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Let $f\colon X \to Y$ be a dominant morphism between normal projective algebraic varieties; assume $f_*\mathcal {O}_X=\mathcal{O}_Y $; let’s also assume that it is defined over the complex numbers, but I do not think this is relevant for the question. I do NOT assume that $f$ is flat.

Let $L$ be a line bundle on $X$ which is trivial along all fibres of $f$. Is $L$ the pull-back of a line bundle from $Y$?

(I would expect that $f_*L$ is a line bundle, and the natural map $$ f^*f_*L\to L $$ is an isomorphism. When $f$ is flat, this is the seesaw principle.)

I am very interested in the case where $f$ is a birational map.

Thanks!

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    $\begingroup$ You should assume that $Y$ is normal and that $f$ is dominant with connected fibers, otherwise there are easy counterexamples. $\endgroup$ Nov 23 '18 at 20:56
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    $\begingroup$ It seems likely that there should be counterexamples even with $X$ a surface and $f$ the contraction of a curve $C$: it suffices to find a line bundle on $X$ which is trivial on $C$ but not on some infinitesimal neighbourhood. (I have not worked out an explicit example...) $\endgroup$
    – naf
    Nov 24 '18 at 9:20
  • $\begingroup$ @ulrich Sounds convincing. Should the answer be positive if we assume $R^1 f_* \mathcal{O}_X = 0$? In this case if $L$ is trivial on the fiber, it should be trivial on all of its infinitesimal neighborhoods. $\endgroup$ Nov 24 '18 at 10:09
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    $\begingroup$ @PiotrAchinger: Indeed: working over $\mathbb{C}$ in the analytic topology, this follows by using the exponential sequence (and the Leray spectral sequence) to compute the Picard group. $\endgroup$
    – naf
    Nov 24 '18 at 12:24
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I have received the following great answer from Vlad Lazić:

In the numerical setup, what you ask (with some assumptions on the base, such as Q-factoriality) is well-understood, see the paper of Lehmann "Numerical triviality and pullbacks" and Nakayama's book on Zariski decomposition. I suppose this should give you the answer under additional hypothesis, such as the vanishing of the first direct image of the structure sheaf. I doubt you can get away without the assumptions on pseudoeffectivity of L: an example should be something like E\times E (for a general elliptic curve) and L=diagonal minus a fibre of one projection, and the map is the second projection (I didn't check the details, although you probably have to take something like E\times P^1 or something similar if you're only interested in linear equivalence). You might also find interesting Lemma 3.1 in my paper with Peternell, "On Generalised Abundance, I".

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