Secant variety

In algebraic geometry, the secant variety $\operatorname {Sect} (V)$, or the variety of chords, of a projective variety $V\subset \mathbb {P} ^{r}$ is the Zariski closure of the union of all secant lines (chords) to V in $\mathbb {P} ^{r}$:

$\operatorname {Sect} (V)=\bigcup _{x,y\in V}{\overline {xy}}$

(for $x=y$, the line ${\overline {xy}}$ is the tangent line.) It is also the image under the projection $p_{3}:(\mathbb {P} ^{r})^{3}\to \mathbb {P} ^{r}$ of the closure Z of the incidence variety

$\{(x,y,r)|x\wedge y\wedge r=0\}$.

Note that Z has dimension $2\dim V+1$ and so $\operatorname {Sect} (V)$ has dimension at most $2\dim V+1$.

More generally, the $k^{th}$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $V$. It may be denoted by $\Sigma _{k}$. The above secant variety is the first secant variety. Unless $\Sigma _{k}=\mathbb {P} ^{r}$, it is always singular along $\Sigma _{k-1}$, but may have other singular points.

If $V$ has dimension d, the dimension of $\Sigma _{k}$ is at most $kd+d+k$. A useful tool for computing the dimension of a secant variety is Terracini's lemma.

Examples

A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space $\mathbb {P} ^{3}$ as follows. Let $C\subset \mathbb {P} ^{r}$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if $r>3$, then there is a point p on $\mathbb {P} ^{r}$ that is not on S and so we have the projection $\pi _{p}$ from p to a hyperplane H, which gives the embedding $\pi _{p}:C\hookrightarrow H\simeq \mathbb {P} ^{r-1}$. Now repeat.

If $S\subset \mathbb {P} ^{5}$ is a surface that does not lie in a hyperplane and if $\operatorname {Sect} (S)\neq \mathbb {P} ^{5}$, then S is a Veronese surface.

References

• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
• Griffiths, P.; Harris, J. (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 617. ISBN 0-471-05059-8.
• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3