A set of points in some space which is intersected by an arbitrary straight line in at most two points, and such that the tangents to 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 , an ovoid is a maximal -cap (cf. Cap) and consists of points, and for odd any ovoid is an elliptic quadric (see ). In a plane of order , an ovoid is called an oval, and consists of points. In a Desarguesian plane of odd order, any oval is uniquely representable as a non-degenerate conic over a Galois field (see ).
|||B. Segre, "Introduction to Galois geometries" Atti Accad. Naz. Lincei , 8 (1967) pp. 133–236|
|||B. Segre, "Ovals in a finite projective plane" Canad. J. Math. , 7 (1955) pp. 414–416|
|||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 is a set of points such that no four lie in a plane and such that at each there is a unique hyperplane through tangent to at that point. Here "tangent" means that the intersection of with the hyperplane consists only of itself.
For a finite field of odd characteristic the ovoids in 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 of points such that every maximal singular subspace intersects in exactly one point. A spread in a generalized quadrangle is a set of lines such that each point is incident with one line of . A spread is an ovoid in the dual generalized quadrangle. An ovoid in a finite generalized quadrangle of order has cardinality .
A (trivial) example of an ovoid is the set of encircled points in the grid (cf. Quadrangle) depicted below:
The connection between the abstract notion of an ovoid in a polar space and an ovoid in is as follows. Consider the classical generalized quadrangle defined by a symplectic bilinear form . I.e. the points are the points of (which are all isotropic) and the lines are the totally isotropic lines of with respect to this form. Then an ovoid in this generalized quadrangle viewed as a subset of is an ovoid in the sense of the geometric version of the concept. (The tangent plane to is .
Let , a finite field, be the (classical) polar space defined by the bilinear form
Ovoids in are used to obtain non-Desarguesian translation planes. From one "master" ovoid in one obtains many ovoids in . It is an open problem whether there are ovoids in . There are none , [a4], or in , [a5].
|[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|
Ovoid(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ovoid(2)&oldid=17115