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.^[a] 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.^[1] It is still unknown if this unital can be embedded in a projective plane of order 36, if such a plane exists.

Unitals[edit]

Classical[edit]

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.^[2]

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

Buekenhout's Constructions[edit]

By examining the classical unital in $PG(2,q^{2})$ 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.^[16]^[17]

The core idea in Buekenhout's construction is that when one looks at $PG(2,q^{2})$ in the higher-dimensional Bruck/Bose model, which lies in $PG(4,q)$ , the equation of the Hermitian curve satisfied by a classical unital becomes a quadric surface in $PG(4,q)$ , 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 $PG(4,q)$ , 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.