The Segre map is an embedding 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 be (finite dimensional) vector spaces; then
In homogeneous coordinates, the pair of points , maps to
If we imagine the target space as the projectivized version of the space of 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 , . Varieties of this form (defined by vanishing of minors in some space of matrices) are usually called determinantal varieties.
|Date of creation||2013-03-22 14:24:45|
|Last modified on||2013-03-22 14:24:45|
|Last modified by||halu (5781)|