Let $\pi: X \to C$ be a fibration (proper, flat with generically smooth irreducible equidimensional fibers) over smooth irreducible $k$-scheme $C$, $k$ field, eg $C$ curve. Let $L$ be a line bundle on $X$ which is assumed to be trivial over generic fiber $X_{\eta}$.
What are the mildest additional assumpions on $X$ and $C$ should be required to guarantee the existence an open $U \subset C$ such that $L$ trivializes over it's preimage $\pi^{-1}(U)$?
A remark on the smoothness / geom. regular assumptions of the base $C$ respectively the fibers: is this assumption really neccessary or does it suffice to require for $C$ to be regular without worsening the situation?
Best Answer
By restricting to an open set in $ C $, we can and will assume that all fibers are smooth and irreducible. The locus of points $ \{ p \in C; L|_{X_p} \cong \mathcal{O}_{X_p} \} $ is closed by semicontinuity - it is given by the intersection of two closed sets $$ \{ p \in C; \dim H^0( X_p, L|_{X_p} ) > 0 \} \cap \{ p \in C; \dim H^0( X_p, L^{\vee}|_{X_p} ) > 0 \} $$ and by assumption the locus contains the generic point, so is all of $ C $.
Because $ L $ is trivial on every fiber, $ \pi_* L $ is a line bundle by cohomology and base change. Let $ U \subset C $ be an open set where $ \pi_* L $ is trivial. Then $ L $ is trivial on $ \pi^{-1} (U) $. To see this, it’s enough to see that $ \pi^* \pi_* L $ is trivial there (being isomorphic to $ L $) and that’s obvious from construction.