= Mumford rigidity lemma
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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.<Abelian Varieties by J. S. Milne (course notes)>[rigidity lemma]{ref}
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