Ovoid (projective geometry)

Definition of an ovoid

In a projective space of dimension d ≥ 3 a set O of points is called an ovoid, if

(1) Any line g meets O in at most 2 points.

In the case of | g ∩ O | = 0 , the line is called a passing (or exterior) line, if | g ∩ O | = 1 the line is a tangent line, and if | g ∩ O | = 2 the line is a secant line.

(2) At any point P ∈ O the tangent lines through P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1).(3) O contains no lines.

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

For an ovoid O and a hyperplane ε , which contains at least two points of O , the subset ε ∩ O is an ovoid (or an oval, if d = 3) within the hyperplane ε .

For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian), the following result is true:

If 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.)

In a finite projective space of order n >2 (i.e. any line contains exactly n + 1 points) and dimension d = 3 any pointset O is an ovoid if and only if | 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 ε not intersecting it, one can call this hyperplane the hyperplane ε ∞ at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to ε ∞ . Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.

In real projective space (inhomogeneous representation)

1. O = { ( x 1 , . . . , x d ) ∈ R d | x 1 2 + ⋯ + x d 2 = 1 }   , (hypersphere)
2. O = { ( x 1 , . . . , x d ) ∈ R d | x d = x 1 2 + ⋯ + x d − 1 2 } ∪ { point at infinity of  x d -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 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.,.

Finite examples

Any ovoid O in a finite projective space of dimension d = 3 over a field K of characteristic ≠ 2 is a quadric.

The last result can not be extended to even characteristic, because of the following non-quadric examples:

For K = G F ( 2 m ) , m odd and σ the automorphism x ↦ x ( 2 m + 1 2 ) ,

the pointset

O = { ( x , y , z ) ∈ K 3 | z = x y + x 2 x σ + y σ } ∪ { point of infinity of the  z -axis } is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).Only when m = 1 is the ovoid O a quadric. 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 O an ovoid in a projective space P of dimension d ≥ 3 and ε a hyperplane. If the ovoid is symmetric to any point P ∈ ε ∖ O (i.e. there is an involutory perspectivity with center P which leaves O invariant), then P is pappian and O a quadric.

An ovoid O in a projective space P is a quadric, if the group of projectivities, which leave O invariant operates 3-transitively on O , i.e. for two triples A 1 , A 2 , A 3 , B 1 , B 2 , B 3 there exists a projectivity π with π ( A i ) = B i , i = 1 , 2 , 3 .

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

Let be O an ovoid in a finite 3-dimensional desarguesian projective space P of odd order, then P is pappian and 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 O of a projective space is called a semi-ovoid if

the following conditions hold:

(SO1) For any point P ∈ O the tangents through point P exactly cover a hyperplane.(SO2) 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 hemitian quadric. (for example)

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