Quadratic set


In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric.

Definition of a quadratic set

Let be a projective space. A quadratic set is a non-empty subset of for which the following two conditions hold:
A quadratic set is called non-degenerate if for every point, the set is a hyperplane.
A Pappian projective space is a projective space in which Pappus's hexagon theorem holds.
The following result, due to Francis Buekenhout, is an astonishing statement for finite projective spaces.

Definition of an oval and an ovoid

Ovals and ovoids are special quadratic sets:

Let be a projective space of dimension. A non-degenerate quadratic set that does not contain lines is called ovoid.
The following equivalent definition of an oval/ovoid are more common:
Definition:
A non-empty point set of a projective plane is called
oval if the following properties are fulfilled:
A line is a exterior or tangent or secant line of the
oval if or or respectively.
For finite planes the following theorem provides a more simple definition.
Theorem: Let be a projective plane of order.
A set of points is an oval if and if no three points
of are collinear.
According to this theorem of Beniamino Segre, for Pappian projective planes of odd order the ovals are just conics:
Theorem:
Let be a Pappian projective plane of odd order.
Any oval in is an oval conic.
Definition:
A non-empty point set of a projective space is called ovoid if the following properties are fulfilled:
Example:
For finite projective spaces of dimension over a field we have:

Theorem:
Counterexamples of the theorem above is not true for :