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 ${\mathfrak P}=({{\mathcal P}},{{\mathcal G}},\in )$ a projective space. A non empty subset ${{\mathcal Q}}$ of ${{\mathcal P}}$ is called quadratic set if

(QS1) Any line

g

{{\mathcal G}}

intersects

{{\mathcal Q}}

in at most 2 points or is contained in

{{\mathcal Q}}

(

g

is called exterior, tangent and secant line if

|g\cap {{\mathcal Q}}|=0,\ |g\cap {{\mathcal Q}}|=1

and

|g\cap {{\mathcal Q}}|=2

respectively.)

(QS2) For any point

P\in {{\mathcal Q}}

the union

{{\mathcal Q}}_{P}

of all tangent lines through

P

is a hyperplane or the entire space

{{\mathcal P}}

A quadratic set ${{\mathcal Q}}$ is called non degenerated if for any point $P$ set ${{\mathcal Q}}_{P}$ is a hyperplane.

The following result is an astonishing statement for finite projective spaces.

Theorem(BUEKENHOUT): Let be ${\mathfrak P}_{n}$ a finite projective space of dimension $n\geq 3$ and ${{\mathcal Q}}$ a non degenerated quadratic set which contains lines. Then: ${\mathfrak P}_{n}$ is pappian and ${{\mathcal Q}}$ is a quadric with index $\geq 2$ .

Definition of an oval and an ovoid

Ovals and ovoids are special quadratic sets:
Let ${\mathfrak P}$ be a projective space of dimension $\geq 2$ . A non degenerated 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

{\mathfrak o}

there is one and only one line

g

such that

g\cap {\mathfrak o}=\{P\}

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.

For pappian projective planes of odd order the ovals are just conics:
Theorem (SEGRE): 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 {{\mathcal O}}|=0,\ |g\cap {{\mathcal O}}|=1

and

|g\cap {{\mathcal O}}|=2

respectively.)

(O2) For any point

P\in {{\mathcal O}}

the union

{{\mathcal O}}_{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

n=3

b) In case of

|K|<\infty ,\ charK\neq 2

an ovoid in

{\mathfrak P}_{n}(K)

is a quadric.

Counter examples (TITS–SUZUKI-ovoid) show that i.g. statement b) of the theorem above is not true for $charK=2$ :

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

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