Ovoid (projective geometry)

Simple examples in a real projective space are hyperspheres (quadrics).

Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).

An ovoid is the spatial analog of an oval in a projective plane.

An ovoid is a special type of a quadratic set.

From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian[1]), the following result is true: Replacing the word projective in the definition of an ovoid by affine, gives the definition of an affine ovoid.

Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.

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 (projective space over

But the following method guarantees many non quadric ovoids: The last result can not be extended to even characteristic, because of the following non-quadric examples: the pointset An ovoidal quadric has many symmetries.

In particular: In the finite case one gets from Segre's theorem: Removing condition (1) 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[10] which is a generalization of a quadratic set.

Examples of semi-ovoids are the sets of isotropic points of an hermitian form.

As for ovoids in literature there are criteria, which make a semi-ovoid to a hermitian quadric.