Segre map

The Segre map is an embedding s:n×mnm+n+m of the product of two projective spacesMathworldPlanetmath 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


In homogeneous coordinates, the pair of points [x0:x1::xn], [y0:y1::ym] maps to


If we imagine the target space as the projectivized version of the space of (n+1)×(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


for all 0i<kn, 0j<lm. VarietiesPlanetmathPlanetmath 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