Secant variety

In algebraic geometry, the Zariski closure of the union of the secant lines to a projective variety X ⊂ P n is the first secant variety to X . It is usually denoted Σ 1 .

The k t h secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on X . It is usually denoted Σ k . Unless Σ k = P n , it is always singular along Σ k − 1 , but may have other singular points.

If X has dimension d, the dimension of Σ k is at most kd+d+k.