# 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 are:

- Any line intersects in at most 2 points,
- The tangents at a point cover a hyperplane (and nothing more), and
- 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 ovoidEdit

- In a projective space of dimension
*d*≥ 3 a set of points is called an**ovoid**, if

- (1) Any line g meets in at most 2 points.

In the case of , the line is called a *passing* (or *exterior*) *line*, if the line is a *tangent line*, and if the line is a *secant line*.

- (2) At any point the tangent lines through P cover a hyperplane, the
*tangent hyperplane*, (i.e., a projective subspace of dimension*d*− 1). - (3) contains no lines.

From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because

- For an ovoid and a hyperplane , which contains at least two points of , the subset is an ovoid (or an oval, if
*d*= 3) within the hyperplane .

For *finite* projective spaces of dimension *d* ≥ 3 (i.e., the point set is finite, the space is pappian^{[1]}), the following result is true:

- If 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.)
^{[2]}

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

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 not intersecting it, one can call this hyperplane the *hyperplane at infinity* and the ovoid becomes an affine ovoid in the affine space corresponding to . Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.

## ExamplesEdit

### In real projective space (inhomogeneous representation)Edit

- (hypersphere)

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 ). 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*.^{[4]}^{[5]}

### Finite examplesEdit

- Any ovoid in a
*finite*projective space of dimension*d*= 3 over a field K of characteristic ≠ 2 is a*quadric*.^{[6]}

The last result can not be extended to even characteristic, because of the following non-quadric examples:

- For odd and the automorphism

the pointset

- is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).
- Only when
*m*= 1 is the ovoid a quadric.^{[7]} - is called the
**Tits-Suzuki-ovoid**.

## Criteria for an ovoid to be a quadricEdit

An ovoidal quadric has many symmetries. In particular:

- Let be an ovoid in a projective space of dimension
*d*≥ 3 and a hyperplane. If the ovoid is symmetric to any point (i.e. there is an involutory perspectivity with center which leaves invariant), then is pappian and a quadric.^{[8]} - An ovoid in a projective space is a quadric, if the group of projectivities, which leave invariant operates 3-transitively on , i.e. for two triples there exists a projectivity with .
^{[9]}

In the finite case one gets from Segre's theorem:

- Let be an ovoid in a
*finite*3-dimensional desarguesian projective space of*odd*order, then is pappian and is a quadric.

## Generalization: semi ovoidEdit

Removing condition (1) from the definition of an ovoid results in the definition of a **semi-ovoid**:

- A point set of a projective space is called a
*semi-ovoid*if

the following conditions hold:

- (SO1) For any point the tangents through point exactly cover a hyperplane.
- (SO2) contains no lines.

A semi ovoid is a special *semi-quadratic set*^{[10]} 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^{[11]}.

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

## See alsoEdit

## NotesEdit

**^**Dembowski 1968, p. 28**^**Dembowski 1968, p. 48**^**Dembowski 1968, p. 48**^**W. Heise:*Bericht über -affine Geometrien*, Journ. of Geometry 1 (1971), S. 197–224, Satz 3.4.**^**F. Buekenhout:*A Characterization of Semi Quadrics*, Atti dei Convegni Lincei 17 (1976), S. 393-421, chapter 3.5**^**Dembowski 1968, p. 49**^**Dembowski 1968, p. 52**^**H. Mäurer:*Ovoide mit Symmetrien an den Punkten einer Hyperebene*, Abh. Math. Sem. Hamburg 45 (1976), S.237-244**^**J. Tits:*Ovoides à Translations*, Rend. Mat. 21 (1962), S. 37–59.**^**F. Buekenhout:*A Characterization of Semi Quadrics*, Atti dei Convegni Lincei 17 (1976), S. 393-421.**^**K.J. Dienst:*Kennzeichnung hermitescher Quadriken durch Spiegelungen*, Beiträge zur geometrischen Algebra (1977), Birkhäuser-Verlag, S. 83-85.

## ReferencesEdit

- Dembowski, Peter (1968),
*Finite geometries*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

## Further readingEdit

- Barlotti, A. (1955), "Un'estensione del teorema di Segre-Kustaanheimo",
*Boll. Un. Mat. Ital.*,**10**: 96–98 - Hirschfeld, J.W.P. (1985),
*Finite Projective Spaces of Three Dimensions*, New York: Oxford University Press, ISBN 0-19-853536-8 - Panella, G. (1955), "Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito",
*Boll. Un. Mat. Ital.*,**10**: 507–513

## External linksEdit

- E. Hartmann:
*Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.*Skript, TH Darmstadt (PDF; 891 kB), S. 121-123.