Ovoid (projective geometry)

Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).

Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries.

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.

Simple examples, which are not quadrics can be obtained by the following constructions:

(a) Glue one half of a hypersphere to a suitable hyperellipsoid in a smooth way.

The last result can not be extended to even characteristic, because of the following non-quadric examples:

Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid:

A semi ovoid is a special semi-quadratic set[10] 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.[11]

Semi-ovoids are used in the construction of examples of Möbius geometries.