Birational proper morphism and global sections

Question

Let $f:X \to Y$ be a proper, surjective, birational morphism of noetherian (connected) projective schemes. Assume that $X$ is non-singular. Let $D$ be a Cartier divisor on $Y$ and $L$ the line bundle $\mathcal{O}_Y(D)$. Is it true that the natural morphism $H^0(Y,L) \to H^0(X,f^*L)$ surjective? (I would imagine this to be true. In particular, take any non-constant global section of $f^*L$, $Z$ be the zero locus of the section. It seems to follows from Fulton that $\mathcal{O}_Y(f(Z)) \cong f_*f^*L \cong L$. But I am not entirely sure of this argument)

Answer 1

This isn't true in the generality you state. Here's a counterexample.

Let $Y \subset \mathbf P^2$ be a nodal cubic curve (over an algebraically closed field), $f : X \rightarrow Y$ the normalisation. Let $L = O(1)_Y$ be the hyperplane bundle on $\mathbf P^2$, restricted to $Y$.

Then $h^0(Y,L)=3$, as one checks with the usual exact sequence, but $f^*L=O_{\mathbf P^1}(3)$ so $h^0(X,f^*L)=4$.

However: if one assumes that $Y$ is normal then your statement is true. The argument is what you sketch in parentheses: if $Y$ is normal and $f$ is birational, then $f_* O_X = O_Y$, so the projection formula gives $f_*f^*L = f_*(f^*L \otimes O_X) = L \otimes f_* O_X = L$. Since $H^0(X,F) \cong H^0(Y, f_* F)$ for any sheaf $F$ on $X$, we are done.

In fact: it's worth noting that the only extra property of $f$ we used here was the condition $f_*O_X=O_Y$. When $Y$ is normal, this is satisfied by any $f$ whose general fibre is connected (in particular, $X$ could have larger dimension than $Y$).