Given a projective variety $X$, say nonsingular over $\mathbb{C}$, what is the precise relation between codimension one irreducible subvarieties, and discrete valuations on the function field $K(X)$?
I was under the impression that the situation worked in the general case as it does for curves, that we have irreducible divisors on $X$ the same thing as invertible subsheaves of the constant $K(X)$, corresponding to discrete valuations on the function field $K(X)$ by taking the local ring of the generic point of the divisor.
My confusion arises in that this picture doesn't seem consistent with the fact that we can freely add new codimension one divisors by blowing up, and since the blowdown is birational, this doesn't change our fraction field.
Something doesn't seem to add up here, so as a precise question, for an arbitrary (say smooth) $X$, is there a classification of the set of discrete valuations contained in $K(X)$, in terms of the geometry of $X$?