Ovoid (polar space)

From Wikipedia, the free encyclopedia

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.

1 Cases
2 See also

[edit] Cases

[edit] 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 $P G (3, q)$ the classical way, it is also an ovoid in the projective geometry sense.

[edit] Hermitian polar space

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

[edit] Hyperbolic quadrics

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

[edit] 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, $Q (2 n, q)$ is isomorphic (as polar space) with $W 2 n - 1 (q)$ , and thus due to the above, it has no ovoid for $n\geq 3$ .