The Segre embedding is a mapping that can be used in algebraic geometry to give the structure of a projective variety to the Cartesian product of two projective varieties . The Segre embed is named after Corrado Segre .
definition
Definition in homogeneous coordinates
Be an algebraically closed field , the - and the dimensional projective space over with homogeneous coordinates and .
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle \ mathbb {P} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle \ mathbb {P} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750bc4d5f674b96948732c50db6d04c56a89df0e)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle X_ {0}, \ dotsc, X_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8545826437b6b5ff3c93735e07b820752d56dda4)
![{\ displaystyle Y_ {0}, \ dotsc, Y_ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ed07581d27769e64d298c1886e233583431fbf)
The Segre embedding of and is defined as
![{\ displaystyle \ sigma _ {n, m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c78c3b501ff959518478a095587b639f8270404a)
![{\ displaystyle \ mathbb {P} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192)
![{\ displaystyle \ mathbb {P} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750bc4d5f674b96948732c50db6d04c56a89df0e)
-
,
which are arranged according to the lexicographical order .
![{\ displaystyle X_ {i} Y_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbd732e429426a9d5f7f2fb0f0f9b1580bbc7ff)
The picture is called the Segre variety .
![{\ displaystyle \ Sigma _ {m, n}: = \ sigma _ {n, m} (\ mathbb {P} ^ {n} \ times \ mathbb {P} ^ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4295b4c28a582d78e8ed40b90d238b7d458fc6)
Coordinate-free definition
It is also possible to define the Segre embedding without coordinates. For finite-dimensional vector spaces and and the associated projective spaces and one defines the Segre embedding with the help of the tensor product as
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
![{\ displaystyle \ mathbb {P} (W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e448d2c0a15d4d6f417cde58e5096a4bad03096a)
![{\ displaystyle \ mathbb {P} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8afe30e988d193afbadbc380123eb6cc6d9007dc)
![\ otimes](https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea)
-
.
properties
The Segre embedding is a well-defined injective mapping, the image of which is a closed, irreducible subset.
![{\ displaystyle \ sigma _ {n, m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c78c3b501ff959518478a095587b639f8270404a)
![{\ displaystyle \ Sigma _ {n, m} \ subseteq \ mathbb {P} ^ {(n + 1) (m + 1) -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/877d06e78b3d8eee07b1fdba60c8024cf4ac7139)
Thus the Segre variety is actually a projective variety . The corresponding homogeneous ideal can be stated explicitly. If we denote the homogeneous coordinates on with , we get
![{\ displaystyle \ mathbb {I} (\ Sigma _ {n, m})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a17400b00f59aafe1a8190c2e07bb4a5e84e7d)
![{\ displaystyle \ mathbb {P} ^ {(n + 1) (m + 1) -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/558837e89095e7f7e78b4d54ab588f7bb0a47123)
![{\ displaystyle Z_ {0,0}, \ dotsc, Z_ {n, m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f89eee26f088ad04f67834983c0806c26dd1f6b5)
-
.
The Segre variety can also be understood as the set of zeros of the minors of the matrix and is therefore a special determinant variety .
![{\ displaystyle (Z_ {ij}) _ {ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/474bb048e939d4cf0b96a61b2ab6d0dedbf2469e)
Products in the category of (quasi-) projective varieties
If , are (locally) closed subsets, then it is also (locally) closed.
![{\ displaystyle X \ subseteq \ mathbb {P} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae42cdb050e9ef27aa6148cc47640091356d0ac5)
![{\ displaystyle Y \ subseteq \ mathbb {P} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe59ffbb7e2ff4ccb469c3945abf6a5cd604f1b)
![{\ displaystyle \ sigma _ {n, m} (X \ times Y) \ subseteq \ mathbb {P} ^ {(n + 1) (m + 1) -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333451634db1226fd46f05152b0f985ec1feb86f)
Since is bijective, the structure of a (quasi-) projective variety can be defined by transferring the structure with the help of bijection .
![{\ displaystyle \ sigma _ {n, m} \ colon X \ times Y \ to \ sigma _ {n, m} (X \ times Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/172c3bd51c27c5694dea43e124ce1f351a6ea755)
![X times Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483)
![{\ displaystyle \ sigma _ {n, m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c78c3b501ff959518478a095587b639f8270404a)
The (quasi) projective variety thus defined is a product in the sense of category theory .
![X times Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483)
Alternatively, if one has defined the products in a different way, one can show that the Segre embedding is a closed embedding , which it is in the above way by definition.
Examples
Quadric
In the simplest case we get for an embedding of the product of the projective straight line . The Segre variety is then a quadric . If one denotes the homogeneous coordinates with , one obtains the quadric as the set of zeros of the determinant![n = m = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd469697e1893c049e1f06846869e0da69aa9fd8)
![{\ mathbb {P}} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb80175bc622b3936c7a0438fc690b2ec410b4f)
![{\ displaystyle \ Sigma _ {1,1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e761b7543801d2113eb55b87a26d7d8a6ac2ec)
![{\ mathbb {P}} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb80175bc622b3936c7a0438fc690b2ec410b4f)
![{\ displaystyle \ det \ left ({\ begin {matrix} Z_ {0} & Z_ {1} \\ Z_ {2} & Z_ {3} \ end {matrix}} \ right) = Z_ {0} Z_ {3} -Z_ {1} Z_ {2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d381397225b71c5a323d91691b8f4e5d0ec845d)
Individual evidence
-
^ Harris: Algebraic Geometry. 1992, Example 2.11.
-
↑ Fiesler, Kaup: Algebraic Geometry. 2005, p. 49.
-
^ Harris: Algebraic Geometry. 1992, Example 2.21.
-
^ Hartshorne: Algebraic Geometry. 1977, Exercise 3.16.
-
↑ Fiesler, Kaup: Algebraic Geometry. 2005, exercise 4.7.
-
^ Harris: Algebraic Geometry. 1992, Example 2.11.
literature