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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

(QS1) Any line $g$ of ${\mathcal {G}}$ 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 ${\mathcal {P}}$ .

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

## 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 $g\cap {\mathfrak {o}}=\{P\}$ .

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

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.
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 $|K|<\infty ,\ charK\neq 2$ 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 $charK=2$ :