# 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:

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.

Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).

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.

## Examples

### 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 x12 by x14.

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:

• For $K=GF(2^{m}),\;m$  odd and $\sigma$  the automorphism $x\mapsto x^{(2^{\frac {m+1}{2}})}\;,$

the pointset

${\mathcal {O}}=\{(x,y,z)\in K^{3}\;|\;z=xy+x^{2}x^{\sigma }+y^{\sigma }\}\;\cup \;\{{\text{point of infinity of the }}z{\text{-axis}}\}$  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.