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| In the [[projective space]] ''PG(3,q)'', with ''q'' a prime power greater than 2, an '''ovoid''' is a set of <math>q^2+1</math> points, no three of which are collinear (the maximum size of such a set).<ref>more properly the term should be ''ovaloid'' and ovoid has a different definition which extends to projective spaces of higher dimension. However, in dimension 3 the two concepts are equivalent and the ovoid terminology is almost universally used, except most notably, in Hirschfeld.</ref> When <math>q = 2</math> the largest set of non-collinear points has size eight and is the complement of a plane.<ref>{{harvnb|Hirschfeld|1985|loc=pg.33, Theorem 16.1.3}}</ref>
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| An important example of an ovoid in any finite projective three-dimensional space are the <math>q^2+1</math> points of an elliptic quadric (all of which are projectively equivalent).
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| When ''q'' is odd or <math>q = 4</math>, no ovoids exist other than the elliptic [[quadric]]s.<ref>{{harvnb|Barlotti|1955}} and {{harvnb|Panella|1955}}</ref>
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| When <math>q=2^{2 h+1}</math> another type of ovoid can be constructed : the [[Jacques Tits|Tits]] ovoid, also known as the [[Michio Suzuki|Suzuki]] ovoid. It is conjectured that no other ovoids exist in ''PG(3,q)''.
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| Through every point ''P'' on the ovoid, there are exactly <math>q+1</math> tangents, and it can be proven that these lines are exactly the lines through ''P'' in one specific plane through ''P''. This means that through every point ''P'' in the ovoid, there is a unique plane intersecting the ovoid in exactly one point.<ref>{{harvnb|Hirschfeld|1985|loc=pg. 34, Lemma 16.1.6}}</ref> Also, if ''q'' is odd or <math>q = 4</math> every plane which is not a tangent plane meets the ovoid in a [[Conic section|conic]].<ref>{{harvnb|Hirschfeld|1985|loc=pg.35, Corollary}}</ref>
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| ==See also==
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| *[[Ovoid (polar space)]]
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| *[[Oval (projective plane)]]
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| *[[Inversive plane]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Barlotti|first=A.|title=Un' estensione del teorema di Segre-Kustaanheimo|journal=Boll. Un. Mat. Ital.|year=1955|volume=10|pages=96–98}}
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| *{{citation|last=Hirschfeld|first=J.W.P.|title=Finite Projective Spaces of Three Dimensions|year=1985|publisher=Oxford University Press|location=New York|isbn=0-19-853536-8}}
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| *{{citation|last=Panella|first=G.|title=Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito|journal=Boll. Un. Mat. Ital.|year=1955|volume=10|pages=507–513}}
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| {{DEFAULTSORT:Ovoid (Projective Geometry)}}
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| [[Category:Projective geometry]]
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| [[Category:Incidence geometry]]
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