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!



Non-flat seesaw
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".


