"Let $f:X\rightarrow Y$ be a proper surjective morphism from a projective variety $X$ to a projective space $Y$ of the same dimension and $A_f=\{y|$ card$f^{-1}\neq [\mathbb{C}(X):\mathbb{C}(Y)]\}$. We let $b(f)$ (the branch locus of $f$) be the union of the irreducible components $X$ of the Zariski-closure of $A_f$ such that cod $X=1$."
I'm a little confused with that part of the text. I understand that the text meant the following:
$b(f)= (\bar{A_f})_1\cup...\cup(\bar{A_f})_r=\bar{A_f}$; $(\bar{A_f})_i$ is irreducible subvarieties and cod$(\bar{A_f})_i=1$.
I'm confused by the use of the letter $ X $ in the definition of $ b(f)$, it seems to me a misunderstanding. Am I right?
Yes, the $X$ in the definition of $b(f)$ should be a $Y$. The branch locus should be on the target of the morphism.