# Ovoid(2)

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

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=48091
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