Unital (geometry)

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. n ≥ 3 is required by some authors to avoid small exceptional cases. This is equivalent to saying that a unital is a 2-(n³ + 1, n + 1, 1) block design. 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 this case of embedded unitals, every line of the plane intersects the unital in either 1 or n + 1 points. In the Desarguesian planes, PG(2,q²), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, n=6, was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order 36, if such a plane exists.

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.

A correlation of order two is called a polarity.

A polarity is called a unitary polarity if its associated sesquilinear form s with companion automorphism α satisfies:

A point is called an absolute point of a polarity if it lies on the image of itself under the polarity.

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.

In PG(2,q²) for some prime power q, the set of points of a nondegenerate Hermitian curve form a unital, which is called a classical unital.

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