Segre embedding

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.

In linear algebra terms there is for given vector spaces U and V, 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

(u,v)

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

(cu,c⁻¹v)

will also be w. In co-ordinate terms, w has co-ordinates formed of all products of a co-ordinate of u with a co-ordinate of v.

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

s: 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 co-ordinates from W, obtained in two different ways as something from U times something from V.

This mapping or morphism s 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.

For example with m = n = 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 quadric.

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