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

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001