The valuative tree of a two-dimensional regular local ring (Q2372669)

Let (\(R,M)\) be a noetherian local integral domain. The valuation space \(\mathcal{V}\) attached to \(R\) consists of the functions \( v:R\rightarrow [ 0,\infty ]\) satisfying the standard axioms of valuations and normalized by \(\min\{v(M)\}=1\). \textit{C. Favre} and \textit{M. Jonsson} [``The Valuative Tree'', Lect. Notes Math. 1853 (2004; Zbl 1064.14024)] showed that \((\mathcal{V},\leq )\) has the structure of a parameterized, rooted, non-metric tree when \(R=\mathbb{C}[[X,Y]]\) and \( \mu \leq \upsilon \Longleftrightarrow \mu (f)\leq \upsilon (f)\) for all \( f\in R\). The paper under review extends this result to any two-dimensional regular local ring \(R\).