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

Theorem 1 Let ${X, Y}$ be proper varieties over an algebraically closed field ${k}$, and let ${Z}$ be a connected variety. Let ${\mathcal{L} \in \mathrm{Pic}(X \times Y \times Z)}$ be a line bundle. Suppose there exist ${k}$-valued points ${x_0, y_0, z_0}$ in ${X, Y, Z}$ such that ${\mathcal{L}}$ is trivial when restricted to ${\left\{x_0\right\} \times Y \times Z, X \times \left\{y_0\right\} \times Z, X \times Y \times \left\{z_0\right\}}$. Then ${\mathcal{L}}$ 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…)