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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.

References

Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

Mathematics Encyclopedia