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:

  1. Any line intersects in at most 2 points,
  2. The tangents at a point cover a hyperplane (and nothing more), and
  3. contains no lines.
To the definition of an ovoid: t tangent, s secant line

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

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  • 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

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In real projective space (inhomogeneous representation)

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  1.   (hypersphere)
  2.  

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.[4][5]

Finite examples

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  • 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

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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

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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

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Notes

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

References

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  • 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

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  • 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
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