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. ^{[lower-alpha 1]} 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

Classical

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

s(u,v) = s(v,u)^α for all vectors u, v of the underlying vector space.

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. ^[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.

Let ${\displaystyle {\mathcal {H}}={\mathcal {H}}(2,q^{2})}$ be a nondegenerate Hermitian curve in ${\displaystyle PG(2,q^{2})}$ for some prime power ${\displaystyle q}$. As all nondegenerate Hermitian curves in the same plane are projectively equivalent, ${\displaystyle {\mathcal {H}}}$ can be described in terms of homogeneous coordinates as follows: ^[5]

$${\displaystyle {\mathcal {H}}=\{(x_{0},x_{1},x_{2})\colon x_{0}^{q+1}+x_{1}^{q+1}+x_{2}^{q+1}=0\}.}$$

Ree unitals

Another family of unitals based on Ree groups was constructed by H. Lüneburg. ^[6] Let Γ = R(q) be the Ree group of type ²G₂ of order (q³ + 1)q³(q − 1) where q = 3^2m+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, Γ_S,T is cyclic of order q - 1, and thus contains a unique involution, μ. Each such involution fixes exactly q + 1 points of P. Construct a block design on the points of P whose blocks are the fixed point sets of these various involutions 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. ^[10]

Unitals with ${\displaystyle n=3}$

In the four projective planes of order 9 (the Desarguesian plane PG(2,9), the Hall plane of order 9, the dual Hall plane of order 9 and the Hughes plane of order 9. ^{[lower-alpha 2]} ), an exhaustive computer search by Penttila and Royle ^[11] found 18 unitals (up to equivalence) with n = 3 in these four planes: two in PG(2,9) (both Buekenhout), four in the Hall plane (two Buekenhout, two not), and so another four in the dual Hall plane, and eight in the Hughes plane. However, one of the Buekenhout unitals in the Hall plane is self-dual, ^[12] and thus gets counted again in the dual Hall plane. Thus, there are 17 distinct embeddable unitals with n = 3. On the other hand, a nonexhaustive computer search found over 900 mutually nonisomorphic designs which are unitals with n = 3. ^[13]

Isomorphic versus equivalent unitals

Since unitals are block designs, two unitals are said to be isomorphic if there is a design isomorphism between them, that is, a bijection between the point sets which maps blocks to blocks. 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. ^[10]

Buekenhout's Constructions

By examining the classical unital in ${\displaystyle PG(2,q^{2})}$ 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 ${\displaystyle PG(2,q^{2})}$ in the higher-dimensional Bruck/Bose model, which lies in ${\displaystyle PG(4,q)}$, the equation of the Hermitian curve satisfied by a classical unital becomes a quadric surface in ${\displaystyle PG(4,q)}$, 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 ${\displaystyle PG(4,q)}$, 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.

Hermitian varieties

Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities.

Definition

Let K be a field with an involutive automorphism ${\displaystyle \theta }$. Let n be an integer ${\displaystyle \geq 1}$ 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.

Representation

Let ${\displaystyle e_{0},e_{1},\ldots ,e_{n}}$ be a basis of V. If a point p in the projective space has homogeneous coordinates ${\displaystyle (X_{0},\ldots ,X_{n})}$ with respect to this basis, it is on the Hermitian variety if and only if :

${\displaystyle \sum _{i,j=0}^{n}a_{ij}X_{i}X_{j}^{\theta }=0}$

where ${\displaystyle a_{ij}=a_{ji}^{\theta }}$ and not all ${\displaystyle a_{ij}=0}$

If one constructs the Hermitian matrix A with ${\displaystyle A_{ij}=a_{ij}}$, the equation can be written in a compact way :

${\displaystyle X^{t}AX^{\theta }=0}$

where ${\displaystyle X={\begin{bmatrix}X_{0}\\X_{1}\\\vdots \\X_{n}\end{bmatrix}}.}$

Tangent spaces and singularity

Let p be a point on the Hermitian variety H. A line L through p is by definition tangent when it is contains only one point (p itself) of the variety or lies completely on the variety. One can prove that these lines form a subspace, either a hyperplane of the full space. In the latter case, the point is singular.

