A moduli problem inspired by Stein factorization

Question

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.?

Answer 1

I don't know a reference but I think what I said in the comment is more or less obvious. Let me prove it though in detail. Again, I can't imagine one can make this into any reasonable moduli space.

Setting: Suppose that $\pi : X \to Y$ is a proper birational map with connected fibers with $X$ nonsingular and $Y$ normal.

Lemma: Suppose that $L$ is a line bundle on $X$ and $\pi_* L \cong O_Y$. Then $L \cong O_X(D)$ for a uniquely determined $D$, an effective $\pi$-exceptional divisor.

Proof: Embed $L \subseteq K(X)$ so that $L = O_X(F)$ for some divisor $F$. Then $\pi_* L \subseteq K(Y)$. Let $M = \pi_* L \subseteq K(Y)$. We know $M \cong O_Y$ is a line bundle and hence $M = O_Y(G)$ for some divisor $G$ (in fact $G = \text{div}_Y(h)$ where $h$ is in $K(Y)$ but that doesn't matter for what follows).

Next consider $D := F - \pi^* G \sim F$ and observe that $O_X(F - \pi^*G) \cong O_X(F) = L$. Also not the $\pi_* O_X(D) = O_Y \subseteq K(Y)$. Since $F$ and $G$ agree where $\pi$ is an isomorphism by construction, we see that $D$ is exceptional. Next observe that if $D$ is non-effective, say $D = \sum a_i D_i$ and $a_1 < 0$, then $1 \notin \Gamma(X, O_X(D))$ (since $1$ does not vanish at $D_1$). But then $\Gamma(X, O_X(D)) = \Gamma(Y, \pi_* O_X(D)) = \Gamma(Y, O_Y)$ and so $1 \notin \Gamma(Y, O_Y)$ which is absurd. This proves existence of an effective $D$ satisfying the property.

For uniqueness, suppose that $O_X(D) \cong O_X(D')$ so that $D - D' = \text{div}_X u$ for some $u \in K(X)$. Also suppose $\pi_* O_X(D) = O_Y = \pi_* O_X(D')$. Then $D - D'$ is $\pi$-exceptional but also the divisor of a rational function in the fraction field. Since $\text{div}_Y(u) = 0$ by construction, we see that $u$ is a unit in $\Gamma(Y, O_Y)$. But then $u$ is also a unit in $\Gamma(X, O_X) = \Gamma(Y, O_Y)$ and hence $\text{div}_X(u) = 0$ as well. Thus $D - D' = 0$ and we have proven uniqueness.