Ovals in translation hyperovals and ovoids
An ovoid in \(\text{PG}(3,q)\) is a set of \(q^2+ 1\) points, no three of which are collinear. A hyperoval in \(\text{PG}(2, q)\) is a set of \(q+2\) points, no three of which are collinear. (A hyperoval cannot exist unless \(q\) is even.) A hyperoval is a translation oval if it is fixed by a group of order \(q\) consisting of elations, all with the same axis. The main result is that if \(O\) is an ovoid of \(\text{PG}(3, q)\), \(q\) even such that every secant plane section is an oval contained in a hyperoval, then \(O\) is either an elliptic quadric or a Tits ovoid.
scientific article; zbMATH DE number 1066021