In mathematics, a quadratic set is a set of points in a projective plane/space which bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
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Definition of a quadratic set
Let
A quadratic set
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.
Theorem: Let beDefinition of an oval and an ovoid
Ovals and ovoids are special quadratic sets:
Let
The following equivalent definition of an oval/ovoid are more common:
Definition: (oval) A non-empty point set
A line
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
Definition: (ovoid) A non-empty point set
Example:
a) Any sphere (quadric of index 1) is an ovoid.b) In case of real projective spaces one can construct ovoids by combining halves of suitable ellipsoids such that they are no quadrics.For finite projective spaces of dimension
Theorem:
Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for