On the classification of low-degree ovoids of $$Q^+(5,q)$$

Ovoids of the Klein quadric Q(+) (5, q) of PG(5, q) have been studied in the last 40 years, also because of their connection with spreads of PG(3, q) and hence translation planes. Beside the classical example given by a three-dimensional elliptic quadric (corresponding to the regular spread of PG(3,q)) many other classes of examples are known. First of all the other examples (beside the elliptic quadric) of ovoids of Q(4, q) give also examples of ovoids of Q(+) (5, q). To every ovoid of Q(+) (5, q) two bivariate polynomials f(1)(x, y) and f(2)(x, y) can be associated. Another important class of ovoids of Q(+) (5, q) is given by the ones associated to a flock of a three-dimensional quadratic cone and in this case f(1)(x, y) = y + g(x). In this paper, we classify such ovoids of Q(+) (5, q) with the additional properties that max{deg(f(1)), deg(f(2))} < (1/6.31q)(3/13) - 1, that is f(1)(x, y) and f(2)(x, y) have "low degree" compared with q.