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.
Here, is understood to be the natural coordinate on the image of the Segre map.
The Segre variety is the categorical product of and .
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 , the map is given by sending to . The equations ensure that these maps agree with each other, because if we have .
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 = 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. Letting