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

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

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

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

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

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.

\varphi :U\times V\to U\otimes 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\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

\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

(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 polynomials

Z_{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 (in the category of projective varieties and homogeneous polynomial maps) 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 $Z_{i_{0},j_{0}}\neq 0$ 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}(\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

[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_{0}Z_{3}-Z_{1}Z_{2}.\

Segre threefold

The map

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

Veronese variety

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

\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

^ McKernan, James (2010). "Algebraic Geometry Course, Lecture 6: Products and fibre products" (PDF). online course material. Lemma 6.3. Retrieved 11 April 2014.
^ 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. hdl:2158/1210686.

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

[1] McKernan, James (2010). "Algebraic Geometry Course, Lecture 6: Products and fibre products" (PDF). online course material. Lemma 6.3. 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. hdl:2158/1210686.

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