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''ovaloid''
 
''ovaloid''
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706701.png" /> of points in some space which is intersected by an arbitrary straight line in at most two points, and such that the tangents to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706702.png" /> at each of its points cover exactly a hyperplane. In projective space a non-ruled [[Quadric|quadric]] is an ovoid. This term is mainly used in finite geometries.
+
A set $  O $
 +
of points in some space which is intersected by an arbitrary straight line in at most two points, and such that the tangents to $  O $
 +
at each of its points cover exactly a hyperplane. In projective space a non-ruled [[Quadric|quadric]] is an ovoid. This term is mainly used in finite geometries.
  
In finite projective spaces of dimension greater than three, ovoids do not exist. In three-dimensional spaces of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706703.png" />, an ovoid is a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706704.png" />-cap (cf. [[Cap|Cap]]) and consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706705.png" /> points, and for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706706.png" /> any ovoid is an elliptic quadric (see [[#References|[1]]]). In a plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706707.png" />, an ovoid is called an oval, and consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706708.png" /> points. In a Desarguesian plane of odd order, any oval is uniquely representable as a non-degenerate conic over a Galois field (see [[#References|[2]]]).
+
In finite projective spaces of dimension greater than three, ovoids do not exist. In three-dimensional spaces of order $  q > 2 $,  
 +
an ovoid is a maximal $  k $-
 +
cap (cf. [[Cap|Cap]]) and consists of $  q  ^ {2} + 1 $
 +
points, and for odd $  q $
 +
any ovoid is an elliptic quadric (see [[#References|[1]]]). In a plane of order $  q $,  
 +
an ovoid is called an oval, and consists of $  q + 1 $
 +
points. In a Desarguesian plane of odd order, any oval is uniquely representable as a non-degenerate conic over a Galois field (see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Segre,  "Introduction to Galois geometries"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1967)  pp. 133–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Segre,  "Ovals in a finite projective plane"  ''Canad. J. Math.'' , '''7'''  (1955)  pp. 414–416</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Tits,  "Ovoids à translations"  ''Rend. Mat. e Appl.'' , '''21'''  (1962)  pp. 37–59</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Segre,  "Introduction to Galois geometries"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1967)  pp. 133–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Segre,  "Ovals in a finite projective plane"  ''Canad. J. Math.'' , '''7'''  (1955)  pp. 414–416</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Tits,  "Ovoids à translations"  ''Rend. Mat. e Appl.'' , '''21'''  (1962)  pp. 37–59</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For Desarguesian planes of even order there are counterexamples to the last statement above.
 
For Desarguesian planes of even order there are counterexamples to the last statement above.
  
An ovoid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o0706709.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067010.png" /> of points such that no four lie in a plane and such that at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067011.png" /> there is a unique hyperplane through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067012.png" /> tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067013.png" /> at that point. Here  "tangent"  means that the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067014.png" /> with the hyperplane consists only of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067015.png" /> itself.
+
An ovoid in $  \mathbf P  ^ {3} $
 +
is a set $  {\mathcal O} $
 +
of points such that no four lie in a plane and such that at each $  A \in {\mathcal O} $
 +
there is a unique hyperplane through $  A $
 +
tangent to $  {\mathcal O} $
 +
at that point. Here  "tangent"  means that the intersection of $  {\mathcal O} $
 +
with the hyperplane consists only of $  A $
 +
itself.
  
For a finite field of odd characteristic the ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067016.png" /> are precisely the zeros of a quadratic form of Witt index 1, [[#References|[a1]]].
+
For a finite field of odd characteristic the ovoids in $  \mathbf P  ^ {3} $
 +
are precisely the zeros of a quadratic form of Witt index 1, [[#References|[a1]]].
  
An ovoid in a [[Polar space|polar space]] (in particular, in a generalized [[Quadrangle|quadrangle]]) is a collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067017.png" /> of points such that every maximal singular subspace intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067018.png" /> in exactly one point. A spread in a generalized quadrangle is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067019.png" /> of lines such that each point is incident with one line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067020.png" />. A spread is an ovoid in the dual generalized quadrangle. An ovoid in a finite generalized quadrangle of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067021.png" /> has cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067022.png" />.
+
An ovoid in a [[Polar space|polar space]] (in particular, in a generalized [[Quadrangle|quadrangle]]) is a collection $  {\mathcal O} $
 +
of points such that every maximal singular subspace intersects $  {\mathcal O} $
 +
in exactly one point. A spread in a generalized quadrangle is a set $  {\mathcal R} $
 +
of lines such that each point is incident with one line of $  {\mathcal R} $.  
 +
A spread is an ovoid in the dual generalized quadrangle. An ovoid in a finite generalized quadrangle of order $  ( s, t) $
 +
has cardinality $  st+ 1 $.
  
