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 intersects O in exactly one point.[1]
Cases[edit]
Symplectic polar space[edit]
An ovoid of (a symplectic polar space of rank n) would contain points. However it only has an ovoid if and only and q is even. In that case, when the polar space is embedded into the classical way, it is also an ovoid in the projective geometry sense.
Hermitian polar space[edit]
Ovoids of and would contain points.
Hyperbolic quadrics[edit]
An ovoid of a hyperbolic quadricwould contain points.
Parabolic quadrics[edit]
An ovoid of a parabolic quadric would contain points. For , 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, is isomorphic (as polar space) with , and thus due to the above, it has no ovoid for .
Elliptic quadrics[edit]
An ovoid of an elliptic quadric would contain points.
See also[edit]
References[edit]
- ^ 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 (eds.), 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, CiteSeerX 10.1.1.487.1198, doi:10.1007/978-3-642-01960-9_11, ISBN 978-3-642-01959-3, MR 2605578.