Summary: Joins and secant varieties
Chris Peterson and Hirotachi Abo
Note. This script is also available at:
There you can find problem sets and Macaulay2 scripts as well.
1. The join of two varieties
Let k be an algebraically closed field, let Pn
be the n-dimensional projective
space over k, and let V and W be two disjoint irreducible projective varieties
. We denote by J(V, W) the union of the lines in Pn
joining V to
W. Then J(V, W) is a projective variety (see pages 69 and 70 in Algebraic
Geometry by J. Harris). This variety is called the join of V and W.
Let A = [a0 : · · · : an] and B = [b0 : · · · : bn] be points of V and W
respectively. Then any point R = [z0 : · · · : zn] of the line passing through P
and Q is given by