Join (algebraic geometry)

Short description: Concept in mathematics

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. 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

• Dickenstein, Alicia; Schreyer, Frank-Olaf; Sommese, Andrew J. (2010-07-10) (in en). Algorithms in Algebraic Geometry. Springer Science & Business Media. ISBN 9780387751559. https://books.google.com/books?id=1PGyO6uQZCYC&dq=Join+%28algebraic+geometry%29&pg=PA2. 
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4 
• Russo, Francesco. "Geometry of Special Varieties". http://www.dmi.unict.it/~frusso/DMI/Note_di_Corso_files/GeometrySpecialVarieties.pdf.