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 (chords) to V in : (for x = y, the line is the tangent line.) 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 k^{th} secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on V. It may be denoted by \Sigma_k. The above secant variety is the first secant variety. Unless, it is always singular along , but may have other singular points. If V has dimension d, the dimension of \Sigma_k is at most kd+d+k. 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 r > 3, then there is a point p on that is not on S and so we have the projection \pi_p 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.