Secant variety

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

[math]\displaystyle{ \operatorname{Sect}(V) = \bigcup_{x, y \in V} \overline{xy} }[/math]

(for [math]\displaystyle{ x = y }[/math], the line [math]\displaystyle{ \overline{xy} }[/math] is the tangent line.) It is also the image under the projection [math]\displaystyle{ p_3: (\mathbb{P}^r)^3 \to \mathbb{P}^r }[/math] of the closure Z of the incidence variety

[math]\displaystyle{ \{ (x, y, r) | x \wedge y \wedge r = 0 \} }[/math].

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

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

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

If [math]\displaystyle{ S \subset \mathbb{P}^5 }[/math] is a surface that does not lie in a hyperplane and if [math]\displaystyle{ \operatorname{Sect}(S) \ne \mathbb{P}^5 }[/math], 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