Ovoid (projective geometry)

In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension $d ≥ 3$. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid $$\mathcal O$$ are: Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).
 * 1)  Any line intersects $$\mathcal O$$ in at most 2 points,
 * 2) The tangents at a point cover a hyperplane (and nothing more), and
 * 3) $$\mathcal O$$ contains no lines.

An ovoid is the spatial analog of an oval in a projective plane.

An ovoid is a special type of a quadratic set.

Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries.

Definition of an ovoid

 * In a projective space of dimension $d ≥ 3$ a set $$\mathcal O$$ of points is called an ovoid, if
 * (1) Any line $g$ meets $$\mathcal O$$ in at most 2 points.

In the case of $$|g\cap\mathcal O|=0$$, the line is called a passing (or exterior) line, if $$|g\cap\mathcal O|=1$$ the line is a tangent line, and if $$|g\cap\mathcal O|=2$$ the line is a secant line.
 * (2) At any point $$P \in \mathcal O$$ the tangent lines through $P$ cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension $d − 1$).
 * (3) $$\mathcal O$$ contains no lines.

From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
 * For an ovoid $$\mathcal O$$ and a hyperplane $$\varepsilon$$, which contains at least two points of $$\mathcal O$$, the subset $$\varepsilon \cap \mathcal O$$ is an ovoid (or an oval, if $d = 3$) within the hyperplane $$\varepsilon$$.

For finite projective spaces of dimension $d ≥ 3$ (i.e., the point set is finite, the space is pappian ), the following result is true:
 * If $$\mathcal O$$ is an ovoid in a finite projective space of dimension $d ≥ 3$, then $d = 3$.
 * (In the finite case, ovoids exist only in 3-dimensional spaces.)


 * In a finite projective space of order $n >2$ (i.e. any line contains exactly $n + 1$ points) and dimension $d = 3$ any pointset $$\mathcal O$$ is an ovoid if and only if $$|\mathcal O|=n^2+1$$ and no three points are collinear (on a common line).

Replacing the word projective in the definition of an ovoid by affine, gives the definition of an affine ovoid.

If for an (projective) ovoid there is a suitable hyperplane $$\varepsilon$$ not intersecting it, one can call this hyperplane the hyperplane $$\varepsilon_\infty$$ at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to $$\varepsilon_\infty$$. Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.

In real projective space (inhomogeneous representation)

 * 1) $$\mathcal O=\{(x_1,...,x_d)\in {\mathbb R}^d \; |\; x_1^2+\cdots +x_d^2=1\}\ ,$$ (hypersphere)
 * 2) $$\mathcal O=\{(x_1,...,x_d)\in {\mathbb R}^d \; | x_d=x_1^2+\cdots +x_{d-1}^2\; \} \; \cup \; \{\text{point at infinity of } x_d\text{-axis}\}$$

These two examples are quadrics and are projectively equivalent.

Simple examples, which are not quadrics can be obtained by the following constructions:
 * (a) Glue one half of a hypersphere to a suitable hyperellipsoid in a smooth way.
 * (b) In the first two examples replace the expression $x_{1}^{2}$ by $x_{1}^{4}$.

Remark: The real examples can not be converted into the complex case (projective space over $${\mathbb C}$$). In a complex projective space of dimension $d ≥ 3$ there are no ovoidal quadrics, because in that case any non degenerated quadric contains lines.

But the following method guarantees many non quadric ovoids:
 * For any non-finite projective space the existence of ovoids can be proven using transfinite induction.

Finite examples

 * Any ovoid $$\mathcal O$$ in a finite projective space of dimension $d = 3$ over a field $K$ of characteristic $≠ 2$ is a quadric.

The last result can not be extended to even characteristic, because of the following non-quadric examples: the pointset
 * For $$K=GF(2^m),\; m $$ odd and $$\sigma$$ the automorphism $$ x \mapsto x^{(2^{\frac{m+1}{2}})}\; ,$$
 * $$\mathcal O=\{(x,y,z)\in K^3 \; |\; z=xy+x^2x^\sigma+y^\sigma \} \; \cup \; \{\text{point of infinity of the } z\text{-axis}\}$$ is an ovoid in the 3-dimensional projective space over $K$ (represented in inhomogeneous coordinates).
 * Only when $m = 1$ is the ovoid $$\mathcal O$$ a quadric.
 * $$\mathcal O$$ is called the Tits-Suzuki-ovoid.

Criteria for an ovoid to be a quadric
An ovoidal quadric has many symmetries. In particular:
 * Let be $$\mathcal O$$ an ovoid in a projective space $$\mathfrak P$$ of dimension $d ≥ 3$ and $$\varepsilon$$ a hyperplane. If the ovoid is symmetric to any point $$P \in \varepsilon \setminus \mathcal O$$ (i.e. there is an involutory perspectivity with center $$P$$ which leaves $$\mathcal O$$ invariant), then $$\mathfrak P$$ is pappian and $$\mathcal O$$ a quadric.
 * An ovoid $$\mathcal O$$ in a projective space $$\mathfrak P$$ is a quadric, if the group of projectivities, which leave $$\mathcal O$$ invariant operates 3-transitively on $$\mathcal O$$, i.e. for two triples $$A_1,A_2,A_3,\; B_1,B_2,B_3$$ there exists a projectivity $$\pi$$ with $$\pi(A_i)=B_i,\; i=1,2,3$$.

In the finite case one gets from Segre's theorem:
 * Let be $$\mathcal O$$ an ovoid in a finite 3-dimensional desarguesian projective space $$\mathfrak P$$ of odd order, then $$\mathfrak P$$ is pappian and $$\mathcal O$$ is a quadric.

Generalization: semi ovoid
Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid:
 * A point set $$ \mathcal O$$ of a projective space is called a semi-ovoid if

the following conditions hold:
 * (SO1) For any point $$P \in \mathcal O$$ the tangents through point $$P$$ exactly cover a hyperplane.
 * (SO2) $$\mathcal O$$ contains no lines.

A semi ovoid is a special semi-quadratic set which is a generalization of a quadratic set. The essential difference between a semi-quadratic set and a quadratic set is the fact, that there can be lines which have 3 points in common with the set and the lines are not contained in the set.

Examples of semi-ovoids are the sets of isotropic points of an hermitian form. They are called hermitian quadrics.

As for ovoids in literature there are criteria, which make a semi-ovoid to a hermitian quadric. See, for example.

Semi-ovoids are used in the construction of examples of Möbius geometries.