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.

Definition

The Segre map may be defined as the map

${\displaystyle \sigma :P^{n}\times P^{m}\to P^{(n+1)(m+1)-1}\ }$

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

${\displaystyle \sigma$ :([X_{0}:X_{1}:\cdots :X_{n}],[Y_{0}:Y_{1}:\cdots :Y_{m}])\mapsto [X_{0}Y_{0}:X_{0}Y_{1}:\cdots :X_{i}Y_{j}:\cdots :X_{n}Y_{m}]\ }

(the X_iY_j are taken in lexicographical order).

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

${\displaystyle [X_{0}:X_{1}:\cdots :X_{n}]\ }$

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 ${\displaystyle \Sigma _{n,m}}$.

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 map their cartesian product to their tensor product.

${\displaystyle \varphi :U\times V\to U\otimes V.\ }$

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

${\displaystyle \varphi (u,v)=u\otimes v=cu\otimes c^{-1}v=\varphi (cu,c^{-1}v).\ }$

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

${\displaystyle \sigma :P(U)\times P(V)\to P(U\otimes V).\ }$

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

${\displaystyle (m+1)(n+1)-1=mn+m+n.\ }$

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

${\displaystyle Z_{i,j}Z_{k,l}-Z_{i,l}Z_{k,j}.\ }$

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

The Segre variety ${\displaystyle \Sigma _{n,m}}$ is the categorical product of ${\displaystyle P^{n}\ }$ and ${\displaystyle P^{m}}$. ^[1] The projection

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

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 ${\displaystyle j_{0}}$, the map is given by sending ${\displaystyle [Z_{i,j}]}$ to ${\displaystyle [Z_{i,j_{0}}]}$. The equations ${\displaystyle Z_{i,j}Z_{k,l}=Z_{i,l}Z_{k,j}\ }$ ensure that these maps agree with each other, because if ${\displaystyle Z_{i_{0},j_{0}}\neq 0}$ we have ${\displaystyle [Z_{i,j_{1}}]=[Z_{i_{0},j_{0}}Z_{i,j_{1}}]=[Z_{i_{0},j_{1}}Z_{i,j_{0}}]=[Z_{i,j_{0}}]}$.

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

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

be the projection to the first factor; and likewise ${\displaystyle \pi _{Y}}$ for the second factor. Then the image of the map

${\displaystyle \sigma (\pi _{X}(\cdot ),\pi _{Y}(p)):\Sigma _{n,m}\to P^{(n+1)(m+1)-1}\ }$

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

${\displaystyle [Z_{0}:Z_{1}:Z_{2}:Z_{3}]\ }$

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

${\displaystyle \det \left({\begin{matrix}Z_{0}&Z_{1}\\Z_{2}&Z_{3}\end{matrix}}\right)=Z_{0}Z_{3}-Z_{1}Z_{2}.\ }$

Segre threefold

The map

${\displaystyle \sigma :P^{2}\times P^{1}\to P^{5}}$

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 ${\displaystyle P^{3}}$ is a twisted cubic curve.

Veronese variety

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

${\displaystyle \nu _{2}:P^{n}\to P^{n^{2}+2n}.\ }$

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.

References

1. McKernan, James (2010). "Algebraic Geometry Course, Lecture 6: Products and fibre products" (PDF). online course material. Retrieved 11 April 2014.
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. doi: 10.1103/PhysRevResearch.2.043003 .

• Harris, Joe (1995), Algebraic Geometry: A First Course, Berlin, New York: Springer-Verlag, ISBN   978-0-387-97716-4
• Hassett, Brendan (2007), Introduction to Algebraic Geometry, Cambridge: Cambridge University Press, p. 154, doi:10.1017/CBO9780511755224, ISBN   978-0-521-69141-3, MR   2324354