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 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), 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.)
- 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).
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.
In real projective space (inhomogeneous representation)
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:
- For any non-finite projective space the existence of ovoids can be proven using transfinite induction.,.
- Any ovoid 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 odd and the automorphism
- is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).
- Only when m = 1 is the ovoid a quadric.
- 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.
- 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 .
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 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 hemitian quadric. (for example)
Semi-ovoids are used in the construction of examples of Möbius geometries.
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