Secant variety
In algebraic geometry, the secant variety , or the variety of chords, of a projective variety is the Zariski closure of the union of all secant lines (chords) to V in :[1]
(for , the line is the tangent line.) It is also the image under the projection of the closure Z of the incidence variety
- .
Note that Z has dimension and so has dimension at most .
More generally, the secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on . It may be denoted by . The above secant variety is the first secant variety. Unless , it is always singular along , but may have other singular points.
If has dimension d, the dimension of is at most . 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 as follows.[2] Let be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if , then there is a point p on that is not on S and so we have the projection from p to a hyperplane H, which gives the embedding . Now repeat.
If is a surface that does not lie in a hyperplane and if , then S is a Veronese surface.[3]
References
- Griffiths–Harris, pg. 173
- Griffiths–Harris, pg. 215
- Griffiths–Harris, pg. 179
- Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
- P. Griffiths; J. Harris (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