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.