Ovoid (polar space)

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.

Cases

Symplectic polar space

An ovoid of W_{2n − 1}(q) (a symplectic polar space of rank n) would contain qⁿ + 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 qⁿ + 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_{2n − 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 qⁿ + 1 points.

See also

• Ovoid (projective geometry)