Support of principal divisor and exceptional locus

Question

Assume we have a birational morphism between smooth varieties $\phi : X \longrightarrow Y$. Let $f$ be a non-zero rational function of $X$, and $E$ be the exceptional locus of $\phi$. I think we should have $\text{supp}(\text{div(f)}) \not\subset E$, but I don't find any argument. Otherwise, maybe we should look at the push-forward, and then we could deduce that $\text{Nm}_{K(X)/K(Y)}(f) \in K^{\times}$ where $K$ is the field of definition of our varieties. Then, I'm tempted to say that we should have $f \in K^\times$. What do you think ?

Answer 1

If the support of the divisor associated to $f$ on $X$ is contained in the exceptional locus, that means the support of the divisor associated to $f$ on $Y$ is contained inside a subset of codimension at least two. But this implies that the divisor associated to $f$ on $Y$ is trivial: the vanishing locus of a function on a normal variety is pure codimension one.

$\operatorname{div} f=0$ does not quite force $f\in K^\times$ without more assumptions: $x/(x-1)$ is a function with trivial divisor on $\Bbb A^1\setminus \{0,1\}$. You'll need at least completeness to ensure that, plus maybe some algebraically closedness hypotheses on the base field.