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

Abstract

Ovoids of the Klein quadric $$Q^+(5,q)$$ Q + ( 5 , q ) of $$\textrm{PG}(5,q)$$ PG ( 5 , q ) have been studied in the last 40 years, also because of their connection with spreads of $$\textrm{PG}(3,q)$$ PG ( 3 , q ) and hence translation planes. Beside the classical example given by a three-dimensional elliptic quadric (corresponding to the regular spread of $$\textrm{PG}(3,q)$$ 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 ) } < ( 1 6.31 q ) 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.