Segre imbedding
The imbedding $ \phi : P ^ {n} \times P ^ {m} \rightarrow P ^ {N} $
of the product $ P ^ {n} \times P ^ {m} $
of projective spaces into the projective space $ P ^ {N} $,
where $ N = nm + n + m $.
If $ x = ( u _ {0} : \dots : u _ {n} ) \in P ^ {n} $,
$ y = ( v _ {0} : \dots : v _ {m} ) \in P ^ {m} $,
and $ w _ {i,j} $(
$ i = 0 \dots n $;
$ j = 0 \dots m $)
are homogeneous coordinates in $ P ^ {N} $,
then the mapping is defined by the formula:
$$ \phi ( x , y) = ( w _ {i,j} ) \in P ^ {N} , $$
where $ w _ {i,j} = u _ {i} v _ {j} $. The mapping $ \phi $ is well-defined and is a closed imbedding. The image $ \phi ( P ^ {n} \times P ^ {m} ) $ of a Segre imbedding is called a Segre variety. The case when $ n = m = 1 $ has a simple geometrical meaning: $ \phi ( P ^ {1} \times P ^ {1} ) $ is the non-singular quadric in $ P ^ {3} $ with equation $ w _ {11} w _ {00} = w _ {01} w _ {10} $. The images $ \phi ( x \times P ^ {1} ) $ and $ \phi ( P ^ {1} \times y) $ give two families of generating lines of the quadric.
The terminology is in honour of B. Segre.
References
[1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001 |
Segre imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_imbedding&oldid=23973