Secant variety


In algebraic geometry, the secant variety, or the variety of chords, of a projective variety is the Zariski closure of the union of all secant lines to V in :
It is also the image under the projection of the closure Z of the incidence variety
Note that Z has dimension and so has dimension at most.
More generally, the secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on. It may be denoted by. The above secant variety is the first secant variety. Unless, it is always singular along, but may have other singular points.
If has dimension d, the dimension of is at most.
A useful tool for computing the dimension of a secant variety is Terracini's lemma.

Examples

A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space as follows. Let be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if, then there is a point p on that is not on S and so we have the projection from p to a hyperplane H, which gives the embedding. Now repeat.
If is a surface that does not lie in a hyperplane and if, then S is a Veronese surface.