Secant variety

In algebraic geometry, the Zariski closure of the union of the secant lines to a projective variety  X\subset\mathbb{P}^n is the first secant variety to  X . It is usually denoted \Sigma_1.

The k^{th} 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 \Sigma_k. Unless \Sigma_k=\mathbb{P}^n, it is always singular along \Sigma_{k-1}, but may have other singular points.

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

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