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  "body": "\n## Statement\n\nLet $k$ be an algebraically closed field.\nLet $X$ and $Y$ be complete [[varieties|HV6YBS0YDH]] over $k$, and let $Z$ be a variety over $k$.\nLet $x_0 \\in X$, $y_0 \\in Y$ and $z_0 \\in Z$ be closed points.\nLet $L$ be a [[line bundle|7JCF2TE3V5]] on $X \\times Y \\times Z$.\nAssume that the restrictions of $L$ to $\\set{x_0} \\times Y \\times Z$, to $X \\times \\set{y_0} \\times Z$ and to $X \\times Y \\times \\set{z_0}$ are trivial.\nThen $L$ is trivial.\n\n## Remarks\n\nThe standard proof uses the seesaw principle, which detects triviality of a line bundle on a product from its restrictions to fibres.\nIt combines this with a reduction to the case where $Z$ is a curve.\n\nApplied to an [[abelian variety|R7W03TXRS9]] $A$, the theorem gives a formula on $A \\times A \\times A$.\nLet $p_i$ be the projections and $p_{ij} = p_i + p_j$, $p_{123} = p_1 + p_2 + p_3$ the sum maps to $A$.\nFor every line bundle $L$ on $A$, the line bundle\n$$p_{123}^{*} L \\otimes p_{12}^{*} L^{-1} \\otimes p_{13}^{*} L^{-1} \\otimes p_{23}^{*} L^{-1} \\otimes p_1^{*} L \\otimes p_2^{*} L \\otimes p_3^{*} L$$\nis trivial.\nLet $t_x$ be translation by a point $x$ of $A$.\nPulling back along maps into $A \\times A \\times A$ gives the theorem of the square: $t_{x+y}^{*} L \\otimes L \\cong t_x^{*} L \\otimes t_y^{*} L$ for all points $x, y$.\nIt also gives $[n]^{*} L \\cong L^{(n^2 + n)/2} \\otimes [-1]^{*} L^{(n^2 - n)/2}$ for multiplication by $n$.\nSo for a symmetric line bundle, with $[-1]^{*} L \\cong L$, one gets $[n]^{*} L \\cong L^{n^2}$.\n\n## Explanations\n\n### L1 human\n\nA line bundle on a product of three spaces is trivial as soon as it is trivial on three coordinate \"faces\" through a common corner.\nThe name comes from this picture of a cube with three faces through one vertex.\nFor abelian varieties this turns into identities for line bundles under addition of points.\n\n### L2 human\n\nThe theorem of the cube is a basic tool in the algebraic theory of abelian varieties.\nThe theorem of the square that follows from it gives a homomorphism $A \\to \\operatorname{Pic}(A)$, $x \\mapsto t_x^{*} L \\otimes L^{-1}$, which lands in the points of the [[dual abelian variety|HZXSESHC8P]].\nFor an ample $L$ this map is an isogeny, the polarization attached to $L$.\nThe formula $[n]^{*} L \\cong L^{n^2}$ for symmetric $L$ is a key step in computing the degree of multiplication by $n$, and in the theory of heights.\n",
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