Theorem of the cube

In mathematics, the theorem of the cube is a foundational result in the algebraic geometry of a complete variety. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The specific result was proved under this name, in the early 1950s, in the course of his fundamental work on abstract algebraic geometry by André Weil; a discussion of the history has been given by Steven Kleiman (lectures "The Picard Scheme", Introduction) . A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given in "Abelian Varieties" (1970) by David Mumford.

The theorem states that for any complete varieties "U", "V" and "W", and given points "u", "v" and "w" on them, any invertible sheaf "L" which has a trivial restriction to each of "U"× "V" × {"w"}, "U"× {"v"} × "W", and {"u"} × "V" × "W", is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete.)

Note: On a ringed space "X", an invertible sheaf "L" is "trivial" if isomorphic to "O"_"X", as "O"_"X"- module. If "L" is taken as a holomorphic line bundle, in the complex manifold case, this is the same here as a trivial bundle, but in a "holomorphic" sense, not just topologically.

The theorem of the square (Mumford p.59) is a corollary applying to an abelian variety "A", defining a group homomorphism from "A" to "Pic"("A"), in terms of the change in "L" by translation on "A".