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.
Read more about this topic: Segre Embedding
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