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 <math>\mathfrak P=({\mathcal P},{\mathcal G},\in)</math> be a projective space. A '''quadratic set''' is a non-empty subset <math>{\mathcal Q}</math> of <math>{\mathcal P}</math> for which the following two conditions hold: :'''(QS1)''' Every line <math>g</math> of <math>{\mathcal G}</math> intersects <math>{\mathcal Q}</math> in at most two points or is contained in <math>{\mathcal Q}</math>. ::(<math>g</math> is called '''exterior''' to <math>{\mathcal Q}</math> if <math>|g\cap {\mathcal Q}|=0</math>, '''tangent''' to <math>{\mathcal Q}</math> if either <math>|g\cap {\mathcal Q}|=1</math> or <math>g\cap {\mathcal Q}=g</math>, and '''secant''' to <math>{\mathcal Q}</math> if <math>|g\cap {\mathcal Q}|=2</math>.) :'''(QS2)''' For any point <math>P\in {\mathcal Q}</math> the union <math>{\mathcal Q}_P</math> of all tangent lines through <math>P</math> is a hyperplane or the entire space <math>{\mathcal P}</math>.

A quadratic set <math>{\mathcal Q}</math> is called '''non-degenerate''' if for every point <math>P\in {\mathcal Q}</math>, the set <math>{\mathcal Q}_P</math> 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 <math>\mathfrak P_n</math> a ''finite'' projective space of dimension <math>n\ge 3</math> and <math>{\mathcal Q}</math> a non-degenerate quadratic set that contains lines. Then: <math>\mathfrak P_n</math> is Pappian and <math>{\mathcal Q}</math> is a ''quadric'' with index <math>\ge 2</math>.

==Definition of an oval and an ovoid== Ovals and ovoids are special quadratic sets:<br /> Let <math>\mathfrak P</math> be a projective space of dimension <math>\ge 2</math>. A non-degenerate quadratic set <math>\mathcal O</math> 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 <math>\mathfrak o</math> of a projective plane is called '''oval''' if the following properties are fulfilled: :'''(o1)''' Any line meets <math>\mathfrak o</math> in at most two points. :('''o2)''' For any point <math>P</math> in <math>\mathfrak o</math> there is one and only one line <math>g</math> such that <math>g\cap \mathfrak o=\{P\}</math>. A line <math>g</math> is a ''exterior'' or ''tangent'' or ''secant'' line of the oval if <math>|g\cap \mathfrak o|=0</math> or <math>|g\cap \mathfrak o|=1</math> or <math>|g\cap \mathfrak o|=2</math> respectively.

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

'''Theorem: (oval in finite plane) '''Let be <math> \mathfrak P</math> a projective plane of order <math>n</math>. A set <math>\mathfrak o</math> of points is an '''oval''' if <math>|\mathfrak o|=n+1</math> and if no three points of <math>\mathfrak o</math> are collinear.

According to this theorem of Beniamino Segre, for ''Pappian'' projective planes of ''odd'' order the ovals are just conics: '''Theorem:''' Let be <math> \mathfrak P</math> a ''Pappian'' projective plane of ''odd'' order. Any oval in <math> \mathfrak P</math> is an oval ''conic'' (non-degenerate quadric).

'''Definition: (ovoid)''' A non-empty point set <math>\mathcal O</math> of a projective space is called '''ovoid''' if the following properties are fulfilled: :'''(O1)''' Any line meets <math>\mathcal O</math> in at most two points. :(<math>g</math> is called '''exterior, tangent''' and '''secant''' line if <math>|g\cap {\mathcal O}|=0, \ |g\cap {\mathcal O}|=1</math> and <math>|g\cap {\mathcal O}|=2</math> respectively.) :'''(O2)''' For any point <math>P\in {\mathcal O}</math> the union <math>{\mathcal O}_P</math> of all tangent lines through <math>P</math> is a hyperplane (tangent plane at <math>P</math>).

'''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 <math>n</math> over a field <math>K</math> we have:<br /> '''Theorem:''' :a) In case of <math>|K| <\infty</math> an ovoid in <math>\mathfrak P_n(K)</math> exists only if <math>n=2</math> or <math>n=3</math>. :b) In case of <math>|K| <\infty,\ \operatorname{char} K \ne 2</math> an ovoid in <math>\mathfrak P_n(K)</math> is a quadric.

Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for <math>\operatorname{char} K=2</math>:

==References== {{Reflist}}

* Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry : from foundations to applications'', Chapter 4: Quadratic Sets, pages 137 to 179, Cambridge University Press {{ISBN|978-0521482776}} * F. Buekenhout (ed.) (1995) ''Handbook of Incidence Geometry'', Elsevier {{ISBN|0-444-88355-X}} * P. Dembowski (1968) ''Finite Geometries'', Springer-Verlag {{ISBN|3-540-61786-8}}, p. 48

==External links== * Eric Hartmann [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes''], from Technische Universität Darmstadt

Category:Geometry