Segre embedding

The Segre map may be defined as the map

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

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

:$\backslash sigma:([X\_0:X\_1:\backslash cdots:X\_n],\; [Y\_0:Y\_1:\backslash cdots:Y\_m])\; \backslash mapsto$

[X_0Y_0: X_0Y_1: \cdots :X_iY_j: \cdots :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

:$[X\_0:X\_1:\backslash cdots:X\_n]\backslash$

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

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.

: $\backslash varphi:\; U\backslash times\; V\; \backslash to\; U\backslash otimes\; V.\backslash$

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

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

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

: $\backslash sigma:\; P(U)\backslash times\; P(V)\; \backslash to\; P(U\backslash otimes\; V).\backslash$

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

:$(m\; +\; 1)(n\; +\; 1)\; -\; 1\; =\; mn\; +\; m\; +\; n.\backslash$

Classical terminology calls the coordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.

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

:$Z\_\{i,j\}\; Z\_\{k,l\}\; -\; Z\_\{i,l\}\; Z\_\{k,j\}.\backslash$

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

The Segre variety $\backslash Sigma\_\{n,m\}$ is the categorical product (in the category of projective varieties and homogeneous polynomial maps) of $P^n\backslash$ and $P^m$.{{cite web|last=McKernan|first=James|title=Algebraic Geometry Course, Lecture 6: Products and fibre products|url=http://math.mit.edu/~mckernan/Teaching/09-10/Autumn/18.725/l_6.pdf|work=online course material|accessdate=11 April 2014|year=2010 |at=Lemma 6.3}}

The projection

:$\backslash pi\_X\; :\backslash Sigma\_\{n,m\}\; \backslash to\; P^n\backslash$

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 $Z\_\{i,j\}\; Z\_\{k,l\}\; =\; Z\_\{i,l\}\; Z\_\{k,j\}\backslash$ ensure that these maps agree with each other, because if $Z\_\{i\_0,j\_0\}\backslash neq\; 0$ 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

:$\backslash pi\_X\; :\backslash Sigma\_\{n,m\}\; \backslash to\; P^n\backslash$

be the projection to the first factor; and likewise $\backslash pi\_Y$ for the second factor. Then the image of the map

:$\backslash sigma\; (\backslash pi\_X\; (\backslash cdot),\; \backslash pi\_Y\; (p)):\backslash Sigma\_\{n,m\}\; \backslash to\; P^\{(n+1)(m+1)-1\}\backslash$

for a fixed point p is a linear subspace of the codomain.

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

:$[Z\_0:Z\_1:Z\_2:Z\_3]\backslash$

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

:

= Z_0Z_3 - Z_1Z_2.\

The map

:$\backslash sigma:\; P^2\; \backslash times\; P^1\; \backslash to\; P^5$

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.

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

:$\backslash nu\_2:P^n\; \backslash to\; P^\{n^2+2n\}.\backslash$

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.{{Cite journal |last=Gharahi |first=Masoud |last2=Mancini |first2=Stefano |last3=Ottaviani |first3=Giorgio |date=2020-10-01 |title=Fine-structure classification of multiqubit entanglement by algebraic geometry |url=https://link.aps.org/doi/10.1103/PhysRevResearch.2.043003 |journal=Physical Review Research |volume=2 |issue=4 |pages=043003 |doi=10.1103/PhysRevResearch.2.043003|doi-access=free |hdl=2158/1210686 |hdl-access=free }}

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.

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Category:Algebraic varieties

Category:Projective geometry