Seesaw theorem
In algebraic geometry, the seesaw theorem, or seesaw principle, says roughly that a limit of trivial line bundles over complete varieties is a trivial line bundle. It was introduced by André Weil in a course at the University of Chicago in 1954–1955, and is related to Severi's theory of correspondences.
The seesaw theorem is proved using proper base change. It can be used to prove the theorem of the cube.
Statement
{{harvtxt|Lang|1959|loc=p.241}} originally stated the seesaw principle in terms of divisors. It is now more common to state it in terms of line bundles as follows {{harv|Mumford|2008|loc=Corollary 6, section 5}}.
Suppose L is a line bundle over X×T, where X is a complete variety and T is an algebraic set. Then the set of points t of T such that L is trivial on X×t is closed. Moreover if this set is the whole of T then L is the pullback of a line bundle on T. {{harvtxt|Mumford|2008|loc=section 10}} also gave a more precise version, showing that there is a largest closed subscheme of T such that L is the pullback of a line bundle on the subscheme.
References
- {{citation|mr=0106225
|last=Lang|first= Serge
|title=Abelian varieties
|series=Interscience Tracts in Pure and Applied Mathematics|volume= 7 |publisher=Interscience Publishers, Inc.|place= New York|year=1959}}
- {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | orig-date=1970 | publisher=American Mathematical Society | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 | year=2008 | volume=5 | mr=0282985}}