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 Andre Weil in a course at the University of Chicago in 1954–1955, and is related to Severi's theory of correspondences.

The seesaw theorem can be used to prove the theorem of the cube.

Statement

Lang (1959, 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 (Mumford 2008, 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. Mumford (2008, 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

Lang, Serge (1959), Abelian varieties, Interscience Tracts in Pure and Applied Mathematics, 7, New York: Interscience Publishers, Inc., MR 0106225
Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

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