 
A (trivial) example of an ovoid is the set of encircled points in the grid (cf. [[Quadrangle|Quadrangle]]) depicted below:
 
A (trivial) example of an ovoid is the set of encircled points in the grid (cf. [[Quadrangle|Quadrangle]]) depicted below:
Line 25: Line 56:
 
Figure: o070670a
 
Figure: o070670a
  
The connection between the abstract notion of an ovoid in a polar space and an ovoid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067023.png" /> is as follows. Consider the classical generalized quadrangle defined by a symplectic bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067024.png" />. I.e. the points are the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067025.png" /> (which are all isotropic) and the lines are the totally isotropic lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067026.png" /> with respect to this form. Then an ovoid in this generalized quadrangle viewed as a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067027.png" /> is an ovoid in the sense of the geometric version of the concept. (The tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067028.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067029.png" />.
+
The connection between the abstract notion of an ovoid in a polar space and an ovoid in $  \mathbf P  ^ {3} $
 +
is as follows. Consider the classical generalized quadrangle defined by a symplectic bilinear form $  Q $.  
 +
I.e. the points are the points of $  \mathbf P  ^ {3} $(
 +
which are all isotropic) and the lines are the totally isotropic lines of $  \mathbf P  ^ {3} $
 +
with respect to this form. Then an ovoid in this generalized quadrangle viewed as a subset of $  \mathbf P  ^ {3} $
 +
is an ovoid in the sense of the geometric version of the concept. (The tangent plane to $  ( y _ {0} :  y _ {1} :  y _ {2} :  y _ {3} ) = A \in {\mathcal O} $
 +
is $  A  ^  \perp  = \{ {x \in \mathbf P  ^ {3} } : {Q( x, y) = 0 } \} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067031.png" /> a finite field, be the (classical) polar space defined by the bilinear form
+
Let $  \Omega  ^ {+} ( 2n, \mathbf F ) $,
 +
$  \mathbf F $
 +
a finite field, be the (classical) polar space defined by the bilinear form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067032.png" /></td> </tr></table>
+
$$
 +
x _ {0} x _ {1} + x _ {2} x _ {3} + \dots + x _ {2n-} 2 x _ {2n-} 1 .
 +
$$
  
Ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067033.png" /> are used to obtain non-Desarguesian translation planes. From one  "master"  ovoid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067034.png" /> one obtains many ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067035.png" />. It is an open problem whether there are ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067036.png" />. There are none <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067037.png" />, [[#References|[a4]]], or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067038.png" />, [[#References|[a5]]].
+
Ovoids in $  \Omega  ^ {+} ( 6, \mathbf F ) $
 +
are used to obtain non-Desarguesian translation planes. From one  "master"  ovoid in $  \Omega  ^ {+} ( 8, \mathbf F ) $
 +
one obtains many ovoids in $  \Omega  ^ {+} ( 6, \mathbf F ) $.  
 +
It is an open problem whether there are ovoids in $  \Omega  ^ {+} ( 10, \mathbf F ) $.  
 +
There are none $  \Omega  ^ {+} ( 10, \mathbf F _ {3} ) $,  
 +
[[#References|[a4]]], or in $  \Omega  ^ {+} ( 10 , \mathbf F _ {2} ) $,  
 +
[[#References|[a5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Barlotti,  "Un' estenzione del teorema di Segre–Kustaanheimo"  ''Boll. Un. Mat. Ital. (3)'' , '''10'''  (1955)  pp. 498–506</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.E. Pagne,  J.A. Thas,  "Finite generalized quadrangles" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Mason,  E.E. Shult,  "The Klein correspondence and the ubiquity of certain translation planes"  ''Geom. Dedicata'' , '''21'''  (1986)  pp. 29–50</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.E. Shult,  "Nonexistence of ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067039.png" />"  ''J. Comb. Theory, Ser. A'' , '''51'''  (1989)  pp. 250–257</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.M. Kantor,  "Ovoids and translation planes"  ''Canad. J. Math.'' , '''34'''  (1982)  pp. 1195–1207</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Finite projective spaces of three dimensions" , Clarendon Press  (1985)  pp. Chapt. 16</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Barlotti,  "Un' estenzione del teorema di Segre–Kustaanheimo"  ''Boll. Un. Mat. Ital. (3)'' , '''10'''  (1955)  pp. 498–506</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.E. Pagne,  J.A. Thas,  "Finite generalized quadrangles" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Mason,  E.E. Shult,  "The Klein correspondence and the ubiquity of certain translation planes"  ''Geom. Dedicata'' , '''21'''  (1986)  pp. 29–50</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.E. Shult,  "Nonexistence of ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067039.png" />"  ''J. Comb. Theory, Ser. A'' , '''51'''  (1989)  pp. 250–257</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.M. Kantor,  "Ovoids and translation planes"  ''Canad. J. Math.'' , '''34'''  (1982)  pp. 1195–1207</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Finite projective spaces of three dimensions" , Clarendon Press  (1985)  pp. Chapt. 16</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


ovaloid

A set $ O $ of points in some space which is intersected by an arbitrary straight line in at most two points, and such that the tangents to $ O $ at each of its points cover exactly a hyperplane. In projective space a non-ruled quadric is an ovoid. This term is mainly used in finite geometries.

