In mathematics, the Segre embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named after Corrado Segre.
Definition We write P n {\displaystyle \mathbb {P} ^{n}} for n {\displaystyle n} -dimensional projective space (over some field). The Segre embedding is defined as the map σ : P n × P m → P ( n + 1 ) ( m + 1 ) − 1 {\displaystyle \sigma :\mathbb {P} ^{n}\times \mathbb {P} ^{m}\to \mathbb {P} ^{(n+1)(m+1)-1}\ } given in homogeneous coordinates by
σ ( [ X 0 : X 1 : ⋯ : X n ] , [ Y 0 : Y 1 : ⋯ : Y m ] ) = [ X 0 Y 0 : X 0 Y 1 : ⋯ : X i Y j : ⋯ : X n Y m ] {\displaystyle \sigma ([X_{0}:X_{1}:\cdots :X_{n}],[Y_{0}:Y_{1}:\cdots :Y_{m}])=[X_{0}Y_{0}:X_{0}Y_{1}:\cdots :X_{i}Y_{j}:\cdots :X_{n}Y_{m}]\ }
(the monomials X i Y j {\displaystyle X_{i}Y_{j}} are taken in lexicographical order). The image of the map is a variety, called a Segre variety. It is sometimes written as Σ n , m {\displaystyle \Sigma _{n,m}} .
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 linearly map their Cartesian product to their tensor product.
φ : U × V → U ⊗ V . {\displaystyle \varphi :U\times V\to U\otimes V.\ }
In general, this need not be injective because, for u ∈ U {\displaystyle u\in U} , v ∈ V {\displaystyle v\in V} and any nonzero c ∈ K {\displaystyle c\in K} ,
φ ( u , v ) = u ⊗ v = c u ⊗ c − 1 v = φ ( c u , c − 1 v ) . {\displaystyle \varphi (u,v)=u\otimes v=cu\otimes c^{-1}v=\varphi (cu,c^{-1}v).\ }
Considering the underlying projective spaces P ( U ) {\displaystyle \mathbb {P} (U)} and P ( V ) {\displaystyle \mathbb {P} (V)} , this mapping becomes a morphism of varieties
σ : P ( U ) × P ( V ) → P ( U ⊗ V ) . {\displaystyle \sigma :\mathbb {P} (U)\times \mathbb {P} (V)\to \mathbb {P} (U\otimes V).\ }
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
( m + 1 ) ( n + 1 ) − 1 = m n + m + n . {\displaystyle (m+1)(n+1)-1=mn+m+n.\ }
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 ( Z i , j ) {\displaystyle (Z_{i,j})} . That is, the Segre variety is the common zero locus of the quadratic polynomials
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