I’d like to use this post to try to understand the “theorem of the cube,” following Mumford’s “Abelian varieties.”

Theorem 1Let be proper varieties over an algebraically closed field , and let be a connected variety. Let be a line bundle. Suppose there exist -valued points in such that is trivial when restricted to . Then is trivial.

This theorem is extremely useful in analyzing the behavior of line bundles on abelian varieties; see these posts for instance.

I’ve found the proof due to Weil and Murre in the second chapter of Mumford to be quite impenetrable; the argument makes sense line by line but I have never been able to see a larger picture. Fortunately, it turns out that the third chapter has a more scheme-theoretic (nilpotents will be used!) argument which is much more transparent. (more…)