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Ovoid (polar space)

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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 {\displaystyle r-1} intersects O in exactly one point.

Cases

Symplectic polar space

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

Hermitian polar space

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

Hyperbolic quadrics

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

Parabolic quadrics

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

Elliptic quadrics

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