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.

References

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