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 $$\mathfrak P=({\mathcal P},{\mathcal G},\in)$$ be a projective space. A quadratic set is a non-empty subset $${\mathcal Q}$$ of $${\mathcal P}$$ for which the following two conditions hold:
 * (QS1) Every line $$g$$ of $${\mathcal G}$$ intersects $${\mathcal Q}$$ in at most two points or is contained in $${\mathcal Q}$$.
 * ($$g$$ is called exterior to $${\mathcal Q}$$ if $$|g\cap {\mathcal Q}|=0$$, tangent to $${\mathcal Q}$$ if either $$|g\cap {\mathcal Q}|=1$$ or $$g\cap {\mathcal Q}=g$$, and secant to $${\mathcal Q}$$ if $$|g\cap {\mathcal Q}|=2$$.)
 * (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-degenerate if for every point $$P\in {\mathcal Q}$$, the set $${\mathcal Q}_P$$ 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 $$\mathfrak P_n$$ a finite projective space of dimension $$n\ge 3$$ and $${\mathcal Q}$$ a non-degenerate quadratic set that contains lines. Then: $$\mathfrak P_n$$ is Pappian and $${\mathcal Q}$$ 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 $$g\cap \mathfrak o=\{P\}$$.

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.

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 $$|g\cap {\mathcal O}|=0, \ |g\cap {\mathcal O}|=1$$ 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 $$n=2$$ or $$n=3$$.
 * b) In case of $$|K| <\infty,\ \operatorname{char} K \ne 2$$ 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 $$\operatorname{char} K=2$$: