I would like to have a clear picture of the notion of pull-back (and push-forward) of a divisor (or a line bundle), under 3 different scenarios: morphism, birational morphism and birational map.
I will always work with normal projective varieties over the field of complex numbers.
The first case is the simplest: let $f:X\to Y$ be a morphism, and consider a line bundle $L$ on $Y$: then we can define a line bundle $f^\star L$, also called the pull-back bundle, on $X$, obtained as a fiber product.
The other two cases are those which I don't fully understand:
Suppose now $f:X\to Y$ is a birational morphism, that is $f$ is an isomorphism on an open subset $U$ of $X$. Take moreover $L$ line bundle on $Y$. Since $f$ is a morphism, I guess can still pull-back the line bundle $L$ via the fiber product, obtaining a line bundle $f^\star L$ on $X$.
If $f:X\dashrightarrow Y$ is a birational map, I think the pullback map $f^\star$ is actually $f^{-1}_{\star}$. I define the push-foward as follows: I choose a divisor $D$ on $X$, and I locally map it either to $0$ (if the image has codimension at least 2), either to a divisor on $Y$. Therefore by using $f^{-1}_{\star}$ I still can obtain a map at the level of divisors from $Y$ to $X$, that is a pull-back.
Is the above reasoning correct? Is there a reference for these topics which describes these differences?
Best Answer
I'll stick to the pullbacks and line bundle sides of things, since this is what you seem to be concentrating on.
Your first two bits are correct. If $f:X\to Y$ is a map of schemes, there is always a homomorphism $\renewcommand{\Pic}{\operatorname{Pic}}\renewcommand{\ClGrp}{\operatorname{Cl}} f^*:\Pic Y \to \Pic X$ given by sending $L\in\Pic Y$ to $f^*L$, a line bundle on $X$. (Though I don't see why you appeal to the fiber product to define $f^*$: the definition is just $f^*(-)=\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}(-)$.)
When $f$ is only a rational map, things may break in interesting ways. If you want a map on Picard groups, then what you might think of doing is pulling back a line bundle $L$ on $Y$ to $U\subset X$ which is a domain of definition for the rational map $f$ and extending this to a line bundle on $X$. But there are obstacles: this line bundle may not extend uniquely (take $X=\Bbb P^1$, $Y=\Bbb A^1$, and $f$ the identity on $U_0$: then $f^*\mathcal{O}_Y$ extends to any line bundle on $X$), or the unique extension of $f^*L$ may not be a line bundle.
If you want to salvage a map of Picard groups here, you need to guarantee that any line bundle on the maximal domain of definition of $f$ has a unique extension. The best general result in this area is that a vector bundle on an open subscheme $U\subset X$ extends uniquely to a vector bundle on $X$ when $X$ is $S_2$ and $\operatorname{codim}(X\setminus U)\geq 2$ (ref). In particular, since you always deal with projective normal varieties over $\Bbb C$, your varieties are $S_2$ by Serre's criterion for normality and the indeterminacy locus of your morphism is at least codimenstion two by Hartshorne lemma V.5.1, for instance. So everything's okay.