Ovoid (projective geometry)

To the definition of an ovoid: t tangent, s secant line

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

(1) Any line

g

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

P

cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension

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 $d = 3$ ) within the hyperplane $\varepsilon$ .

For finite projective spaces of dimension $d \geq 3$ (i.e., the point set is finite, the space is pappian^[1]), the following result is true:

If ${\mathcal {O}}$ is an ovoid in a finite projective space of dimension $d \geq 3$ , 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 ${\mathcal {O}}$ is an ovoid if and only if $|{\mathcal {O}}|=n^{2}+1$ 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 $\varepsilon$ 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

x 12

x 14

Remark: The real examples can not be converted into the complex case (projective space over $\mathbb {C}$ ). In a complex projective space of dimension $d \geq 3$ 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

Any ovoid ${\mathcal {O}}$ in a finite projective space of dimension $d = 3$ over a field $K$ of characteristic $\neq 2$ is a quadric.^[6]

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^{2}x^{\sigma }+y^{\sigma }\}\;\cup \;\{{\text{point of infinity of the }}z{\text{-axis}}\}

is an ovoid in the 3-dimensional projective space over

K

(represented in inhomogeneous coordinates).

Only when

m = 1

is the ovoid

{\mathcal {O}}

a quadric.^[7]

{\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 $d \geq 3$ 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.^[8]
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 $\pi (A_{i})=B_{i},\;i=1,2,3$ .^[9]

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^[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 hemitian quadric. (for example^[11])

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

Notes

↑ Dembowski 1968, p. 28
↑ Dembowski 1968, p. 48
↑ Dembowski 1968, p. 48
↑ 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
↑ 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

External links

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

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