For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It seems that codimension $1$ irreducible subschemes of these models yield valuations of $F$. My question is: which valuations are obtained this way? Would it be appropriate to call them 'geometric'; is there any canonical text that treats this question (an introduces some terms of sort)? The problem is that I do not want to restrict myself to the case when all of these subschemes are defined over a fixed subfield of $F$.



Which valuations of a field yield codimension $1$ subschemes of their 'models'
Suppose you are given a ~~proper~~ noetherian integral scheme $X$ whose field of rational functions is $F$. Let $A$ be a discrete valuation of $F$ with residue field $k\_A$ and suppose it has a center $x\in X$. Then the existence of a model $Y$ (of finite type over $X$) of $F$ such that $A$ is induced by a codimension $1$ subscheme of $Y$ is equivalent to the condition 
$$\dim O\_{X,x} -1 =\mathrm{trdeg}\_{k(x)}k\_A$$ 
(transcendence degree), under the assumption that *$X$ is excellent*. The statement is apparently due to Zariski. You can find a proof in Artin: "Néron models", §5, in Arithmetic Geometry (ed. Cornell and Silverman). See also Liu : "Algebraic geometry and arithmetic curves", Theorem 8.3.26 (where the condition 
$X$ is excellent is explicitly used).


