my-server
← Wiki

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 be a projective space. A quadratic set is a non-empty subset of for which the following two conditions hold:

(QS1) Every line of intersects in at most two points or is contained in .
:( is called exterior to if , tangent to if either or , and secant to if .)
(QS2) For any point the union of all tangent lines through is a hyperplane or the entire space .

A quadratic set is called non-degenerate if for every point , the set 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 a finite projective space of dimension and a non-degenerate quadratic set that 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:<br /> Let be a projective space of dimension . A non-degenerate 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 in 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.

According to this theorem of Beniamino Segre, for Pappian projective planes of odd order the ovals are just conics:

Theorem: 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:<br /> Theorem:

a) In case of an ovoid in exists only if or .
b) In case of an ovoid in is a quadric.

Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for :

References

External links