Let $f\colon X\to Y$ be a surjective morphism of smooth projective complex algebraic varieties. Assume $f$ has connected fibres. Let $\eta$ be the generic point of $Y$ and let $X_\eta$ be the generic fibre of $f$. Consider the pullback map $$\imath^*\colon {\rm Pic}(X)\to {\rm Pic}(X_\eta)$$ induced by the morphism $\imath\colon X_\eta\to X$. Questions:

- Under which conditions is $\imath^*$ surjective?
- Is the kernel of $\imath^*$ generated by line bundles corresponding to divisors whose support does not dominate $Y$?