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). Definition of a quadratic set. Let formula_1 be a projective space. A quadratic set is a non-empty subset formula_2 of formula_3 for which the following two conditions hold: A quadratic set formula_2 is called non-degenerate if for every point formula_16, the set formula_17 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 formula_29 be a projective space of dimension formula_28. A non-degenerate quadratic set formula_31 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 formula_32 of a projective plane is called oval if the following properties are fulfilled: A line formula_4 is a "exterior" or  tangent or "secant" line of the oval if formula_39 or formula_40 or formula_41 respectively. For "finite" planes the following theorem provides a more simple definition. Theorem: (oval in finite plane) Let be formula_42 a projective plane of order formula_43. A set formula_32 of points is an oval if formula_45 and if no three points of formula_32 are collinear. According to this theorem of Beniamino Segre, for "Pappian" projective planes of "odd" order the ovals are just conics: Theorem: Let be formula_42 a "Pappian" projective plane of "odd" order. Any oval in formula_42 is an oval "conic" (non-degenerate quadric). Definition : (ovoid) A non-empty point set formula_31 of a projective space is called ovoid if the following properties are fulfilled: Example: For "finite" projective spaces of dimension formula_43 over a field formula_59 we have: Theorem: Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for formula_66: