In mathematics, the Segre embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named after Corrado Segre.
Definition
We write \mathbb{P}^n for n-dimensional projective space (over some field). The Segre embedding is defined as the map \sigma: \mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^{(n+1)(m+1)-1}\given in homogeneous coordinates by
\sigma([X_0:X_1:\cdots:X_n], [Y_0:Y_1:\cdots:Y_m]) = [X_0Y_0: X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m]\
(the monomials X_i Y_j are taken in lexicographical order). 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 \mathbb{P}(U) and \mathbb{P}(V), this mapping becomes a morphism of varieties
\sigma: \mathbb{P}(U)\times \mathbb{P}(V) \to \mathbb{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,[clarification needed] 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 \mathbb{P^n} and \mathbb{P^m}.[1] The projection
\pi_X :\Sigma_{n,m} \to \mathbb{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 \mathbb{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 \mathbb{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 \mathbb{P}^3. 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 \mathbb{P}^3, 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.\
Segre threefold
The map
\sigma: \mathbb{P}^2 \times \mathbb{P}^1 \to \mathbb{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 \mathbb{P}^3 is a twisted cubic curve.
Veronese variety
The image of the diagonal \Delta \subset \mathbb{P}^n \times \mathbb{P}^n under the Segre map is the Veronese variety of degree two
\nu_2:\mathbb{P}^n \to \mathbb{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 \mathbb{P}^2 \times \mathbb{P}^2 in \mathbb{P}^8 is the only Severi variety of dimension 4.
References
- ^ McKernan, James (2010). "Algebraic Geometry Course, Lecture 6: Products and fibre products". 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). arXiv:1910.09665. 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