Segre Embedding - Discussion

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 in, in and any nonzero in ,

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.