Ovoid (projective geometry)

• 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=\{(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 by . Remark: The real examples can not be converted into the complex case (projective space over {\mathbb C}). 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^{(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 (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. Semi-ovoids are used in the construction of examples of Möbius geometries. == See also ==