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 ${\displaystyle {\mathcal {O}}}$ are:

1. Any line intersects ${\displaystyle {\mathcal {O}}}$ in at most 2 points,
2. The tangents at a point cover a hyperplane (and nothing more), and
3. ${\displaystyle {\mathcal {O}}}$ 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.

• In a projective space of dimension d ≥ 3 a set ${\displaystyle {\mathcal {O}}}$ of points is called an ovoid, if

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

In the case of ${\displaystyle |g\cap {\mathcal {O}}|=0}$, the line is called a passing (or exterior) line, if ${\displaystyle |g\cap {\mathcal {O}}|=1}$ the line is a tangent line, and if ${\displaystyle |g\cap {\mathcal {O}}|=2}$ the line is a secant line.

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

(3) ${\displaystyle {\mathcal {O}}}$ contains no lines.

From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because

• For an ovoid ${\displaystyle {\mathcal {O}}}$ and a hyperplane ${\displaystyle \varepsilon }$, which contains at least two points of ${\displaystyle {\mathcal {O}}}$, the subset ${\displaystyle \varepsilon \cap {\mathcal {O}}}$ is an ovoid (or an oval, if d = 3) within the hyperplane ${\displaystyle \varepsilon }$.

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 ${\displaystyle {\mathcal {O}}}$ 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 ${\displaystyle {\mathcal {O}}}$ is an ovoid if and only if ${\displaystyle |{\mathcal {O}}|=n^{2}+1}$ 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 ${\displaystyle \varepsilon }$ not intersecting it, one can call this hyperplane the hyperplane ${\displaystyle \varepsilon _{\infty }}$ at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to ${\displaystyle \varepsilon _{\infty }}$. Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.

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

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 x₁² by x₁⁴.

Remark: The real examples can not be converted into the complex case (projective space over ${\displaystyle {\mathbb {C} }}$). 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]

• Any ovoid ${\displaystyle {\mathcal {O}}}$ 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 ${\displaystyle K=GF(2^{m}),\;m}$ odd and ${\displaystyle \sigma }$ the automorphism ${\displaystyle x\mapsto x^{(2^{\frac {m+1}{2}})}\;,}$

the pointset

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

Only when m = 1 is the ovoid ${\displaystyle {\mathcal {O}}}$ a quadric.^[7]

${\displaystyle {\mathcal {O}}}$ is called the Tits-Suzuki-ovoid.

An ovoidal quadric has many symmetries. In particular:

• Let be ${\displaystyle {\mathcal {O}}}$ an ovoid in a projective space ${\displaystyle {\mathfrak {P}}}$ of dimension d ≥ 3 and ${\displaystyle \varepsilon }$ a hyperplane. If the ovoid is symmetric to any point ${\displaystyle P\in \varepsilon \setminus {\mathcal {O}}}$ (i.e. there is an involutory perspectivity with center ${\displaystyle P}$ which leaves ${\displaystyle {\mathcal {O}}}$ invariant), then ${\displaystyle {\mathfrak {P}}}$ is pappian and ${\displaystyle {\mathcal {O}}}$ a quadric.^[8]
• An ovoid ${\displaystyle {\mathcal {O}}}$ in a projective space ${\displaystyle {\mathfrak {P}}}$ is a quadric, if the group of projectivities, which leave ${\displaystyle {\mathcal {O}}}$ invariant operates 3-transitively on ${\displaystyle {\mathcal {O}}}$, i.e. for two triples ${\displaystyle A_{1},A_{2},A_{3},\;B_{1},B_{2},B_{3}}$ there exists a projectivity ${\displaystyle \pi }$ with ${\displaystyle \pi (A_{i})=B_{i},\;i=1,2,3}$.^[9]

In the finite case one gets from Segre's theorem:

• Let be ${\displaystyle {\mathcal {O}}}$ an ovoid in a finite 3-dimensional desarguesian projective space ${\displaystyle {\mathfrak {P}}}$ of odd order, then ${\displaystyle {\mathfrak {P}}}$ is pappian and ${\displaystyle {\mathcal {O}}}$ is a quadric.

Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid:

A point set ${\displaystyle {\mathcal {O}}}$ of a projective space is called a semi-ovoid if

the following conditions hold:

(SO1) For any point ${\displaystyle P\in {\mathcal {O}}}$ the tangents through point ${\displaystyle P}$ exactly cover a hyperplane.

(SO2) ${\displaystyle {\mathcal {O}}}$ 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 hermitian quadric. See, for example.^[11]

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

• Ovoid (polar space)
• Möbius plane

• 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

• 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

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