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.
![](../I/Ovoid-definition.svg.png.webp)
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 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.
Examples
In real projective space (inhomogeneous representation)
- (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 x12 by x14.
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:
Finite examples
- 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 quadric
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 ovoid
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 also
Notes
- 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.
References
- 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 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.