Segre embedding

In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. The Segre map may be defined as the map taking a pair of points ( [ X ] , [ Y ] ) ∈ P n × P m {displaystyle (,)in P^{n} imes P^{m}} to their product (the XiYj are taken in lexicographical order). Here, P n {displaystyle P^{n}} and P m {displaystyle P^{m}} 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 Σ n , m {displaystyle Sigma _{n,m}} . 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 u {displaystyle u} in U {displaystyle U} , v {displaystyle v} in V {displaystyle V} and any nonzero c {displaystyle c} in K {displaystyle K} , Considering the underlying projective spaces P(U) and P(V), this mapping becomes a morphism of varieties