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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

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