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.

The Segre map may be defined as the map

taking a pair of points ${\displaystyle ([X],[Y])\in P^{n}\times P^{m}}$ to their product

(the X_iY_j are taken in lexicographical order).

Here, ${\displaystyle P^{n}}$ and ${\displaystyle P^{m}}$ 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 ${\displaystyle \Sigma _{n,m}}$.

In the language of linear algebra, for given vector spaces U and V over the same field K, there is a natural way to linearly map their Cartesian product to their tensor product.

In general, this need not be injective because, for ${\displaystyle u\in U}$, ${\displaystyle v\in V}$ and any nonzero ${\displaystyle c\in K}$,