Ruled join

In algebraic geometry, given irreducible subvarieties V, W of a projective space Pⁿ, the ruled join of V and W is the union of all lines from V to W in P²ⁿ⁺¹, where V, W are embedded into P²ⁿ⁺¹ so that the last (resp. first) n + 1 coordinates on V (resp. W) vanish.^[1] It is denoted by J(V, W). For example, if V and W are linear subspaces, then their join is the linear span of them, the smallest linear subcontaining them.

The join of several subvarieties is defined in a similar way.

See also

• Secant variety

References

1. Fulton 1998, Example 8.4.5.

• Dickenstein, Alicia; Schreyer, Frank-Olaf; Sommese, Andrew J. (2010-07-10). Algorithms in Algebraic Geometry. Springer Science & Business Media. ISBN 9780387751559.
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Flenner, H.; O'Carroll, L.; Vogel, W. (29 June 2013). Joins and Intersections. ISBN 9783662038178.
• Russo, Francesco. "Geometry of Special Varieties" (PDF). University of Catania. Retrieved 7 March 2018.