Segre embed

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 . ${\displaystyle K}$ ${\displaystyle \mathbb {P} ^{n}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {P} ^{m}}$ ${\displaystyle m}$ ${\displaystyle K}$ ${\displaystyle X_{0},\dotsc ,X_{n}}$ ${\displaystyle Y_{0},\dotsc ,Y_{m}}$

The Segre embedding of and is defined as ${\displaystyle \sigma _{n,m}}$ ${\displaystyle \mathbb {P} ^{n}}$ ${\displaystyle \mathbb {P} ^{m}}$

${\displaystyle \sigma _{n,m}\colon \mathbb {P} ^{n}\times \mathbb {P} ^{m}\to \mathbb {P} ^{(n+1)(m+1)-1},([X_{0}:X_{1}:\cdots :X_{n}],[Y_{0}:Y_{1}:\cdots :Y_{m}])\mapsto [X_{0}Y_{0}:X_{0}Y_{1}:\cdots :X_{i}Y_{j}:\cdots :X_{n}Y_{m}]}$,

which are arranged according to the lexicographical order . ${\displaystyle X_{i}Y_{j}}$

The picture is called the Segre variety . ${\displaystyle \Sigma _{m,n}:=\sigma _{n,m}(\mathbb {P} ^{n}\times \mathbb {P} ^{n})}$

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 ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle \mathbb {P} (W)}$ ${\displaystyle \mathbb {P} (V)}$ ${\displaystyle \sigma _{V,W}}$ ${\displaystyle \otimes }$

${\displaystyle \sigma _{V,W}\colon \mathbb {P} (V)\times \mathbb {P} (W)\to \mathbb {P} (V\otimes W),([v],[w])\mapsto [v\otimes w]}$.

properties

The Segre embedding is a well-defined injective mapping, the image of which is a closed, irreducible subset. ${\displaystyle \sigma _{n,m}}$ ${\displaystyle \Sigma _{n,m}\subseteq \mathbb {P} ^{(n+1)(m+1)-1}}$

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 ${\displaystyle \Sigma _{n,m}}$ ${\displaystyle \mathbb {I} (\Sigma _{n,m})}$ ${\displaystyle \mathbb {P} ^{(n+1)(m+1)-1}}$ ${\displaystyle Z_{0,0},\dotsc ,Z_{n,m}}$

${\displaystyle \mathbb {I} (\Sigma _{n,m})=\langle Z_{a,b}Z_{c,d}-Z_{a,d}Z_{c,b}\mid 0\leq a,c\leq n,0\leq b,d\leq m\rangle }$.

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 . ${\displaystyle 2\times 2}$ ${\displaystyle (Z_{ij})_{ij}}$

Products in the category of (quasi-) projective varieties

If , are (locally) closed subsets, then it is also (locally) closed. ${\displaystyle X\subseteq \mathbb {P} ^{n}}$ ${\displaystyle Y\subseteq \mathbb {P} ^{m}}$ ${\displaystyle \sigma _{n,m}(X\times Y)\subseteq \mathbb {P} ^{(n+1)(m+1)-1}}$

Since is bijective, the structure of a (quasi-) projective variety can be defined by transferring the structure with the help of bijection . ${\displaystyle \sigma _{n,m}\colon X\times Y\to \sigma _{n,m}(X\times Y)}$ ${\displaystyle X\times Y}$ ${\displaystyle \sigma _{n,m}}$

The (quasi) projective variety thus defined is a product in the sense of category theory . ${\displaystyle X\times Y}$

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 ${\displaystyle n=m=1}$ ${\displaystyle \mathbb {P} ^{3}}$ ${\displaystyle \Sigma _{1,1}}$ ${\displaystyle \mathbb {P} ^{3}}$ ${\displaystyle Z_{0},\dotsc ,Z_{3}}$

${\displaystyle \det \left({\begin{matrix}Z_{0}&Z_{1}\\Z_{2}&Z_{3}\end{matrix}}\right)=Z_{0}Z_{3}-Z_{1}Z_{2}.}$

Individual evidence

1. ^ Harris: Algebraic Geometry. 1992, Example 2.11.
2. ↑ Fiesler, Kaup: Algebraic Geometry. 2005, p. 49.
3. ^ Harris: Algebraic Geometry. 1992, Example 2.21.
4. ^ Hartshorne: Algebraic Geometry. 1977, Exercise 3.16.
5. ↑ Fiesler, Kaup: Algebraic Geometry. 2005, exercise 4.7.
6. ^ Harris: Algebraic Geometry. 1992, Example 2.11.

literature

• Joe Harris : Algebraic Geometry. A first course. Springer, New Your 1992, ISBN 3-540-97716-3 .
• Karl-Heinz Fiesler, Ludger Kaup: Algebraic Geometry. Heldermann Verlag, Lemgo 2005, ISBN 3-88538-113-3 .
• Robin Hartshorne : Algebraic Geometry. Springer, New York 1977, ISBN 978-1-4419-2807-8 , Exercises 2.10., 3.16.