Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ such that $f_*\mathcal{L}$ is the trivial sheaf $\mathcal{O}_Y$? If so, is there any literature on this topic, for example how can I compute the dimension of such a space, is it non-singular, etc.?

Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ such that $f_*\mathcal{L}$ is isomorphic to the trivial sheaf $\mathcal{O}_Y$? If so, is there any literature on this topic, for example how can I compute the dimension of such a space, is it non-singular, etc.?