In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Segre embedding is used in projective geometry

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

 to consider the cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

 of two or more projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

s as a projective variety. It is named after Corrado Segre

Corrado Segre

Corrado Segre was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry....

.

Definition

The Segre map may be defined as the map $\backslash sigma:\; P^n\; \backslash times\; P^m\; \backslash to\; P^\{(n+1)(m+1)-1\}\backslash$ taking a pair of points $([X],[Y])\; \backslash in\; P^n\; \backslash times\; P^m$ to their product $\backslash sigma:([X\_0:X\_1:\backslash cdots:X\_n],\; [Y\_0:Y\_1:\backslash cdots:Y\_m])\; \backslash mapsto\; [X\_0Y\_0:\; X\_0Y\_1:\; \backslash cdots\; :X\_iY\_j:\; \backslash cdots\; :X\_nY\_m]\backslash$ (the X_iY_j are taken in lexicographical order

Lexicographical order

In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...

). Here, $P^n$ and $P^m$ are projective vector spaces over some arbitrary field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, and the notation $[X\_0:X\_1:\backslash cdots:X\_n]\backslash$ is that of homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

 on the space. The image of the map is a variety, called a Segre variety. It is sometimes written as $\backslash Sigma\_\{n,m\}$.

Discussion

In the language of linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, for given vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s U and V over the same field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K, there is a natural way to map their cartesian product to their tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

. NEWLINE

NEWLINE

$\backslash varphi:\; U\backslash times\; V\; \backslash to\; U\backslash otimes\; V.\backslash$

NEWLINE In general, this need not be injective because, for $u$ in $U$, $v$ in $V$ and any nonzero $c$ in $K$, NEWLINE

NEWLINE

$\backslash varphi(u,v)\; =\; u\backslash otimes\; v\; =\; cu\backslash otimes\; c^\{-1\}v\; =\; \backslash varphi(cu,\; c^\{-1\}v).\backslash$

NEWLINE Considering the underlying projective spaces P(U) and P(V), this mapping becomes a morphism of varieties NEWLINE

NEWLINE

$\backslash sigma:\; P(U)\backslash times\; P(V)\; \backslash to\; P(U\backslash otimes\; V).\backslash$

NEWLINE This is not only injective in the set-theoretic sense: it is a closed immersion in the sense of algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. 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 $(m\; +\; 1)(n\; +\; 1)\; -\; 1\; =\; mn\; +\; m\; +\; n.\backslash$ 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

Determinantal variety

In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the Segre embedding of a product of two projective spaces.-Definition:Given m and...

; it is the zero locus of the 2×2 minors of the matrix $(Z\_\{i,j\})$. That is, the Segre variety is the common zero locus of the quadratic polynomial

Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...

s $Z\_\{i,j\}\; Z\_\{k,l\}\; -\; Z\_\{i,l\}\; Z\_\{k,j\}.\backslash$ Here, $Z\_\{i,j\}$ is understood to be the natural coordinate on the image of the Segre map. The fibers of the product are linear subspaces. That is, let $\backslash pi\_X\; :P^\{(n+1)(m+1)-1\}\; \backslash to\; P^n\backslash$ be the projection to the first factor; and likewise $\backslash pi\_Y$ for the second factor. Then the image of the map $\backslash sigma\; (\backslash pi\_X\; (\backslash cdot),\; \backslash pi\_Y\; (p)):P^\{(n+1)(m+1)-1\}\; \backslash to\; P^\{(n+1)(m+1)-1\}\backslash$ for a fixed point p is a linear subspace of the codomain

Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...

.

Quadric

For example with m = n = 1 we get an embedding of the product of the projective line

Projective line

In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...

 with itself in P³. The image is a quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

, and is easily seen to contain two one-parameter families of lines. Over the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s this is a quite general non-singular quadric. Letting $[Z\_0:Z\_1:Z\_2:Z\_3]\backslash$ be the homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

 on P³, this quadric is given as the zero locus of the quadratic polynomial given by the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

Segre threefold

The map $\backslash sigma:\; P^2\; \backslash times\; P^1\; \backslash to\; P^5$ is known as the Segre threefold. It is an example of a rational normal scroll. The intersection of the Segre threefold and a three-plane $P^3$ is a twisted cubic curve.

Veronese variety

The image of the diagonal $\backslash Delta\; \backslash subset\; P^n\; \backslash times\; P^n$ under the Segre map is the Veronese variety of degree two$\backslash nu\_2:P^n\; \backslash to\; P^\{n^2+2n\}.\backslash$

Applications

Because the Segre map is to the categorical product of projective spaces, it is a natural mapping for describing entangled states in quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert space

Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P of a complex Hilbert space H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by...

s. In algebraic statistics

Algebraic statistics

Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing....

, Segre varieties correspond to independence models.