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 quadratics

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.

[edit] References

  • Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3
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