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 [math]\displaystyle{ r-1 }[/math] intersects O in exactly one point.[1]
Cases
Symplectic polar space
An ovoid of [math]\displaystyle{ W_{2 n-1}(q) }[/math] (a symplectic polar space of rank n) would contain [math]\displaystyle{ q^n+1 }[/math] points. However it only has an ovoid if and only [math]\displaystyle{ n=2 }[/math] and q is even. In that case, when the polar space is embedded into [math]\displaystyle{ PG(3,q) }[/math] the classical way, it is also an ovoid in the projective geometry sense.
Hermitian polar space
Ovoids of [math]\displaystyle{ H(2n,q^2)(n\geq 2) }[/math] and [math]\displaystyle{ H(2n+1,q^2)(n\geq 1) }[/math] would contain [math]\displaystyle{ q^{2n+1}+1 }[/math] points.
Hyperbolic quadrics
An ovoid of a hyperbolic quadric[math]\displaystyle{ Q^{+}(2n-1,q)(n\geq 2) }[/math]would contain [math]\displaystyle{ q^{n-1}+1 }[/math] points.
Parabolic quadrics
An ovoid of a parabolic quadric [math]\displaystyle{ Q(2 n,q)(n\geq 2) }[/math] would contain [math]\displaystyle{ q^n+1 }[/math] points. For [math]\displaystyle{ n=2 }[/math], 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, [math]\displaystyle{ Q(2n,q) }[/math] is isomorphic (as polar space) with [math]\displaystyle{ W_{2 n-1}(q) }[/math], and thus due to the above, it has no ovoid for [math]\displaystyle{ n\geq 3 }[/math].
Elliptic quadrics
An ovoid of an elliptic quadric [math]\displaystyle{ Q^{-}(2n+1,q)(n\geq 2) }[/math]would contain [math]\displaystyle{ q^{n}+1 }[/math] points.
See also
References
- ↑ Moorhouse, G. Eric (2009), "Approaching some problems in finite geometry through algebraic geometry", in Klin, Mikhail; Jones, Gareth A.; Jurišić, Aleksandar et al., 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, ISBN 978-3-642-01959-3, https://books.google.com/books?id=sstt1cj7Nv8C&pg=PA285.
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