Ovoids with a pencil of translation ovals
Let \({\mathcal O}\) be an ovoid of \(PG (3,q)\), \(q>2\): that is, \({\mathcal O}\) is a set of \(q^2 +1\) points in \(PG (3,q)\) with no three collinear. Take some tangent line \(L\) to \({\mathcal O}\), and consider the \(q\) nontangent planes to \({\mathcal O}\) through \(L\). Each such plane clearly meets \({\mathcal O}\) in an oval \((q+1\) points in \(PG (2,q)\) with no three collinear). Such a collection of ovals will be called a pencil of ovals for the given ovoid. Based on some recently completed work of Penttila and Praeger, the authors of the work under review show that an ovoid of \(PG (3,q)\), where \(q>2\) is even, has a pencil of translation ovals if and only if the ovoid is either an elliptic quadric or a Tits ovoid. Recall that a translation oval is an oval fixed by an elation group of order \(q\), each nonidentity element having the same axis (necessarily tangent to the oval). The proof techniques used include finite field computations, especially trace computations, and a plane representation of ovoids due to Glynn and Penttila. This research is part of a huge body of work attempting to show that the only ovoids of \(PG (3,q)\) are the known ones: namely, the elliptic quadrics and the Tits ovoids. In particular, the result of this paper is an extension of a theorem by \textit{D. G. Glynn} [Simon Stevin 58, 319-353 (1984; Zbl 0563.51005)] which states that an ovoid is an elliptic quadric if and only if it admits a pencil of conics.
scientific article; zbMATH DE number 929905