Ovoid (projective Geometry)

In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension . Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid $\mathcal O$ are:
# Any line intersects $\mathcal O$ in at most 2 points,
# The tangents at a point cover a hyperplane (and nothing more), and
# $\mathcal O$ 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 a set $\mathcal O$ of points is called an ovoid, if
: (1) Any line meets $\mathcal O$ in at most 2 points.
In the case of $, g\cap\mathcal O, =0$, the line is called a ''passing'' (or ''exterior'') ''line'', if $, g\cap\mathcal O, =1$ the line is a ''tangent line'', and if $, g\cap\mathcal O, =2$ the line is a ''secant line''.
: (2) At any point $P \in \mathcal O$ the tangent lines through cover a hyperplane, the ''tangent hyperplane'', (i.e., a projective subspace of dimension ).
: (3) $\mathcal O$ contains no lines.

From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
*For an ovoid $\mathcal O$ and a hyperplane $\varepsilon$, which contains at least two points of $\mathcal O$, the subset $\varepsilon \cap \mathcal O$ is an ovoid (or an oval, if ) within the hyperplane $\varepsilon$.

For ''finite'' projective spaces of dimension (i.e., the point set is finite, the space is pappian), the following result is true:
* If $\mathcal O$ is an ovoid in a ''finite'' projective space of dimension , then .
:(In the finite case, ovoids exist only in 3-dimensional spaces.)
*In a finite projective space of order (i.e. any line contains exactly points) and dimension any pointset $\mathcal O$ is an ovoid if and only if $, \mathcal O, =n^2+1$ 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 $\varepsilon$ not intersecting it, one can call this hyperplane the ''hyperplane $\varepsilon_\infty$ at infinity'' and the ovoid becomes an affine ovoid in the affine space corresponding to $\varepsilon_\infty$. 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)

#$\mathcal O=\\ ,$ (hypersphere)
#$\mathcal O=\ \; \cup \; \$

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 by .

''Remark:'' The real examples can not be converted into the complex case (projective space over $$). In a complex projective space of dimension 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 $\mathcal O$ in a ''finite'' projective space of dimension over a field of characteristic 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 $x \mapsto x^\; ,$
the pointset
:$\mathcal O=\ \; \cup \; \$ is an ovoid in the 3-dimensional projective space over (represented in inhomogeneous coordinates).
:Only when is the ovoid $\mathcal O$ a quadric.
:$\mathcal O$ is called the Tits-Suzuki-ovoid.

Criteria for an ovoid to be a quadric

An ovoidal quadric has many symmetries. In particular:
* Let be $\mathcal O$ an ovoid in a projective space $\mathfrak P$ of dimension and $\varepsilon$ a hyperplane. If the ovoid is symmetric to any point $P \in \varepsilon \setminus \mathcal O$ (i.e. there is an involutory perspectivity with center $P$ which leaves $\mathcal O$ invariant), then $\mathfrak P$ is pappian and $\mathcal O$ a quadric.
*An ovoid $\mathcal O$ in a projective space $\mathfrak P$ is a quadric, if the group of projectivities, which leave $\mathcal O$ invariant operates 3-transitively on $\mathcal O$, 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 $\mathcal O$ an ovoid in a ''finite'' 3-dimensional desarguesian projective space $\mathfrak P$ of ''odd'' order, then $\mathfrak P$ is pappian and $\mathcal O$ 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 $\mathcal O$ of a projective space is called a ''semi-ovoid'' if
the following conditions hold:
:(SO1) For any point $P \in \mathcal O$ the tangents through point $P$ exactly cover a hyperplane.
: (SO2) $\mathcal O$ 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.K.J. Dienst: ''Kennzeichnung hermitescher Quadriken durch Spiegelungen'', Beiträge zur geometrischen Algebra (1977), Birkhäuser-Verlag, S. 83-85.

Semi-ovoids are used in the construction of examples of Möbius geometries.

See also

*Ovoid (polar space)
* Möbius plane

Notes

References

*

Further reading

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*
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External links

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

{{DEFAULTSORT:Ovoid (Projective Geometry)
Projective geometry
Incidence geometry