In geometry, a unital is a set of n3 + 1 points arranged into subsets of size n + 1 so that every pair of distinct points of the set are contained in exactly one subset.
Some unitals may be embedded in a projective plane of order n2 (the subsets of the design become sets of collinear points in the projective plane).
In the Desarguesian planes, PG(2,q2), the classical examples of unitals are given by nondegenerate Hermitian curves.
We review some terminology used in projective geometry.
A correlation of a projective geometry is a bijection on its subspaces that reverses containment.
In particular, a correlation interchanges points and hyperplanes.
The absolute points of a unitary polarity of the projective geometry PG(d,F), for some d ≥ 2, is a nondegenerate Hermitian variety, and if d = 2 this variety is called a nondegenerate Hermitian curve.
[3] In PG(2,q2) for some prime power q, the set of points of a nondegenerate Hermitian curve form a unital,[4] which is called a classical unital.
As all nondegenerate Hermitian curves in the same plane are projectively equivalent,
Another family of unitals based on Ree groups was constructed by H.
[6] Let Γ = R(q) be the Ree group of type 2G2 of order (q3 + 1)q3(q − 1) where q = 32m+1.
Let P be the set of all q3 + 1 Sylow 3-subgroups of Γ. Γ acts doubly transitively on this set by conjugation (it will be convenient to think of these subgroups as points that Γ is acting on.)
For any S and T in P, the pointwise stabilizer, ΓS,T is cyclic of order q - 1, and thus contains a unique involution, μ.
Since Γ acts doubly transitively on P, this will be a 2-design with parameters 2-(q3 + 1, q + 1, 1) called a Ree unital.
[7] Lüneburg also showed that the Ree unitals can not be embedded in projective planes of order q2 (Desarguesian or not) such that the automorphism group Γ is induced by a collineation group of the plane.
[8] For q = 3, Grüning[9] proved that a Ree unital can not be embedded in any projective plane of order 9.
On the other hand, a nonexhaustive computer search found over 900 mutually nonisomorphic designs which are unitals with n = 3.
This concept does not take into account the property of embeddability, so to do so we say that two unitals, embedded in the same ambient plane, are equivalent if there is a collineation of the plane which maps one unital to the other.
in the Bruck/Bose model, Buekenhout[14] provided two constructions, which together proved the existence of an embedded unital in any finite 2-dimensional translation plane.
Metz[15] subsequently showed that one of Buekenhout's constructions actually yields non-classical unitals in all finite Desarguesian planes of square order at least 9.
These Buekenhout-Metz unitals have been extensively studied.
, the equation of the Hermitian curve satisfied by a classical unital becomes a quadric surface in
, either a point-cone over a 3-dimensional ovoid if the line represented by the spread of the Bruck/Bose model meets the unital in one point, or a non-singular quadric otherwise.
Because these objects have known intersection patterns with respect to planes of
, the resulting point set remains a unital in any translation plane whose generating spread contains all of the same lines as the original spread within the quadric surface.
In the ovoidal cone case, this forced intersection consists of a single line, and any spread can be mapped onto a spread containing this line, showing that every translation plane of this form admits an embedded unital.
Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities.
and V be an (n+1)-dimensional vector space over K. A Hermitian variety H in PG(V) is a set of points of which the representing vector lines consisting of isotropic points of a non-trivial Hermitian sesquilinear form on V. Let
be a basis of V. If a point p in the projective space has homogeneous coordinates
One can prove that these lines form a subspace, either a hyperplane of the full space.