Segre embedding

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In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety.

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[edit] 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_0Y_0: X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m]

Here, Pn and Pm 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. Notationally, it is sometimes written as Σn,m.

[edit] Discussion

In linear algebra terms there is for given vector spaces U and V, over the same field, a natural way to map

U × V

to the tensor product space W. This not in general injective, because it takes the pair

(u,v)

to the pure tensor w formed from u and v. For any non-zero scalar c, the image of

(cu,c−1v)

will also be w. In co-ordinate terms, w has co-ordinates formed of all products of a co-ordinate of u with a co-ordinate of v.

Considering now the underlying projective spaces P(U) and P(V), the mapping passes to a morphism of varieties

σ: P(U) × P(V) → P(W).

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 co-ordinates from W, 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 co-ordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.

[edit] Properties

The Segre variety is an example of a determinantal variety; it is the zero locus of the 2×2 minors of the matrix (Zi,j). That is, the Segre variety is the common zero locus of the quadratic polynomials

Zi,jZk,l − Zi,lZk,j

Here, Zi,j is understood to be the natural coordinate on the image of the Segre map.

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

\pi_X :P^{(n+1)(m+1)-1} \to P^n

be the projection to the first factor; and likewise πY for the second factor. Then the image of the map

\sigma (\pi_X (\cdot), \pi_Y (p)):P^{(n+1)(m+1)-1} \to P^{(n+1)(m+1)-1}

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

[edit] Examples

[edit] 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

[Z0:Z1:Z2:Z3]

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

[edit] 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 P3 is a twisted cubic curve.

[edit] 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}.

[edit] Applications

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

In algebraic statistics, Segre varieties correspond to independence models.

[edit] References