 Seesaw theorem

A useful form of the Seesaw theorem determines a line bundle on a product from its fibre restrictions. Let $X$ be a complete geometrically integral variety with a base point $x_0$, and let $L$ be a line bundle on $X\times Y$ trivial on each geometric fibre $X\times \{y\}$. For a reduced variety $Y$, constancy of the fibrewise section dimension and cohomology and base change imply that $L$ is pulled back from $Y$; restricting to $\{x_0\}\times Y$ identifies the bundle being pulled back. If that transverse restriction is also trivial, then $L$ is trivial. This is useful for verifying normalized universal bundle identities by checking fibres; matching only one fibre in each direction is insufficient.