×

A plane representation of ovoids. (English) Zbl 0952.51006

Let \(\{{\mathcal O}_s\mid s\in GF(q)\}\) be a set of \(q\) ovals in \(PG(2,q)\), \(q\) even, all with nucleus \((0,1,0)\), satisfying \({\mathcal O}_s \cap{ \mathcal O}_t= (0,0,1)\) for all \(s,t\) in \(GF(q)\) with \(s\neq t\), and such that every secant line to \({\mathcal O}_s\) on \((0,1,s+t)\) is external to \({\mathcal O}_t\). A fan of ovals is the image under a collineation of such a set of ovals.
A pencil of an ovoid \({\mathcal O}\) in \(PG(3,q)\) is the set of secant plane sections arising from the secant planes to \({\mathcal O}\) on a fixed tangent line \(l\). If \(\{\pi_s \mid s\in GF(q)\}\) are the planes on the pencil, there are homographies \(M_s\) fixing pointwise the line \(l\) and mapping \(\pi_s\) to \(\pi_0\).
The author proves that \[ \bigl\{ {\mathcal O}_s= M_s({\mathcal O}\cap \pi_s)\mid s\in GF(q) \bigr\} \] is a fan of ovals of \(\pi_0\) and, conversely, a fan of ovals defines an ovoid of \(PG(3,q)\).
This theorem has been an useful tool for studying the existence of ovoids in \(PG(3,q)\), \(q\) even.

MSC:

51E20 Combinatorial structures in finite projective spaces
05B25 Combinatorial aspects of finite geometries
PDFBibTeX XMLCite