Notes

1. Some authors, such as Barwick & Ebert 2008, p. 28, further require that n ≥ 3 to avoid small exceptional cases.
2. PG(2,9) and the Hughes plane are both self-dual.

Citations

1. Bagchi & Bagchi 1989, pp. 51–61.
2. Barwick & Ebert 2008, p. 15.
3. Barwick & Ebert 2008, p. 18.
4. Dembowski 1968, p. 104.
5. Barwick & Ebert 2008, p. 21.
6. Lüneburg 1966, pp. 256–259.
7. Assmus & Key 1992, p. 209.
8. Dembowski 1968, p. 105.
9. Grüning 1986, pp. 473–480.
10. Barwick & Ebert 2008, p. 29.
11. Penttila & Royle 1995, pp. 229–245.
12. Grüning, Klaus (1987-06-01). "A class of unitals of order ${\displaystyle q}$ which can be embedded in two different planes of order ${\displaystyle q^{2}}$". Journal of Geometry. 29 (1): 61–77. doi:10.1007/BF01234988. ISSN   1420-8997. S2CID   117872040.
13. Betten, Betten & Tonchev 2003, pp. 23–33.
14. Buekenhout, F. (1976-07-01). "Existence of unitals in finite translation planes of order ${\displaystyle q^{2}}$ with a kernel of order ${\displaystyle q}$". Geometriae Dedicata. 5 (2): 189–194. doi:10.1007/BF00145956. ISSN   1572-9168. S2CID   123037502.
15. Metz, Rudolf (1979-03-01). "On a class of unitals". Geometriae Dedicata. 8 (1): 125–126. doi:10.1007/BF00147935. ISSN   1572-9168. S2CID   119595725.
16. Baker, R.D; Ebert, G.L (1992-05-01). "On Buekenhout-Metz unitals of odd order". Journal of Combinatorial Theory, Series A. 60 (1): 67–84. doi: 10.1016/0097-3165(92)90038-V . ISSN   0097-3165.
17. Ebert, G.L. (1992-03-01). "On Buekenhout-Metz unitals of even order". European Journal of Combinatorics. 13 (2): 109–117. doi:10.1016/0195-6698(92)90042-X. ISSN   0195-6698.

Sources

• Assmus, E. F. Jr; Key, J. D. (1992), Designs and Their Codes, Cambridge Tracts in Mathematics #103, Cambridge University Press, ISBN   0-521-41361-3
• Bagchi, S.; Bagchi, B. (1989), "Designs from pairs of finite fields. A cyclic unital U(6) and other regular steiner 2-designs", Journal of Combinatorial Theory, Series A, 52: 51–61, doi:10.1016/0097-3165(89)90061-7
• Barwick, Susan; Ebert, Gary (2008), Unitals in Projective Planes, Springer, doi:10.1007/978-0-387-76366-8, ISBN   978-0-387-76364-4
• Betten, A.; Betten, D.; Tonchev, V.D. (2003), "Unitals and codes", Discrete Mathematics, 267 (1–3): 23–33, doi:10.1016/s0012-365x(02)00600-3
• Dembowski, Peter (1968), Finite geometries , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN   3-540-61786-8, MR   0233275 – via Internet Archive
• Grüning, K. (1986), "Das Kleinste Ree-Unital", Archiv der Mathematik, 46 (5): 473–480, doi:10.1007/bf01210788, S2CID   115302560
• Lüneburg, H. (1966), "Some remarks concerning the Ree group of type (G₂)", Journal of Algebra, 3 (2): 256–259, doi: 10.1016/0021-8693(66)90014-7
• Penttila, T.; Royle, G.F. (1995), "Sets of type (m,n) in the affine and projective planes of order nine", Designs, Codes and Cryptography, 6 (3): 229–245, doi:10.1007/bf01388477, S2CID   43638589