Generalized quadrangles of order \((s,s^ 2)\). II
This paper, which is self-contained and does not really depend on part I [ibid. 67, No. 2, 140-160 (1994; Zbl 0592.51003)], studies essentially the finite generalized quadrangles of order \((s,s^2)\), \(s\) odd, which are translation generalized quadrangles. It is well-known that these quadrangles can be constructed in \(\text{PG} (4n,q)\) for a suitable \(n\), by using a so-called generalized ovoid or egg \(O\) in \(\text{PG} (4n-1,q)\). The author now defines the egg \(O\) to be good (as opposed to Kantor's bad eggs) at some element \(U\) (which is an \((n-1)\)-dimensional subspace) if every \((3n-1)\)-dimensional subspace containing \(U\) and at least two further elements of \(O\) contains exactly \(q^n+1\) elements of \(O\). Denote the corresponding quadrangle by \(T(O)\). There is a notion of the dual \(O^*\) of \(O\), giving rise to the translation dual \(T(O^*)\) of \(T(O)\). Now \(O\) is good at one of its elements if and only if \(T(O^*)\) satisfies property (G) at a corresponding flag \(\{p,L\}\), i.e., every triple of pairwise non-collinear points \(\{p,q,z\}\) of \(T(O^*)\) such that \(y\) and \(z\) are collinear with the same point on \(L\), is 3-regular. The main result of the paper under review now says that, if \(T(O)\) is a translation generalized quadrangle arising from the egg \(O\), and \(O\) is good in at least one of its elements \(U\), then there are three possibilities. We describe these roughly. (1) \(T(O)\) is classical, (2) \(T(O)\) is nonclassical, and there is a 4-dimensional space \(W\) meeting \(U\) in a line and every other element \(U'\) of \(O\) in a point \(x(U')\). The set of all such \(x(U')\) is contained in the projection of a quadratic Veronesean \({\mathcal V}^4_2\), (3) we are not in the previous cases, and there exists a 5-dimensional space \(W\) meeting \(U\) in a plane and every other element \(U'\) of \(O\) in a point \(x(U')\). The set of all such \(x(U')\) is contained in a quadratic Veronesean \({\mathcal V}^4_2\). In case (2), Thas shows that if \(T(O^*)\) arises from a flock \(F\), then \(F\) is a Kantor flock. He also conjectures that every translation generalized quadrangle satisfying property (G) at some flag arises from a flock. That conjecture is proved in the meantime by the same author (presumably giving rise to part III of the paper). Hence case (2) of the main result of the paper can be restated as: (2) \(T(O^*)\) arises from a Kantor flock. It is also conjectured by Thas that in case (3) \(T(O^*)\) is a Roman generalized quadrangle. The reviewer conjectures that this will be proved in Part IV by Thas.
scientific article; zbMATH DE number 1051479
scientific article