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

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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

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Ovals and ovoids are special quadratic sets:
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:
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

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  • 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
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