The Segre map may be defined as the map taking a pair of points to their product . 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. It is sometimes written as.
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 and P, 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.
Properties
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 Segre variety is the categorical product of and. The projection 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 pointp is a linear subspace of the codomain.
Examples
Quadric
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 be the homogeneous coordinates on P3, this quadric is given as the zero locus of the quadratic polynomial given by the determinant
The map 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.
