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 (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).

A quadratic set is a non-empty subset

for which the following two conditions hold: A quadratic set

is called non-degenerate if for every point

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.

Ovals and ovoids are special quadratic sets: Let

that does not contain lines is called ovoid (or oval in plane case).

The following equivalent definition of an oval/ovoid are more common: Definition: (oval) 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

For finite planes the following theorem provides a more simple definition.

Theorem: (oval in finite plane) Let be

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.

is an oval conic (non-degenerate quadric).

Definition: (ovoid) 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

we have: Theorem: Counterexamples (Tits–Suzuki ovoid) show that i.g.