Quadratic set

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title: "Quadratic set" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["geometry"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Quadratic_set" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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| an ovoid in \mathfrak P_n(K) exists only if n=2 or n=3. :b) In case of |K| 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:

References

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