Secant variety

In algebraic geometry, the secant variety formula_1, or the variety of chords, of a projective variety formula_2 is the Zariski closure of the union of all secant lines (chords) to "V" in formula_3:
(for formula_5, the line formula_6 is the tangent line.) It is also the image under the projection formula_7 of the closure "Z" of the incidence variety
Note that "Z" has dimension formula_9 and so formula_1 has dimension at most formula_9.
More generally, the formula_12 secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on formula_13. It may be denoted by formula_14. The above secant variety is the first secant variety. Unless formula_15, it is always singular along formula_16, but may have other singular points.
If formula_13 has dimension "d", the dimension of formula_14 is at most formula_19.
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 formula_20 as follows. Let formula_21 be a smooth curve. Since the dimension of the secant variety "S" to "C" has dimension at most 3, if formula_22, then there is a point "p" on formula_3 that is not on "S" and so we have the projection formula_24 from "p" to a hyperplane "H", which gives the embedding formula_25. Now repeat.
If formula_26 is a surface that does not lie in a hyperplane and if formula_27, then "S" is a Veronese surface.