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

This paper deals with the classification of ovoids of the Klein quadric Q+(5,q) having low degree with respect to q. Let P be a finite classical polar space of PG(m,q). A subset O of P is said to be an ovoid if O intersects each generator (i.e., a subspace of maximal dimension) of P in exactly one point. The polar space studied in the paper is the Klein quadric Q+(5,q):X0X5+X1X4+X2X3=0. Its ovoids consist of q2+1 pairwise non-collinear points of Q+(5,q). One can always assume that the points (1,0,0,0,0,0) and (0,0,0,0,0,1) are contained in an ovoid O5 of Q+(5,q), so that it can be written as O5(f1,f2)={(1,x,y,f1(x,y),f2(x,y),−xf2(x,y)−yf1(x,y)):x,y∈Fq}∪{(0,0,0,0,0,1)} for f1,f2:F2q→Fq such that f1(0,0)=f2(0,0)=0. The set O5(f1,f2) is in an ovoid if and only if, for all (x1,y1)≠(x2,y2)∈F2q, (x1−x2)(f2(x2,y2)−f2(x1,y1))+(y1−y2)(f1(x2,y2)−f1(x1,y1))≠0. Using the above characterization, a hypersurface Sf1,f2 of PG(4,q) is attached to O5(f1,f2), defined as Sf1,f2:F(X0,x1,X2,X3,X4)=0, with F(X0,x1,X2,X3,X4)=(X1−X3)(f~2(X3,X4,X0)−f~2(X1,X2,X0))+(X2−X4)(f~1(X3,X4,X0)−f~1(X1,X2,X0)), where f1,f2 are replaced by their homogenized versions f~1,f~2. The first result obtained is the following: If q>6.31(d+1)13/3, where d=max{deg(f1),deg(f2)} and Sf1,f2 contains an absolutely irreducible component defined over Fq, then O5(f1,f2) is not an ovoid of the Klein quadric. Then, several cases for f1 and f2 are studied, in order to prove that the corresponding hypersurface contains an absolutely irreducible component defined over Fq, and therefore providing non-existence results. By putting all the results together, the main result of the paper is obtained, which reads as follows. If q>6.31(d+1)13/3 and O5(f1,f2) is an ovoid of Q+(5,q), then Sf1,f2 does not contain an absolutely irreducible component defined over Fq, and all admissible values of deg(f1),deg(f2) and p (the field characteristic) are listed.