 Mumford rigidity lemma

A local form of the Mumford rigidity lemma says: if $X$ is a complete integral variety over an algebraically closed field, and a morphism $f:X\times Y\to Z$ sends one fibre $X\times \{y_0\}$ to a point, then on $X\times U$ for some open neighbourhood $U$ of $y_0$, it factors through the projection to $U$. To see the mechanism, choose an affine neighbourhood of that image point. Properness of the projection supplies $U$ whose whole product maps into that neighbourhood. Regular functions on the complete integral $X$ are constant, forcing the factorization there. Pointed global versions add further hypotheses; the local conclusion alone should not be silently replaced by a global one.^[ref]