Segre embedding について

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Segre mapping ：ウィキペディア英語版

Segre embedding
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
==Definition==
The Segre map may be defined as the map
:

\sigma: P^n \times P^m \to P^\

taking a pair of points

)) \in P^n \times P^m

to their product
:

\sigma:((), ()) \mapsto   (X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m)\

(the ''X_iY_j'' are taken in lexicographical order).
Here,

P^n

and

P^m

are projective vector spaces over some arbitrary field, and the notation
:

()\

is that of homogeneous coordinates on the space. The image of the map is a variety, called a Segre variety. It is sometimes written as

\Sigma_

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