Segre map

The Segre map is an embedding $s:{\mathbb{P}}^n\times {\mathbb{P}}^m\to {\mathbb{P}}^{nm+n+m}$ of the product oftwo projective spaces into a larger projective space. It is important since it makes theproduct of two projective varieties into a projective variety again.Invariantly, it can described as follows. Let $V,W$ be (finite dimensional) vector spaces; then

In homogeneous coordinates, the pair of points $[x_0:x_1:\mathrm{\cdots }:x_n]$,$[y_0:y_1:\mathrm{\cdots }:y_m]$ maps to

$$[x_0{y}_0:x_1{y}_0:\mathrm{\cdots }:x_n{y}_0:x_0{y}_1:x_1{y}_1:\mathrm{\cdots }:x_n{y}_m].$$

If we imagine the target space as the projectivized version of the spaceof $(n+1)\times (m+1)$ matrices, then the image is exactly the set of matriceswhich have rank 1; thus it is the common zero locus of the equations

for all , . Varieties of this form (defined by vanishingof minors in some space of matrices) are usually called determinantal varieties.