OVOIDS AND TRANSLATION OVALS

Abstract

An ovoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q²+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.