In
mathematics, the Segre embedding is 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 ...
to consider the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
(of sets) of two
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
s as a
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. It is named after
Corrado Segre
Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry.
Early life
Corrado's parents were Abramo Segre and Estella De Be ...
.
Definition
The Segre map may be defined as the map
:
taking a pair of points
to their product
:
(the ''X
iY
j'' are taken in
lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
).
Here,
and
are projective
vector spaces
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 ca ...
over some arbitrary
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 ...
, and the notation
:
is that 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 ...
on the space. The image of the map is a variety, called a Segre variety. It is sometimes written as
.
Discussion
In the language of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, for given
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 ca ...
s ''U'' and ''V'' over the same
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 ...
''K'', there is a natural way to map their cartesian product to their
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
.
:
In general, this need not be
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposit ...
because, for
in
,
in
and any nonzero
in
,
:
Considering the underlying projective spaces ''P''(''U'') and ''P''(''V''), this mapping becomes a morphism of varieties
:
This is not only injective in the set-theoretic sense: it is a
closed immersion in the sense of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
. That is, one can give a set of equations for the image. Except for notational trouble, it is easy to say what such equations are: they express two ways of factoring products of coordinates from the tensor product, obtained in two different ways as ''something from U times something from V''.
This mapping or morphism ''σ'' is the Segre embedding. Counting dimensions, it shows how the product of projective spaces of dimensions ''m'' and ''n'' embeds in dimension
:
Classical terminology calls the coordinates on the product multihomogeneous, and the product generalised to ''k'' factors k-way projective space.
Properties
The Segre variety is an example of a
determinantal variety; it is the zero locus of the 2×2 minors of the matrix
. That is, the Segre variety is the common zero locus of the
quadratic polynomial
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomi ...
s
:
Here,
is understood to be the natural coordinate on the image of the Segre map.
The Segre variety
is the categorical product of
and
.
The projection
:
to the first factor can be specified by m+1 maps on open subsets covering the Segre variety, which agree on intersections of the subsets. For fixed
, the map is given by sending