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 with 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).

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).