Segre embedding explained

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

\sigma: P ^{n} x P ^{m} \to P ^{(n+1)(m+1)-1}

taking a pair of points

([X],[Y]) \in P ^{n} x P ^{m} to their product\sigma:([X _{0:X} _{1: … :X} _{n],} [Y _{0:Y} _{1: … :Y} _{m])} \mapsto [X _{0Y} _{0:} X _{0Y} _{1:} … :X _{iY} _{j:} … :X _{nY} _{m]} (the X_iY_j are taken in lexicographical order).

Here,

P ^{n} and P ^{m} are projective vector spaces over some arbitrary field, and the notation[X _{0:X} _{1: … :X} _{n]}

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

\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 map their cartesian product to their tensor product.

\varphi: U x V \to U ⊗ V.

In general, this need not be injective because, for

u\in U , v\in V and any nonzero c\in K ,\varphi(u,v) = u ⊗ v = cu ⊗ c ^{-1} v = \varphi(cu, c ^{-1} v).

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

\sigma: P(U) x P(V) \to P(U ⊗ 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 = 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} ) . That is, the Segre variety is the common zero locus of the quadratic polynomialsZ _{i,j} Z _{k,l} - Z _{i,l} Z _{k,j} .

Here,

Z _{i,j} is understood to be the natural coordinate on the image of the Segre map. The Segre variety \Sigma _{n,m} is the categorical product of P ^{n} and P ^{m} .^[1] The projection\pi _{X} :\Sigma _{n,m} \to P ^{n}

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

j _{0} , the map is given by sending [Z _{i,j} ] to [Z _{i,j _{0}} ] . The equations Z _{i,j} Z _{k,l} = Z _{i,l} Z _{k,j} ensure that these maps agree with each other, because if Z _{i _{0,j} _{0}} ≠ 0 we have [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}} ] .

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

\pi _{X} :\Sigma _{n,m} \to P ^{n}

be the projection to the first factor; and likewise

\pi _{Y} for the second factor. Then the image of the map\sigma (\pi _{X} ( ⋅ ), \pi _{Y} (p)):\Sigma _{n,m} \to P ^{(n+1)(m+1)-1}

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

[Z _{0:Z} _{1:Z} _{2:Z} _{3]}

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

\det \left(\begin{matrix}Z _{0&Z} _{1\\Z} _{2&Z} _{3\end{matrix}\right)} = Z _{0Z} _{3} - Z _{1Z} _{2.}

Segre threefold

The map

\sigma: P ^{2} x P ^{1} \to P ^{5}

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

P ^{3} is a twisted cubic curve.

Veronese variety

The image of the diagonal

\Delta \subset P ^{n} x P ^{n} under the Segre map is the Veronese variety of degree two\nu ^{n} _{2 :P} \to P ^{n ^{2+2n}} .

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.

Notes and References

1. Web site: McKernan. James. Algebraic Geometry Course, Lecture 6: Products and fibre products. online course material. 11 April 2014. 2010.
2. Gharahi . Masoud . Mancini . Stefano . Ottaviani . Giorgio . 2020-10-01 . Fine-structure classification of multiqubit entanglement by algebraic geometry . Physical Review Research . 2 . 4 . 043003 . 10.1103/PhysRevResearch.2.043003. free . 2158/1210686 . free .