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 of two projective spaces into a larger projective space. It is important since it makes the product of two projective varieties into a projective variety again. Invariantly, it can described as follows. Let $V, W$ be (finite dimensional) vector spaces; then

$\begin{array}[]{rrclcc}s:&\mathbb{P}V&\times&\mathbb{P}W&\longrightarrow&% \mathbb{P}(V\otimes W)\\ &[x]&,&[y]&\longmapsto&[x\otimes y]\end{array}$

In homogeneous coordinates, the pair of points $[x_{0}:x_{1}:\cdots:x_{n}]$ , $[y_{0}:y_{1}:\cdots:y_{m}]$ maps to

$[x_{0}y_{0}:x_{1}y_{0}:\cdots:x_{n}y_{0}:x_{0}y_{1}:x_{1}y_{1}:\cdots:x_{n}y_{% m}].$

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

$\left|\begin{array}[]{cc}a_{ij}&a_{il}\\ a_{kj}&a_{kl}\end{array}\right|=a_{ij}a_{kl}-a_{il}a_{kj}=0$

for all $0\leq i<k\leq n$ , $0\leq j<l\leq m$ . Varieties of this form (defined by vanishing of minors in some space of matrices) are usually called determinantal varieties.

Title	Segre map
Canonical name	SegreMap
Date of creation	2013-03-22 14:24:45
Last modified on	2013-03-22 14:24:45
Owner	halu (5781)
Last modified by	halu (5781)
Numerical id	4
Author	halu (5781)
Entry type	Definition
Classification	msc 14A25
Classification	msc 14M12
Synonym	Segre embedding