Segre embedding

In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
Definition.
The Segre map may be defined as the map
taking a pair of points formula_2 to their product
(the "XiYj" are taken in lexicographical order).
Here, formula_4 and formula_5 are projective vector spaces over some arbitrary field, and the notation
is that of homogeneous coordinates on the space. The image of the map is a variety, called a Segre variety. It is sometimes written as formula_7.
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 map their cartesian product to their tensor product.
In general, this need not be injective because, for formula_9 in formula_10, formula_11 in formula_12 and any nonzero formula_13 in formula_14,
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. 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 formula_18. That is, the Segre variety is the common zero locus of the quadratic polynomials
Here, formula_20 is understood to be the natural coordinate on the image of the Segre map.
The Segre variety formula_7 is the categorical product of formula_22 and formula_5.
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 formula_25, the map is given by sending formula_26 to formula_27. The equations formula_28 ensure that these maps agree with each other, because if formula_29 we have formula_30.
The fibers of the product are linear subspaces. That is, let
be the projection to the first factor; and likewise formula_32 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 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
be the homogeneous coordinates on "P"3, 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. The intersection of the Segre threefold and a three-plane formula_37 is a twisted cubic curve.
Veronese variety.
The image of the diagonal formula_38 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.
The Segre embedding of P2×P2 in P8 is the only Severi variety of dimension 4.