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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 <math>r-1</math> intersects ''O'' in exactly one point.
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| ==Cases==
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| ===Symplectic polar space===
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| 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.
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| 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.
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| ===Hermitian polar space===
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| 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.
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| ===Hyperbolic quadrics===
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| An ovoid of a hyperbolic quadric<math> Q^{+}(2n-1,q)(n\geq 2)</math>would contain <math>q^{n-1}+1</math> points.
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| ===Parabolic quadrics===
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| 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.
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| 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>.
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| ===Elliptic quadrics===
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| An ovoid of an elliptic quadric <math>Q^{-}(2n+1,q)(n\geq 2)</math>would contain <math>q^{n}+1</math> points.
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| ==See also==
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| * [[Ovoid (projective geometry)]]
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| [[Category:Incidence geometry]]
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| {{geometry-stub}}
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