Ovoids of Q(6, q) of low degree

Abstract

AbstractOvoids of the parabolic quadric Q(6, q) of $$\textrm{PG}(6,q)$$ PG ( 6 , q ) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials $$f_1(X,Y,Z)$$ f 1 ( X , Y , Z ) , $$f_2(X,Y,Z)$$ f 2 ( X , Y , Z ) can be associated. In this paper we classify ovoids of Q(6, q) with $$\max \{\deg (f_1),\deg (f_2)\}<(\frac{1}{6.3}q)^{\frac{3}{13}}-1$$ max { deg ( f 1 ) , deg ( f 2 ) } < ( 1 6.3 q ) 3 13 - 1 .