Quadratic set

Let $\backslash mathfrak\; P=(\{\backslash mathcal\; P\},\{\backslash mathcal\; G\},\backslash in)$ be a projective space. A quadratic set is a non-empty subset $\{\backslash mathcal\; Q\}$ of $\{\backslash mathcal\; P\}$ for which the following two conditions hold:

:(QS1) Every line $g$ of $\{\backslash mathcal\; G\}$ intersects $\{\backslash mathcal\; Q\}$ in at most two points or is contained in $\{\backslash mathcal\; Q\}$.

::($g$ is called exterior to $\{\backslash mathcal\; Q\}$ if $|g\backslash cap\; \{\backslash mathcal\; Q\}|=0$, tangent to $\{\backslash mathcal\; Q\}$ if either $|g\backslash cap\; \{\backslash mathcal\; Q\}|=1$ or $g\backslash cap\; \{\backslash mathcal\; Q\}=g$, and secant to $\{\backslash mathcal\; Q\}$ if $|g\backslash cap\; \{\backslash mathcal\; Q\}|=2$.)

:(QS2) For any point $P\backslash in\; \{\backslash mathcal\; Q\}$ the union $\{\backslash mathcal\; Q\}\_P$ of all tangent lines through $P$ is a hyperplane or the entire space $\{\backslash mathcal\; P\}$.

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

Ovals and ovoids are special quadratic sets:

Let $\backslash mathfrak\; P$ be a projective space of dimension $\backslash ge\; 2$. A non-degenerate quadratic set $\backslash mathcal\; O$ 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 $\backslash mathfrak\; o$ of a projective plane is called

oval if the following properties are fulfilled:

:(o1) Any line meets $\backslash mathfrak\; o$ in at most two points.

:(o2) For any point $P$ in $\backslash mathfrak\; o$ there is one and only one line $g$ such that $g\backslash cap\; \backslash mathfrak\; o=\backslash \{P\backslash \}$.

A line $g$ is a exterior or tangent or secant line of the

oval if $|g\backslash cap\; \backslash mathfrak\; o|=0$ or $|g\backslash cap\; \backslash mathfrak\; o|=1$ or $|g\backslash cap\; \backslash mathfrak\; o|=2$ respectively.

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

Theorem: (oval in finite plane) Let be $\backslash mathfrak\; P$ a projective plane of order $n$.

A set $\backslash mathfrak\; o$ of points is an oval if $|\backslash mathfrak\; o|=n+1$ and if no three points

of $\backslash mathfrak\; o$ are collinear.

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

Theorem:

Let be $\backslash mathfrak\; P$ a Pappian projective plane of odd order.

Any oval in $\backslash mathfrak\; P$ is an oval conic (non-degenerate quadric).

Definition: (ovoid)

A non-empty point set $\backslash mathcal\; O$ of a projective space is called ovoid if the following properties are fulfilled:

:(O1) Any line meets $\backslash mathcal\; O$ in at most two points.

:($g$ is called exterior, tangent and secant line if $|g\backslash cap\; \{\backslash mathcal\; O\}|=0,\; \backslash \; |g\backslash cap\; \{\backslash mathcal\; O\}|=1$ and $|g\backslash cap\; \{\backslash mathcal\; O\}|=2$ respectively.)

:(O2) For any point $P\backslash in\; \{\backslash mathcal\; O\}$ the union $\{\backslash mathcal\; O\}\_P$ 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 an ovoid in $\backslash mathfrak\; P\_n(K)$ exists only if $n=2$ or $n=3$.

:b) In case of an ovoid in $\backslash mathfrak\; P\_n(K)$ is a quadric.

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

{{Reflist}}

• 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

• Eric Hartmann [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes], from Technische Universität Darmstadt

Category:Geometry