In finite projective spaces of dimension greater than three, ovoids do not exist. In three-dimensional spaces of order $ q > 2 $, an ovoid is a maximal $ k $- cap (cf. Cap) and consists of $ q ^ {2} + 1 $ points, and for odd $ q $ any ovoid is an elliptic quadric (see [1]). In a plane of order $ q $, an ovoid is called an oval, and consists of $ q + 1 $ points. In a Desarguesian plane of odd order, any oval is uniquely representable as a non-degenerate conic over a Galois field (see [2]).

References

[1] B. Segre, "Introduction to Galois geometries" Atti Accad. Naz. Lincei , 8 (1967) pp. 133–236
[2] B. Segre, "Ovals in a finite projective plane" Canad. J. Math. , 7 (1955) pp. 414–416
[3] J. Tits, "Ovoids à translations" Rend. Mat. e Appl. , 21 (1962) pp. 37–59

Comments

For Desarguesian planes of even order there are counterexamples to the last statement above.

An ovoid in $ \mathbf P ^ {3} $ is a set $ {\mathcal O} $ of points such that no four lie in a plane and such that at each $ A \in {\mathcal O} $ there is a unique hyperplane through $ A $ tangent to $ {\mathcal O} $ at that point. Here "tangent" means that the intersection of $ {\mathcal O} $ with the hyperplane consists only of $ A $ itself.

For a finite field of odd characteristic the ovoids in $ \mathbf P ^ {3} $ are precisely the zeros of a quadratic form of Witt index 1, [a1].

An ovoid in a polar space (in particular, in a generalized quadrangle) is a collection $ {\mathcal O} $ of points such that every maximal singular subspace intersects $ {\mathcal O} $ in exactly one point. A spread in a generalized quadrangle is a set $ {\mathcal R} $ of lines such that each point is incident with one line of $ {\mathcal R} $. A spread is an ovoid in the dual generalized quadrangle. An ovoid in a finite generalized quadrangle of order $ ( s, t) $ has cardinality $ st+ 1 $.

A (trivial) example of an ovoid is the set of encircled points in the grid (cf. Quadrangle) depicted below:

Figure: o070670a

The connection between the abstract notion of an ovoid in a polar space and an ovoid in $ \mathbf P ^ {3} $ is as follows. Consider the classical generalized quadrangle defined by a symplectic bilinear form $ Q $. I.e. the points are the points of $ \mathbf P ^ {3} $( which are all isotropic) and the lines are the totally isotropic lines of $ \mathbf P ^ {3} $ with respect to this form. Then an ovoid in this generalized quadrangle viewed as a subset of $ \mathbf P ^ {3} $ is an ovoid in the sense of the geometric version of the concept. (The tangent plane to $ ( y _ {0} : y _ {1} : y _ {2} : y _ {3} ) = A \in {\mathcal O} $ is $ A ^ \perp = \{ {x \in \mathbf P ^ {3} } : {Q( x, y) = 0 } \} $.

Let $ \Omega ^ {+} ( 2n, \mathbf F ) $, $ \mathbf F $ a finite field, be the (classical) polar space defined by the bilinear form

$$ x _ {0} x _ {1} + x _ {2} x _ {3} + \dots + x _ {2n-} 2 x _ {2n-} 1 . $$

Ovoids in $ \Omega ^ {+} ( 6, \mathbf F ) $ are used to obtain non-Desarguesian translation planes. From one "master" ovoid in $ \Omega ^ {+} ( 8, \mathbf F ) $ one obtains many ovoids in $ \Omega ^ {+} ( 6, \mathbf F ) $. It is an open problem whether there are ovoids in $ \Omega ^ {+} ( 10, \mathbf F ) $. There are none $ \Omega ^ {+} ( 10, \mathbf F _ {3} ) $, [a4], or in $ \Omega ^ {+} ( 10 , \mathbf F _ {2} ) $, [a5].

References

[a1] A. Barlotti, "Un' estenzione del teorema di Segre–Kustaanheimo" Boll. Un. Mat. Ital. (3) , 10 (1955) pp. 498–506
[a2] S.E. Pagne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1984)
[a3] G. Mason, E.E. Shult, "The Klein correspondence and the ubiquity of certain translation planes" Geom. Dedicata , 21 (1986) pp. 29–50
[a4] E.E. Shult, "Nonexistence of ovoids in " J. Comb. Theory, Ser. A , 51 (1989) pp. 250–257
[a5] W.M. Kantor, "Ovoids and translation planes" Canad. J. Math. , 34 (1982) pp. 1195–1207
[a6] J.W.P. Hirschfeld, "Finite projective spaces of three dimensions" , Clarendon Press (1985) pp. Chapt. 16
How to Cite This Entry:
Ovoid(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ovoid(2)&oldid=17115
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article