Segre embedding

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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.

Definition

The Segre map may be defined as the map

[math]\displaystyle{ \sigma: P^n \times P^m \to P^{(n+1)(m+1)-1}\ }[/math]

taking a pair of points [math]\displaystyle{ ([X],[Y]) \in P^n \times P^m }[/math] to their product

[math]\displaystyle{ \sigma:([X_0:X_1:\cdots:X_n], [Y_0:Y_1:\cdots:Y_m]) \mapsto [X_0Y_0: X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m]\ }[/math]

(the XiYj are taken in lexicographical order).

Here, [math]\displaystyle{ P^n }[/math] and [math]\displaystyle{ P^m }[/math] are projective vector spaces over some arbitrary field, and the notation

[math]\displaystyle{ [X_0:X_1:\cdots:X_n]\ }[/math]

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 [math]\displaystyle{ \Sigma_{n,m} }[/math].

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.

[math]\displaystyle{ \varphi: U\times V \to U\otimes V.\ }[/math]

In general, this need not be injective because, for [math]\displaystyle{ u\in U }[/math], [math]\displaystyle{ v\in V }[/math] and any nonzero [math]\displaystyle{ c\in K }[/math],

[math]\displaystyle{ \varphi(u,v) = u\otimes v = cu\otimes c^{-1}v = \varphi(cu, c^{-1}v).\ }[/math]

Considering the underlying projective spaces P(U) and P(V), this mapping becomes a morphism of varieties

[math]\displaystyle{ \sigma: P(U)\times P(V) \to P(U\otimes V).\ }[/math]

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

[math]\displaystyle{ (m + 1)(n + 1) - 1 = mn + m + n.\ }[/math]

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 [math]\displaystyle{ (Z_{i,j}) }[/math]. That is, the Segre variety is the common zero locus of the quadratic polynomials

[math]\displaystyle{ Z_{i,j} Z_{k,l} - Z_{i,l} Z_{k,j}.\ }[/math]

Here, [math]\displaystyle{ Z_{i,j} }[/math] is understood to be the natural coordinate on the image of the Segre map.

The Segre variety [math]\displaystyle{ \Sigma_{n,m} }[/math] is the categorical product of [math]\displaystyle{ P^n\ }[/math] and [math]\displaystyle{ P^m }[/math].[1] The projection

[math]\displaystyle{ \pi_X :\Sigma_{n,m} \to P^n\ }[/math]

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 [math]\displaystyle{ j_0 }[/math], the map is given by sending [math]\displaystyle{ [Z_{i,j}] }[/math] to [math]\displaystyle{ [Z_{i,j_0}] }[/math]. The equations [math]\displaystyle{ Z_{i,j} Z_{k,l} = Z_{i,l} Z_{k,j}\ }[/math] ensure that these maps agree with each other, because if [math]\displaystyle{ Z_{i_0,j_0}\neq 0 }[/math] we have [math]\displaystyle{ [Z_{i,j_1}]=[Z_{i_0,j_0}Z_{i,j_1}]=[Z_{i_0,j_1}Z_{i,j_0}]=[Z_{i,j_0}] }[/math].

The fibers of the product are linear subspaces. That is, let

[math]\displaystyle{ \pi_X :\Sigma_{n,m} \to P^n\ }[/math]

be the projection to the first factor; and likewise [math]\displaystyle{ \pi_Y }[/math] for the second factor. Then the image of the map

[math]\displaystyle{ \sigma (\pi_X (\cdot), \pi_Y (p)):\Sigma_{n,m} \to P^{(n+1)(m+1)-1}\ }[/math]

for a fixed point p 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

[math]\displaystyle{ [Z_0:Z_1:Z_2:Z_3]\ }[/math]

be the homogeneous coordinates on P3, this quadric is given as the zero locus of the quadratic polynomial given by the determinant

[math]\displaystyle{ \det \left(\begin{matrix}Z_0&Z_1\\Z_2&Z_3\end{matrix}\right) = Z_0Z_3 - Z_1Z_2.\ }[/math]

Segre threefold

The map

[math]\displaystyle{ \sigma: P^2 \times P^1 \to P^5 }[/math]

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 [math]\displaystyle{ P^3 }[/math] is a twisted cubic curve.

Veronese variety

The image of the diagonal [math]\displaystyle{ \Delta \subset P^n \times P^n }[/math] under the Segre map is the Veronese variety of degree two

[math]\displaystyle{ \nu_2:P^n \to P^{n^2+2n}.\ }[/math]

Applications

Because the Segre map is to the categorical product of projective spaces, it is a natural mapping for describing non-entangled states in quantum mechanics and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert spaces.[2]

In algebraic statistics, Segre varieties correspond to independence models.

The Segre embedding of P2×P2 in P8 is the only Severi variety of dimension 4.

References