The universal covers of a family of extended generalized quadrangles
A construction of extended generalized quadrangles of order \((q+1,q-1)\) from families of \(q+3\) subplanes of \(\text{PG}(5,q)\) subject to certain conditions was introduced by the author in [Eur. J. Comb. 18, 835-848 (1997; Zbl 0890.51011)]. Two constructions of such families of subplanes are known. One starts with a hyperoval in \(\text{PG}(2,q)\), the other with a \((q+1)\)-arc in \(\text{PG}(3,q)\). The paper under review presents a condition on the family of subplanes that implies that the associated extended generalized quadrangle is simply connected. It then is discussed, which of the known families of subplanes satisfy this condition. The main results state that an extended generalized quadrangle constructed from a \((q+1)\)-arc in \(\text{PG}(3,q)\) is simply connected if and only if \(q\neq 4\) and an extended generalized quadrangle constructed from a hyperoval in \(\text{PG}(2,q)\) is simply connected except when \(q=4\) or the polynomial describing the hyperoval belongs to a certain family discovered by Payne.
scientific article; zbMATH DE number 1235006