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theorem of the cube is a theorem in algebraic geometry which says an invertible sheaf(line bundle ) on the product of three complete varieties is trivial iff its restriction on the product of each two of them is trivial.I understand this theorem but I don't understand why it is important.can you give some application of it in geometry or algebraic geometry

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A classical application is the study of the map $$\varphi_{\mathcal L} \colon A \longrightarrow \mathrm{Pic}^0(A),\quad x \mapsto t_x^* \mathcal L \otimes \mathcal L^{-1},$$ from an abelian variety $A$ to its dual abelian variety, associated to a line bundle $\mathcal{L}$ on $A$.

See David Zureick-Brown's answer to this MathOverflow question and the references given therein.

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