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In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety.
## Definition

The Segre map may be defined as the map## Discussion

In linear algebra terms there is for given vector spaces U and V, over the same field, a natural way to map## Properties

The Segre variety is an example of a determinantal variety; 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 polynomials ## Examples

### Quadric

For example with m = n = 1 we get an embedding of the product of the projective line with itself in P^{3}. The image is a quadric, and is easily seen to contain two one-parameter families of lines. Over the complex numbers this is a quite general non-singular quadric. Letting ### Segre threefold

The map### Veronese variety

The image of the diagonal $Delta\; subset\; P^n\; times\; P^n$ under the Segre map is the Veronese variety of degree two $nu\_2:P^n\; to\; P^\{n^2+2n\}$.
## 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 and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert spaces. ## References

- $sigma:\; P^n\; times\; P^m\; to\; P^\{(n+1)(m+1)-1\}$

taking a pair of points $([X],[Y])\; in\; P^n\; times\; P^m$ to their product

- $sigma:([X\_0:X\_1:cdots:X\_n],\; [Y\_0:Y\_1:cdots:Y\_m])\; mapsto$

[X_0Y_0: X_0Y_1: cdots :X_iY_j: cdots :X_nY_m]

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

- $[X\_0:X\_1:cdots:X\_n]$

is that of homogeneous coordinates on the space. The image of the map is a variety, called a Segre variety. Notationally, it is sometimes written as $Sigma\_\{n,m\}$.

- U × V

to the tensor product space W. This not in general injective, because it takes the pair

- (u,v)

to the pure tensor w formed from u and v. For any non-zero scalar c, the image of

- (cu,c
^{−1}v)

will also be w. In coordinate terms, w has coordinates formed of all products of a coordinate of u with a coordinate of v.

Considering now the underlying projective spaces P(U) and P(V), the mapping passes to a morphism of varieties

- σ: P(U) × P(V) → P(W).

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

Classical terminology calls the coordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.

- $Z\_\{i,j\}\; Z\_\{k,l\}\; -\; Z\_\{i,l\}\; Z\_\{k,j\}$

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

- $pi\_X\; :P^\{(n+1)(m+1)-1\}\; to\; P^n$

be the projection to the first factor; and likewise $pi\_Y$ for the second factor. Then the image of the map

- $sigma\; (pi\_X\; (cdot),\; pi\_Y\; (p)):P^\{(n+1)(m+1)-1\}\; to\; P^\{(n+1)(m+1)-1\}$

for a fixed point p is a linear subspace of the codomain.

- $[Z\_0:Z\_1:Z\_2:Z\_3]$

be the homogeneous coordinates on P^{3}, this quadric is given as the zero locus of the quadratic polynomial given by the determinant

- $det\; left(begin\{matrix\}Z\_0\&Z\_1Z\_2\&Z\_3end\{matrix\}right)$

- $sigma:\; P^2\; times\; P^1\; to\; P^5$

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.

In algebraic statistics, Segre varieties correspond to independence models.

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Last updated on Saturday July 12, 2008 at 19:37:46 PDT (GMT -0700)

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Last updated on Saturday July 12, 2008 at 19:37:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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