OVOIDS OF PG(3, q), q EVEN, WITH A CONIC SECTION

Throughout this paper we will assume that q is even. We will be considering ovoids of PG(3, q) which contain a conic as a plane section by using different representations of the classical generalized quadrangle of order q. An oŠal / of PG(2, q) is a set of q­1 points of PG(2, q), with no three collinear. Let F be a line of PG(2, q) ; then F is incident with zero, one or two points of / and is accordingly called an external line, a tangent or a secant to /. An elementary count shows that there is a unique tangent to / incident with a given point of /. If F is tangent to / and incident with the point P `/, then we say that F is the tangent to / at P. If q is even, then the tangents to / are coincident in a fixed point, the nucleus of / (see [9, Lemma 8.6]). The classical example of an oval in PG(2, q) is the set of points satisfying an irreducible quadratic equation, called a conic (more precisely this is a non-degenerate conic). A hyperoŠal of PG(2, q) is a set of q­2 points, with no three collinear. An oval of PG(2, q) together with its nucleus is a hyperoval of PG(2, q). An oŠoid Ω of PG(3, q) is a set of q#­1 points of PG(3, q), with no three collinear. From this point we will assume that q" 2, so that an ovoid is a maximal-sized set of points, with no three collinear. Let π be a plane of PG(3, q) ; then π meets Ω in a single point or in an oval of π and is called accordingly a tangent plane or a secant plane. The intersection of Ω and a secant plane of Ω is called a secant plane section of Ω. There is a unique tangent plane to Ω containing a given point P `Ω. This plane is the tangent plane to Ω at P. (See [1, 2, 18] for the above.) If q is odd, then ovoids of PG(3, q) have been classified as the non-degenerate elliptic quadrics of PG(3, q) (see [1, 18]). For q even, q ̄ 2h, the two known isomorphism classes of ovoids are the non-degenerate elliptic quadrics, which exist for all h& 1, and the Tits ovoids (see [10, Chapter 16]) which exist for h odd, h& 3. Most results characterizing ovoids of PG(3, q) have been in terms of the secant plane sections. Barlotti [2] proved that if every secant plane section of an ovoid is a conic, then the ovoid must be an elliptic quadric. In [22] Segre strengthened this result (for q& 8) to say that if at least (q$®q#­2q)}2 secant plane sections of an ovoid are conics, then the ovoid must be an elliptic quadric. The motivation for further results of this type characterizing ovoids is the connection of ovoids with inversive planes