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 P = ( P , G , ∈ ) {\displaystyle {\mathfrak {P}}=({\mathcal {P}},{\mathcal {G}},\in )} be a projective space. A quadratic set is a non-empty subset Q {\displaystyle {\mathcal {Q}}} of P {\displaystyle {\mathcal {P}}} for which the following two conditions hold:
(QS1) Every line g {\displaystyle g} of G {\displaystyle {\mathcal {G}}} intersects Q {\displaystyle {\mathcal {Q}}} in at most two points or is contained in Q {\displaystyle {\mathcal {Q}}} . ( g {\displaystyle g} is called exterior to Q {\displaystyle {\mathcal {Q}}} if | g ∩ Q | = 0 {\displaystyle |g\cap {\mathcal {Q}}|=0} , tangent to Q {\displaystyle {\mathcal {Q}}} if either | g ∩ Q | = 1 {\displaystyle |g\cap {\mathcal {Q}}|=1} or g ∩ Q = g {\displaystyle g\cap {\mathcal {Q}}=g} , and secant to Q {\displaystyle {\mathcal {Q}}} if | g ∩ Q | = 2 {\displaystyle |g\cap {\mathcal {Q}}|=2} .) (QS2) For any point P ∈ Q {\displaystyle P\in {\mathcal {Q}}} the union Q P {\displaystyle {\mathcal {Q}}_{P}} of all tangent lines through P {\displaystyle P} is a hyperplane or the entire space P {\displaystyle {\mathcal {P}}} . A quadratic set Q {\displaystyle {\mathcal {Q}}} is called non-degenerate if for every point P ∈ Q {\displaystyle P\in {\mathcal {Q}}} , the set Q P {\displaystyle {\mathcal {Q}}_{P}} 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.
Theorem: Let be P n {\displaystyle {\mathfrak {P}}_{n}} a finite projective space of dimension n ≥ 3 {\displaystyle n\geq 3} and Q {\displaystyle {\mathcal {Q}}} a non-degenerate quadratic set that contains lines. Then: P n {\displaystyle {\mathfrak {P}}_{n}} is Pappian and Q {\displaystyle {\mathcal {Q}}} is a quadric with index ≥ 2 {\displaystyle \geq 2} .
Definition of an oval and an ovoid Ovals and ovoids are special quadratic sets: Let P {\displaystyle {\mathfrak {P}}} be a projective space of dimension ≥ 2 {\displaystyle \geq 2} . A non-degenerate quadratic set O {\displaystyle {\mathcal {O}}} 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 o {\displaystyle {\mathfrak {o}}} of a projective plane is called oval if the following properties are fulfilled:
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