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 ${\displaystyle r-1}$ intersects O in exactly one point.^[1]

Cases

Symplectic polar space

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

Hermitian polar space

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

Hyperbolic quadrics

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

Parabolic quadrics

An ovoid of a parabolic quadric ${\displaystyle Q(2n,q)(n\geq 2)}$ would contain ${\displaystyle q^{n}+1}$ points. For ${\displaystyle n=2}$, 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, ${\displaystyle Q(2n,q)}$ is isomorphic (as polar space) with ${\displaystyle W_{2n-1}(q)}$, and thus due to the above, it has no ovoid for ${\displaystyle n\geq 3}$.

Elliptic quadrics

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

See also

• Ovoid (projective geometry)