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

Ovoids of the Klein quadric \(Q^+(5,q)\) $Q^{+}(5,q)$ of \(\textrm{PG}(5,q)\) $\text{PG}(5,q)$ have been studied in the last 40 years, also because of their connection with spreads of \(\textrm{PG}(3,q)\) $\text{PG}(3,q)$ and hence translation planes. Beside the classical example given by a three-dimensional elliptic quadric (corresponding to the regular spread of \(\textrm{PG}(3,q)\) $\text{PG}(3,q)$ ) 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 \(Q^+(5,q)\) $Q^{+}(5,q)$ . To every ovoid of \(Q^+(5,q)\) $Q^{+}(5,q)$ two bivariate polynomials \(f_1(x,y)\) $f_1(x,y)$ and \(f_2(x,y)\) $f_2(x,y)$ can be associated. Another important class of ovoids of \(Q^+(5,q)\) $Q^{+}(5,q)$ is given by the ones associated to a flock of a three-dimensional quadratic cone and in this case \(f_1(x,y)=y+g(x)\) $f_1(x,y)=y+g(x)$ . In this paper, we classify such ovoids of \(Q^+(5,q)\) $Q^{+}(5,q)$ with the additional properties that \(\max \{\deg (f_1),\deg (f_2)\}<(\frac{1}{6.31}q)^{\frac{3}{13}}-1\) $max\{deg(f_1),deg(f_2)\}<{(\frac{1}{6.31}q)}^{\frac{3}{13}}-1$ , that is \(f_1(x,y)\) $f_1(x,y)$ and \(f_2(x,y)\) $f_2(x,y)$ have “low degree" compared with q.