On the Classification of Low-Degree Ovoids of Q(4,q)

Ovoids of Q(6, q) of low degree

“…be the surface associated with an ovoid in Q(4, q); see [3]. As already observed in [3], up to a change of variables we can always suppose that a 0,1 = 0. Suppose that this is not the case, then consider S…”

“…defined as in (3). Suppose that q > 6.3(d +1) 13/3 , max{deg( f 1 ), deg( f 2 )} = d, and W f 1 , f 2 contains an absolutely irreducible component V defined over F q .…”

“…A starting point in our investigation is the main result of [3] which characterizes ovoids of Q(4, q) of "small degree". Similarly to what happens in the dimension-6 case, any ovoid O 4 of Q(4, q) can be written in the following form:…”

“…x, y, z, f 1 (x, y, z), f 2 (x, y, z), −z 2 − y f 1 (x, y, z) − x f 2 (x, y, z))} x,y,z∈F q ∪{(0, 0, 0, 0, 0, 0, 1)}, for some functions f i : F 3 q −→ F q with f i (0, 0, 0) = 0 for i ∈ {1, 2}. The set O 6 ( f 1 , f 2 ) is an ovoid if and only if (1) for every (x 1 , y 1 , z 1 ) = (x 2 , y 2 , z 2 ) in F 3 q , where •, • is the symmetric form associated to the quadratic form of the quadric Q. The only known ovoids of Q are the two classes of ovoids listed in the following table.…”