In
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
* Point ...
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 b ...
. Some unitals may be
embedded in a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
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, pr ...
.
A correlation of a projective geometry is a
bijection 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
* Point ...
and
hyperplanes.
A correlation of order two is called a polarity.
A polarity is called a unitary polarity if its associated
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
''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 ...
.
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
be a nondegenerate Hermitian curve in
for some prime power
. As all nondegenerate Hermitian curves in the same plane are projectively equivalent,
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 geometry. ...
as follows:
Ree unitals
Another family of unitals based on
Ree groups was constructed by H. Lüneburg. Let Γ = R(''q'') be the Ree group of type
2G
2 of order (''q''
3 + 1)''q''
3(''q'' − 1) where ''q'' = 3
2''m''+1. Let ''P'' be the set of all ''q''
3 + 1
Sylow 3-subgroups of Γ. Γ
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
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 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 b ...
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 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
In the
four projective planes of order 9 (the
Desarguesian plane PG(2,9), the
Hall plane In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4.
Algebraic con ...
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 In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4.
Algebraic con ...
(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 b ...
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 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
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
in the higher-dimensional Bruck/Bose model, which lies in
, 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
Hermitian varieties are in a sense a generalisation of
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...
s, 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 with an involutive
automorphism . Let ''n'' be an integer
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 ...
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
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows ...
on ''V''.
Representation
Let
be a basis of ''V''. If a point ''p'' in the
projective space has homogeneous coordinates
with respect to this basis, it is on the Hermitian variety if and only if :
where
and not all
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
, the equation can be written in a compact way :
where
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
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{{refend
Combinatorial design
Finite geometry
Incidence geometry
Projective geometry
Algebraic varieties