Theorem of the cube

The theorem states that for any complete varieties ${\displaystyle U}$, ${\displaystyle V}$ and ${\displaystyle W}$ over an algebraically closed field, and given points ${\displaystyle u}$, ${\displaystyle v}$ and ${\displaystyle w}$ on them, any invertible sheaf ${\displaystyle L}$ which has a trivial restriction to each of ${\displaystyle U\times V\times \{w\}}$, ${\displaystyle U\times \{v\}\times W}$, and ${\displaystyle \{u\}\times V\times W}$ is itself trivial.^[4]

The theorem of the square^[1][6] is a corollary (also due to Weil) applying to an abelian variety ${\displaystyle A}$. One version of it states that the function ${\displaystyle \phi _{L}}$ taking ${\displaystyle x\in A}$ to ${\displaystyle T_{x}^{*}L\otimes L^{-1}}$ is a group homomorphism from ${\displaystyle A}$ to ${\displaystyle \operatorname {Pic} (A)}$ (where ${\displaystyle T_{x}^{*}}$ is translation by ${\displaystyle x}$ on line bundles).