Ovoids of Q(6, q) of low degree

The parabolic quadric Q(6,q) is one of the finite classical polar spaces. It is a rank 3 geometry and consists of points, lines and planes embedded in the projective space PG(6,q) that are totally singular with relation to a non-singular quadratic form. An ovoid of Q(6,q) is a set O of points of Q(6,q) with the property that every plane of Q(6,q) contains exactly one point of O. There are several reasons ovoids of Q(6,q) (and other polar spaces of low rank) are studied. The polar space Q(6,q) is one of the notorious cases, since the existence or non-existence of ovoids is still not settled completely. It is known that ovoids of Q(6,q) cannot exist when q is even or when q is equal to an odd prime p>3. However, for q=3h, there are two families of ovoids known, the so-called Thas-Kantor ovoids and the Ree-Tits ovoids, while for q=ph, p>3, h>1, existence or non-existence is open. The nice paper under review contributes to the characterization of particular ovoids. It is known (and recalled in the paper) that to any ovoid of Q(6,q), q a prime power, two polynomials f1(X,Y,Z) and f2(X,Y,Z) can be associated. The main theorem can be summarized as follows. Given two polynomials f1 and f2 with d=max{deg(f1),def(f2)}, suppose that q>6.3(d+1)(13/3). If f1 and f2 determine an ovoid O of Q(6,q), then q=3h and O is a Thas-Kantor ovoid. Note that, as the authors mention, the Ree-Tits ovoids give rise to polynomials (f1,f2) that do not satisfy the degree condition of the main theorem.