In algebraic geometry, the secant variety Sect ( V ) {\displaystyle \operatorname {Sect} (V)} , or the variety of chords, of a projective variety V ⊂ P r {\displaystyle V\subset \mathbb {P} ^{r}} is the Zariski closure of the union of all secant lines (chords) to V in P r {\displaystyle \mathbb {P} ^{r}} :
Sect ( V ) = ⋃ x , y ∈ V x y ¯ {\displaystyle \operatorname {Sect} (V)=\bigcup _{x,y\in V}{\overline {xy}}}
(for x = y {\displaystyle x=y} , the line x y ¯ {\displaystyle {\overline {xy}}} is the tangent line.) It is also the image under the projection p 3 : ( P r ) 3 → P r {\displaystyle p_{3}:(\mathbb {P} ^{r})^{3}\to \mathbb {P} ^{r}} of the closure Z of the incidence variety
{ ( x , y , l ) | x ∧ y ∧ l = 0 } {\displaystyle \{(x,y,l)|x\wedge y\wedge l=0\}} . Note that Z has dimension 2 dim V + 1 {\displaystyle 2\dim V+1} and so Sect ( V ) {\displaystyle \operatorname {Sect} (V)} has dimension at most 2 dim V + 1 {\displaystyle 2\dim V+1} . More generally, the k t h {\displaystyle k^{th}} secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on V {\displaystyle V} . It may be denoted by Σ k {\displaystyle \Sigma _{k}} . The above secant variety is the first secant variety. Unless Σ k = P r {\displaystyle \Sigma _{k}=\mathbb {P} ^{r}} , it is always singular along Σ k − 1 {\displaystyle \Sigma _{k-1}} , but may have other singular points. If V {\displaystyle V} has dimension d, the dimension of Σ k {\displaystyle \Sigma _{k}} is at most k d + d + k {\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 P 3 {\displaystyle \mathbb {P} ^{3}} as follows. Let C ⊂ P r {\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 r > 3 {\displaystyle r>3} , then there is a point p on P r {\displaystyle \mathbb {P} ^{r}} that is not on S and so we have the projection π p {\displaystyle \pi _{p}} from p to a hyperplane H, which gives the embedding π p : C ↪ H ≃ P r − 1 {\displaystyle \pi _{p}:C\hookrightarrow H\simeq \mathbb {P} ^{r-1}} . Now repeat. If S ⊂ P 5 {\displaystyle S\subset \mathbb {P} ^{5}} is a surface that does not lie in a hyperplane and if Sect ( S ) ≠ P 5 {\displaystyle \operatorname {Sect} (S)\neq \mathbb {P} ^{5}} , then S is a Veronese surface.
Notes
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
