Ovoid (projective geometry)

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In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension {{math|d ≥ 3}}. 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 {{math|d ≥ 3}} a set $\mathcal O$ of points is called an ovoid, if

: (1) Any line {{mvar|g}} meets $\mathcal O$ in at most 2 points.

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

: (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 {{math|1=d = 3}}) within the hyperplane $\varepsilon$ .

For finite projective spaces of dimension {{math|d ≥ 3}} (i.e., the point set is finite, the space is pappian{{harvnb|Dembowski|1968|page=28}}), the following result is true:

If $\mathcal O$ is an ovoid in a finite projective space of dimension {{math|d ≥ 3}}, then {{math|1=d = 3}}.

:(In the finite case, ovoids exist only in 3-dimensional spaces.){{harvnb|Dembowski|1968|page=48}}

In a finite projective space of order {{math|n >2}} (i.e. any line contains exactly {{math|n + 1}} points) and dimension {{math|1=d = 3}} 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).{{harvnb|Dembowski|1968|page=48}}

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) =

$\mathcal O=\{(x_1,...,x_d)\in {\mathbb R}^d \; |\; x_1^2+\cdots +x_d^2=1\}\ ,$ (hypersphere)
$\mathcal O=\{(x_1,...,x_d)\in {\mathbb R}^d \; | x_d=x_1^2+\cdots +x_{d-1}^2\; \} \; \cup \; \{\text{point at infinity of } x_d\text{-axis}\}$

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 {{math|x₁²}} by {{math|x₁⁴}}.

Remark: The real examples can not be converted into the complex case (projective space over ${\mathbb C}$ ). In a complex projective space of dimension {{math|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.W. Heise: Bericht über $\kappa$ -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

= Finite examples =

Any ovoid $\mathcal O$ in a finite projective space of dimension {{math|1=d = 3}} over a field {{mvar|K}} of characteristic {{math|≠ 2}} is a quadric.{{harvnb|Dembowski|1968|page=49}}

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 $x \mapsto x^{(2^{\frac{m+1}{2}})}\; ,$

the pointset

: $\mathcal O=\{(x,y,z)\in K^3 \; |\; z=xy+x^2x^\sigma+y^\sigma \} \; \cup \; \{\text{point of infinity of the } z\text{-axis}\}$ is an ovoid in the 3-dimensional projective space over {{mvar|K}} (represented in inhomogeneous coordinates).

:Only when {{math|1=m = 1}} is the ovoid $\mathcal O$ a quadric.{{harvnb|Dembowski|1968|page=52}}

: $\mathcal O$ 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 {{math|d ≥ 3}} 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.H. Mäurer: Ovoide mit Symmetrien an den Punkten einer Hyperebene, Abh. Math. Sem. Hamburg 45 (1976), S.237-244
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 $\pi(A_i)=B_i,\; i=1,2,3$ .J. Tits: Ovoides à Translations, Rend. Mat. 21 (1962), S. 37–59.

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 setF. Buekenhout: A Characterization of Semi Quadrics, Atti dei Convegni Lincei 17 (1976), S. 393-421. 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.

Notes

References

{{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}

External links

E. Hartmann: [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.] Skript, TH Darmstadt (PDF; 891 kB), S. 121-123.

Category:Projective geometry

Category:Incidence geometry