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 d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid [math]\displaystyle{ \mathcal O }[/math] are:

1. Any line intersects [math]\displaystyle{ \mathcal O }[/math] in at most 2 points,
2. The tangents at a point cover a hyperplane (and nothing more), and
3. [math]\displaystyle{ \mathcal O }[/math] 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 [math]\displaystyle{ \mathcal O }[/math] of points is called an ovoid, if

(1) Any line g meets [math]\displaystyle{ \mathcal O }[/math] in at most 2 points.

In the case of [math]\displaystyle{ |g\cap\mathcal O|=0 }[/math], the line is called a passing (or exterior) line, if [math]\displaystyle{ |g\cap\mathcal O|=1 }[/math] the line is a tangent line, and if [math]\displaystyle{ |g\cap\mathcal O|=2 }[/math] the line is a secant line.

(2) At any point [math]\displaystyle{ P \in \mathcal O }[/math] the tangent lines through P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1).

(3) [math]\displaystyle{ \mathcal O }[/math] contains no lines.

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

• For an ovoid [math]\displaystyle{ \mathcal O }[/math] and a hyperplane [math]\displaystyle{ \varepsilon }[/math], which contains at least two points of [math]\displaystyle{ \mathcal O }[/math], the subset [math]\displaystyle{ \varepsilon \cap \mathcal O }[/math] is an ovoid (or an oval, if d = 3) within the hyperplane [math]\displaystyle{ \varepsilon }[/math].

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 [math]\displaystyle{ \mathcal O }[/math] 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 [math]\displaystyle{ \mathcal O }[/math] is an ovoid if and only if [math]\displaystyle{ |\mathcal O|=n^2+1 }[/math] 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 [math]\displaystyle{ \varepsilon }[/math] not intersecting it, one can call this hyperplane the hyperplane [math]\displaystyle{ \varepsilon_\infty }[/math] at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to [math]\displaystyle{ \varepsilon_\infty }[/math]. 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. [math]\displaystyle{ \mathcal O=\{(x_1,...,x_d)\in {\mathbb R}^d \; |\; x_1^2+\cdots +x_d^2=1\}\ , }[/math] (hypersphere)
2. [math]\displaystyle{ \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}\} }[/math]

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 [math]\displaystyle{ {\mathbb C} }[/math]). 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 [math]\displaystyle{ \mathcal O }[/math] 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 [math]\displaystyle{ K=GF(2^m),\; m }[/math] odd and [math]\displaystyle{ \sigma }[/math] the automorphism [math]\displaystyle{ x \mapsto x^{(2^{\frac{m+1}{2}})}\; , }[/math]

the pointset

[math]\displaystyle{ \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}\} }[/math] is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).

Only when m = 1 is the ovoid [math]\displaystyle{ \mathcal O }[/math] a quadric.

[math]\displaystyle{ \mathcal O }[/math] is called the Tits-Suzuki-ovoid.

Criteria for an ovoid to be a quadric

An ovoidal quadric has many symmetries. In particular:

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

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

• Let be [math]\displaystyle{ \mathcal O }[/math] an ovoid in a finite 3-dimensional desarguesian projective space [math]\displaystyle{ \mathfrak P }[/math] of odd order, then [math]\displaystyle{ \mathfrak P }[/math] is pappian and [math]\displaystyle{ \mathcal O }[/math] 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 [math]\displaystyle{ \mathcal O }[/math] of a projective space is called a semi-ovoid if

the following conditions hold:

(SO1) For any point [math]\displaystyle{ P \in \mathcal O }[/math] the tangents through point [math]\displaystyle{ P }[/math] exactly cover a hyperplane.

(SO2) [math]\displaystyle{ \mathcal O }[/math] 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

• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, https://archive.org/details/finitegeometries0000demb 

Further reading

• 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 links

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