Ovoid (projective geometry)

To the definition of an ovoid: t tangent, s secant line

In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension Template:Math. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid $𝒪$ are:

1. Any line intersects $𝒪$ in at most 2 points,
2. The tangents at a point cover a hyperplane (and nothing more), and
3. $𝒪$ 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 Template:Math a set $𝒪$ of points is called an ovoid, if

(1) Any line Template:Mvar meets $𝒪$ in at most 2 points.

In the case of $|g\cap 𝒪|=0$, the line is called a passing (or exterior) line, if $|g\cap 𝒪|=1$ the line is a tangent line, and if $|g\cap 𝒪|=2$ the line is a secant line.

(2) At any point $P\in 𝒪$ the tangent lines through Template:Mvar cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension Template:Math).

(3) $𝒪$ contains no lines.

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

• For an ovoid $𝒪$ and a hyperplane $\epsilon$, which contains at least two points of $𝒪$, the subset $\epsilon \cap 𝒪$ is an ovoid (or an oval, if Template:Math) within the hyperplane $\epsilon$.

For finite projective spaces of dimension Template:Math (i.e., the point set is finite, the space is pappian), the following result is true:

• If $𝒪$ is an ovoid in a finite projective space of dimension Template:Math, then Template:Math.

(In the finite case, ovoids exist only in 3-dimensional spaces.)

• In a finite projective space of order Template:Math (i.e. any line contains exactly Template:Math points) and dimension Template:Math any pointset $𝒪$ is an ovoid if and only if $|𝒪|=n^2+1$ 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 $\epsilon$ not intersecting it, one can call this hyperplane the hyperplane ${\epsilon }_{\mathrm{\infty }}$ at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to ${\epsilon }_{\mathrm{\infty }}$. 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. $𝒪=\{(x_1,...,x_d)\in {ℝ}^d|x_1^2+\cdots +x_d^2=1\},$ (hypersphere)
2. $𝒪=\{(x_1,...,x_d)\in {ℝ}^d|x_d=x_1^2+\cdots +x_{d-1}^2\}\cup \{\text{point at infinity of }{x}_d\text{-axis}\}$

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 Template:Math by Template:Math.

Remark: The real examples can not be converted into the complex case (projective space over $ℂ$). In a complex projective space of dimension Template:Math 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 $𝒪$ in a finite projective space of dimension Template:Math over a field Template:Mvar of characteristic Template:Math 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

$𝒪=\{(x,y,z)\in K^3|z=xy+x^2{x}^{\sigma }+y^{\sigma }\}\cup \{\text{point of infinity of the }z\text{-axis}\}$ is an ovoid in the 3-dimensional projective space over Template:Mvar (represented in inhomogeneous coordinates).

Only when Template:Math is the ovoid $𝒪$ a quadric.

$𝒪$ is called the Tits-Suzuki-ovoid.

Criteria for an ovoid to be a quadric

An ovoidal quadric has many symmetries. In particular:

• Let be $𝒪$ an ovoid in a projective space $𝔓$ of dimension Template:Math and $\epsilon$ a hyperplane. If the ovoid is symmetric to any point $P\in \epsilon \setminus 𝒪$ (i.e. there is an involutory perspectivity with center $P$ which leaves $𝒪$ invariant), then $𝔓$ is pappian and $𝒪$ a quadric.
• An ovoid $𝒪$ in a projective space $𝔓$ is a quadric, if the group of projectivities, which leave $𝒪$ invariant operates 3-transitively on $𝒪$, 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 $𝒪$ an ovoid in a finite 3-dimensional desarguesian projective space $𝔓$ of odd order, then $𝔓$ is pappian and $𝒪$ 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 $𝒪$ of a projective space is called a semi-ovoid if

the following conditions hold:

(SO1) For any point $P\in 𝒪$ the tangents through point $P$ exactly cover a hyperplane.

(SO2) $𝒪$ 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.

See also

• Ovoid (polar space)
• Möbius plane

References

• Template:Citation

Further reading

• Template:Citation
• Template:Citation
• Template:Citation

External links

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