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.^[1]

Cases

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.

See also

• Ovoid (projective geometry)

References

1. Moorhouse, G. Eric (2009), "Approaching some problems in finite geometry through algebraic geometry", in Klin, Mikhail; Jones, Gareth A.; Jurišić, Aleksandar; Muzychuk, Mikhail; Ponomarenko, Ilia, Algorithmic Algebraic Combinatorics and Gröbner Bases: Proceedings of the Workshop D1 "Gröbner Bases in Cryptography, Coding Theory and Algebraic Combinatorics” held in Linz, May 1–6, 2006, Berlin: Springer, pp. 285–296, doi:10.1007/978-3-642-01960-9_11, MR 2605578.