In mathematics, the Segre embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named after Corrado Segre.

Definition

-dimensional projective space (over some field). The Segre embeddinggiven in homogeneous coordinates by

are taken in lexicographical order). The image of the map is a variety, called a Segre variety.

Discussion

In the language of linear algebra, for given vector spaces U and V over the same field K, there is a natural way to linearly map their Cartesian product to their tensor product.

In general, this need not be injective,

, 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. 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 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. That is, the Segre variety is the common zero locus of the quadratic polynomials

is understood to be the natural coordinate on the image of the Segre map.

is the categorical product. The projection

.

The fibers of the product are linear subspaces. That is, let

for the second factor. Then the image of the map

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

Examples

Quadric

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

be the homogeneous coordinates, this quadric is given as the zero locus of the quadratic polynomial given by the determinant

Segre threefold

The map

is known as the Segre threefold. It is an example of a rational normal scroll is a twisted cubic curve.

Veronese variety

under the Segre map is the Veronese variety of degree two

Applications

Because the Segre map is to the categorical product of projective spaces, it is a natural mapping for describing non-entangled states in quantum mechanics and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert spaces.

In algebraic statistics, Segre varieties correspond to independence models.

is the only Severi variety of dimension 4.

References