (5,q)\).