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 .







The Segre embedding of and is defined as



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,
which are arranged according to the lexicographical order .

The picture is called the Segre variety .

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






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.
properties
The Segre embedding is a well-defined injective mapping, the image of which is a closed, irreducible subset.


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



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.
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 .

Products in the category of (quasi-) projective varieties
If , are (locally) closed subsets, then it is also (locally) closed.



Since is bijective, the structure of a (quasi-) projective variety can be defined by transferring the structure with the help of bijection .



The (quasi) projective variety thus defined is a product in the sense of category theory .

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




Individual evidence
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^ Harris: Algebraic Geometry. 1992, Example 2.11.
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↑ Fiesler, Kaup: Algebraic Geometry. 2005, p. 49.
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^ Harris: Algebraic Geometry. 1992, Example 2.21.
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^ Hartshorne: Algebraic Geometry. 1977, Exercise 3.16.
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↑ Fiesler, Kaup: Algebraic Geometry. 2005, exercise 4.7.
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^ Harris: Algebraic Geometry. 1992, Example 2.11.
literature