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

Abstract

Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a three dimensional elliptic quadric (corresponding to the regular spread of $\mathrm{PG}(3,q)$) 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 $Q^+(5,q)$. Another important class of ovoids of $Q^+(5,q)$ is given by the ones associated to a flock of a three dimensional quadratic cone. To every ovoid of $Q^+(5,q)$ two bivariate polynomials $f_1(x,y)$ and $f_2(x,y)$ can be associated. In this paper, we classify ovoids of $Q^+(5,q)$ such that $f_1(x,y)=y+g(x)$ and $\max\{deg(f_1),deg(f_2)\}<(\frac{1}{6.3}q)^{\frac{3}{13}}-1$, that is $f_1(x,y)$ and $f_2(x,y)$ have "low degree" compared with $q$.