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

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

Definition of an oval and an ovoid

Ovals and ovoids are special quadratic sets:

Let $\mathfrak P$ be a projective space of dimension $\ge 2$. A non-degenerate quadratic set $\mathcal O$ 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 $\mathfrak o$ of a projective plane is called
oval if the following properties are fulfilled:
:(o1) Any line meets $\mathfrak o$ in at most two points.
:(o2) For any point $P$ in $\mathfrak o$ there is one and only one line $g$ such that $g\cap \mathfrak o=\$.
A line $g$ is a ''exterior'' or ''tangent'' or ''secant'' line of the
oval if $, g\cap \mathfrak o, =0$ or $, g\cap \mathfrak o, =1$ or $, g\cap \mathfrak o, =2$ respectively.

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

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

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

Theorem:
Let be $\mathfrak P$ a ''Pappian'' projective plane of ''odd'' order.
Any oval in $\mathfrak P$ is an oval ''conic'' (non-degenerate quadric).

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

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 $n$ over a field $K$ we have:

Theorem:
:a) In case of $, K, <\infty$ an ovoid in $\mathfrak P_n(K)$ exists only if $n=2$ or $n=3$.
:b) In case of $, K, <\infty,\ \operatorname K \ne 2$ an ovoid in $\mathfrak P_n(K)$ is a quadric.

Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for $\operatorname K=2$:

References

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

External links

* Eric Hartman
Lecture Note ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes''
from Technische Universität Darmstadt

Geometry