Quadratic set
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).
Definition of a quadratic set
Let be a projective space. A non empty subset of is called quadratic set if
- (QS1) Any line of intersects in at most 2 points or is contained in .
- ( is called exterior, tangent and secant line if and respectively.)
- (QS2) For any point the union of all tangent lines through is a hyperplane or the entire space .
A quadratic set is called non degenerated if for any point set is a hyperplane.
The following result is an astonishing statement for finite projective spaces.
Theorem(BUEKENHOUT): Let be a finite projective space of dimension and a non degenerated quadratic set which contains lines. Then: is pappian and is a quadric with index .
Definition of an oval and an ovoid
Ovals and ovoids are special quadratic sets:
Let be a projective space of dimension . A non degenerated quadratic set 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:
- (o1) Any line meets in at most two points.
- (o2) For any point there is one and only one line such that .
A line is a exterior or tangent or secant line of the oval if \ or or respectively.
For finite planes the following theorem provides a more simple definition.
Theorem: (oval in finite plane) Let be a projective plane of order . A set of points is an oval if and if no three points of are collinear.
For pappian projective planes of odd order the ovals are just conics:
Theorem (SEGRE):
Let be a pappian projective plane of odd order.
Any oval in 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:
- (O1) Any line meets in at most two points.
- ( is called exterior, tangent and secant line if and respectively.)
- (O2) For any point the union of all tangent lines through is a hyperplane (tangent plane at ).
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 over a field we have:
Theorem:
- a) In case of an ovoid in exists only if or .
- b) In case of an ovoid in is a quadric.
Counter examples (TITS–SUZUKI-ovoid) show that i.g. statement b) of the theorem above is not true for :
References
- Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry : from foundations to applications, § 4.3 Quadratic sets in spaces of small dimension, page 144, § 4.4 Quadratic sets in finite projective spaces, page 147, 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 Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, from Technische Universität Darmstadt