Characterisations of ovoids of PG(3,q) by plane section

Matthew Brown

An ovoid of the projective space PG(3,q) is a set of q^2+1 points, no three of which are collinear. An ovoid has the properties that at each point of the ovoid there is a unique tangent plane and any other plane of PG(3,q) meets the ovoid in an oval (a set of q+1 points, no three of which are collinear). The classical model of the ovoid is a non-singular quadric of elliptic type and the only other known construction is due to Tits in 1960. I will speak on my work which includes two characterisation results for ovoids where the hypothesis is on a single plane section of the ovoid. In particular an ovoid containing a single conic will be shown to the be the elliptic quadric; and an ovoid containing a pointed conic will be shown to be either an elliptic quadric in PG(3,4) or a Tits ovoid in PG(3,8).