Ovoids and Translation Ovals

An oŠoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q#­1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields ; for from an ovoid one can construct an inversive plane [3], a generalised quadrangle [11], and if q is even, a translation plane [13]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [2]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [3, 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids. For q" 2 ovoids in PG(3, q) are sets of points with no three points collinear of maximal size [8, Theorem 16.1.5]. A plane π intersects an ovoid / in a set πf/ of points with no three points of πf/ collinear in π ; in fact the set πf/ either consists of a single point and π is called a tangent plane to /, or has size q­1 and π is called a secant plane to / see [8, Lemma 16.1.6]. In the latter case the set πf/ is an oval of π. There have been several results which characterise ovoids by the nature of their (secant) plane sections πf/. The first of these was due to Barlotti [2] ; he showed that an ovoid is an elliptic quadric if and only if all of its plane sections are conics. This was later strengthened by Prohaska and Walker [12] ; they showed that elliptic quadrics were characterised by the nature of the plane sections of the set of planes on a single secant line. (Note that a line in PG(3, q) is called an external, tangent, or secant line to an oŠoid if it meets the ovoid in zero, one or two points respectively.)