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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a unital is a set of ''n''3 + 1
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' + 1, 1)
block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
. Some unitals may be embedded in a projective plane of order ''n''2 (the subsets of the design become sets of
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
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,''q''2), 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.


Unitals


Classical

We review some terminology used in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
. A correlation of a projective geometry is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
on its subspaces that reverses containment. In particular, a correlation interchanges
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * 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 :: ''s''(''u'',''v'') = ''s''(''v'',''u'')''α'' for all vectors ''u'', ''v'' of the underlying
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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,''q''2) for some prime power ''q'', the set of points of a nondegenerate Hermitian curve form a unital, which is called a ''classical unital''. Let \mathcal = \mathcal(2,q^2) be a nondegenerate Hermitian curve in PG(2,q^2) for some prime power q. As all nondegenerate Hermitian curves in the same plane are projectively equivalent, \mathcal can be described in terms of
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
as follows: \mathcal = \.


Ree unitals

Another family of unitals based on
Ree group In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a ...
s was constructed by H. Lüneburg. Let Γ = R(''q'') be the Ree group of type 2G2 of order (''q''3 + 1)''q''3(''q'' − 1) where ''q'' = 32''m''+1. Let ''P'' be the set of all ''q''3 + 1 Sylow 3-subgroups of Γ. Γ acts doubly transitively on this set by
conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change ...
(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 Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
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 In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
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-(''q''3 + 1, ''q'' + 1, 1) called a Ree unital. Lüneburg also showed that the Ree unitals can not be embedded in projective planes of order ''q''2 ( Desarguesian or not) such that the automorphism group Γ is induced by a
collineation group In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is ...
of the plane. For ''q'' = 3, Grüning proved that a Ree unital can not be embedded in any projective plane of order 9.


Unitals with 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.), an exhaustive computer search by Penttila and Royle 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, 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.


Isomorphic versus equivalent unitals

Since unitals are
block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
s, two unitals are said to be ''isomorphic'' if there is a design
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between them, that is, a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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.


Buekenhout's Constructions

By examining the classical unital in PG(2,q^2) in the Bruck/Bose model, Buekenhout provided two constructions, which together proved the existence of an embedded unital in any finite 2-dimensional translation plane. Metz 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 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.


Hermitian varieties

Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the
theory of polarities A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
.


Definition

Let ''K'' be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
with an involutive automorphism \theta. Let ''n'' be an integer \geq 1 and ''V'' be an ''(n+1)''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 e_0,e_1,\ldots,e_n be a basis of ''V''. If a point ''p'' in the projective space has homogeneous coordinates (X_0,\ldots,X_n) with respect to this basis, it is on the Hermitian variety if and only if : \sum_^ a_ X_ X_^ =0 where a_=a_^ and not all a_=0 If one constructs the
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
''A'' with A_=a_, the equation can be written in a compact way : X^t A X^=0 where X= \begin X_0 \\ X_1 \\ \vdots \\ X_n \end.


Tangent spaces and singularity

Let ''p'' be a point on the Hermitian variety ''H''. A line ''L'' through ''p'' is by definition
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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


Citations


Sources

* * * * * * * * {{refend Combinatorial design Finite geometry Incidence geometry Projective geometry Algebraic varieties