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