Secant variety

In algebraic geometry, the secant variety ${\displaystyle \operatorname {Sect} (V)}$, or the variety of chords, of a projective variety ${\displaystyle V\subset \mathbb {P} ^{r}}$ is the Zariski closure of the union of all secant lines (chords) to V in ${\displaystyle \mathbb {P} ^{r}}$:^[1]

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

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

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

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

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

If ${\displaystyle V}$ has dimension d, the dimension of ${\displaystyle \Sigma _{k}}$ is at most ${\displaystyle kd+d+k}$. 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 ${\displaystyle \mathbb {P} ^{3}}$ as follows.^[2] Let ${\displaystyle C\subset \mathbb {P} ^{r}}$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if ${\displaystyle r>3}$, then there is a point p on ${\displaystyle \mathbb {P} ^{r}}$ that is not on S and so we have the projection ${\displaystyle \pi _{p}}$ from p to a hyperplane H, which gives the embedding ${\displaystyle \pi _{p}:C\hookrightarrow H\simeq \mathbb {P} ^{r-1}}$. Now repeat.

If ${\displaystyle S\subset \mathbb {P} ^{5}}$ is a surface that does not lie in a hyperplane and if ${\displaystyle \operatorname {Sect} (S)\neq \mathbb {P} ^{5}}$, then S is a Veronese surface.^[3]

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