, the Segre embedding
is used in projective geometry
to consider the cartesian product
of two or more projective spaces
as a projective variety
The Segre map
may be defined as the map
taking a pair of points to their product
[X_0Y_0: X_0Y_1: cdots :X_iY_j: cdots :X_nY_m]
Here, and 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. Notationally, it is sometimes written as .
In linear algebra
terms there is for given vector spaces U
, over the same field
, a natural way to map
- U × V
to the tensor product space W. This not in general injective, because it takes the pair
to the pure tensor w formed from u and v. For any non-zero scalar c, the image of
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.
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 polynomials
Here, 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
be the projection to the first factor; and likewise for the second factor. Then the image of the map
for a fixed point p is a linear subspace of the codomain.
For example with m
= 1 we get an embedding of the product of the projective line
with itself in P3
. 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
be the homogeneous coordinates on P3, this quadric is given as the zero locus of the quadratic polynomial given by the determinant
= Z_0Z_3 - Z_1Z_2
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 is a twisted cubic curve.
The image of the diagonal
under the Segre map is the Veronese variety
of degree two
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
In algebraic statistics, Segre varieties correspond to independence models.