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

Symplectic polar space

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

Ovoids of and would contain points.

Hyperbolic quadrics

An ovoid of a hyperbolic quadricwould contain points.

Parabolic quadrics

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

An ovoid of an elliptic quadric would contain points.

See also

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 (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.
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