Property 2) excludes degenerated cases excludes ruled surfaces. An ovoid is the spatial analog of an oval in a projective plane. An ovoid is a special type of a quadratic set. Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries.
Definition of an ovoid
In a projective space of dimension a set of points is called an ovoid, if
In the case of, the line is called a passingline, if the line is a tangent line, and if the line is a secant line. From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
For an ovoid and a hyperplane, which contains at least two points of, the subset is an ovoid within the hyperplane.
For finiteprojective spaces of dimension , the following result is true:
If is an ovoid in a finite projective space of dimension, then.
In a finite projective space of order and dimension any pointset is an ovoid if and only if and no three points are collinear.
Replacingthe wordprojective in the definition of an ovoid by affine, gives the definition of an affine ovoid. If for an ovoid there is a suitable hyperplane not intersecting it, one can call this hyperplane the hyperplane at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to. Also, any affine ovoid can be considered a projective ovoid in the projective closure of the affine space.
Examples
In real projective space (inhomogeneous representation)
These two examples are quadrics and are projectively equivalent. Simple examples, which are not quadrics can be obtained by the following constructions: Remark: The real examples can not be converted into the complex case. In a complex projective space of dimension there are no ovoidal quadrics, because in that case any non degenerated quadric contains lines. But the following method guarantees many non quadric ovoids:
For any non-finite projective space the existence of ovoids can be proven using transfinite induction.
Finite examples
Any ovoid in a finite projective space of dimension over a field of characteristic is a quadric.
The last result can not be extended to even characteristic, because of the following non-quadric examples:
For odd and the automorphism
the pointset
Criteria for an ovoid to be a quadric
An ovoidal quadric has many symmetries. In particular:
Let be an ovoid in a projective space of dimension and a hyperplane. If the ovoid is symmetric to any point , then is pappian and a quadric.
An ovoid in a projective space is a quadric, if the group of projectivities, which leave invariant operates 3-transitively on, i.e. for two triples there exists a projectivity with.
Let be an ovoid in a finite 3-dimensional desarguesian projective space of odd order, then is pappian and is a quadric.
Generalization: semi ovoid
Removing condition from the definition of an ovoid results in the definition of a semi-ovoid: the following conditions hold: A semi ovoid is a special semi-quadratic set which is a generalization of a quadratic set. The essential difference between a semi-quadratic set and a quadratic set is the fact, that there can be lines which have 3 points in common with the set and the lines are not contained in the set. Examples of semi-ovoids are the sets of isotropic points of an hermitian form. They are called hermitian quadrics. As for ovoids in literature there are criteria, which make a semi-ovoid to a hermitian quadric. See, for example. Semi-ovoids are used in the construction of examples of Möbius geometries.
