Ovoid (polar space)

In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank $$r-1$$ intersects O in exactly one point.

Symplectic polar space
An ovoid of $$W_{2 n-1}(q)$$ (a symplectic polar space of rank n) would contain $$q^n+1$$ points. However it only has an ovoid if and only $$n=2$$ and q is even. In that case, when the polar space is embedded into $$PG(3,q)$$ the classical way, it is also an ovoid in the projective geometry sense.

Hermitian polar space
Ovoids of $$H(2n,q^2)(n\geq 2)$$ and $$H(2n+1,q^2)(n\geq 1)$$ would contain $$q^{2n+1}+1$$ points.

Hyperbolic quadrics
An ovoid of a hyperbolic quadric$$ Q^{+}(2n-1,q)(n\geq 2)$$would contain $$q^{n-1}+1$$ points.

Parabolic quadrics
An ovoid of a parabolic quadric $$Q(2 n,q)(n\geq 2)$$ would contain $$q^n+1$$ points. For $$n=2$$, it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid. If q is even, $$Q(2n,q)$$ is isomorphic (as polar space) with $$W_{2 n-1}(q)$$, and thus due to the above, it has no ovoid for $$n\geq 3$$.

Elliptic quadrics
An ovoid of an elliptic quadric $$Q^{-}(2n+1,q)(n\geq 2)$$would contain $$q^{n}+1$$ points.