View source for 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>r-1</math> intersects ''O'' in exactly one point.<ref>{{citation
 | last = Moorhouse | first = G. Eric
 | editor1-last = Klin | editor1-first = Mikhail
 | editor2-last = Jones | editor2-first = Gareth A.
 | editor3-last = Jurišić | editor3-first = Aleksandar
 | editor4-last = Muzychuk | editor4-first = Mikhail
 | editor5-last = Ponomarenko | editor5-first = Ilia
 | contribution = Approaching some problems in finite geometry through algebraic geometry
 | doi = 10.1007/978-3-642-01960-9_11
 | location = Berlin
 | mr = 2605578
 | pages = 285–296
 | publisher = Springer
 | title = 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
 | url = https://books.google.com/books?id=sstt1cj7Nv8C&pg=PA285
 | year = 2009| isbn = 978-3-642-01959-3
 | citeseerx = 10.1.1.487.1198
 }}.</ref>

==Cases==
===Symplectic polar space===

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

===Hermitian polar space===

Ovoids of <math>H(2n,q^2)(n\geq 2)</math> and <math>H(2n+1,q^2)(n\geq 1)</math> would contain <math>q^{2n+1}+1</math> points.

===Hyperbolic quadrics===

An ovoid of a hyperbolic quadric<math> Q^{+}(2n-1,q)(n\geq 2)</math>would contain <math>q^{n-1}+1</math> points.

===Parabolic quadrics===

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

===Elliptic quadrics===

An ovoid of an elliptic quadric <math>Q^{-}(2n+1,q)(n\geq 2)</math>would contain <math>q^{n}+1</math> points.

==See also==

* [[Ovoid (projective geometry)]]

==References==
{{reflist}}

[[Category:Incidence geometry]]