Convergence of Rees valuations of sequences of one-fibered domains (Q2417791)

Let \(\{(R_n, \mathbf{m}_n)\}_{n \geq 0}\) be an infinite sequence of normalized local quadratic transforms of analytically unramified Noetherian local domains whose maximal ideals \(\mathbf{m}_n\) are one-fibered and let \(v_n\) be the unique Rees valuation of \(\mathbf{m}_n\). In this paper the author showed that the sequence \(\{v_n\}_{n \geq 0}\) converges. Note that every regular local ring is analytically unramified, every local quadratic transform of a regular local ring is normal, and every regular local ring \((R, \mathbf{m})\) is one-fibered with Rees \(\mathbf{m} = \{\mathrm{ord}_R\}\). Thus, this extends a result of [\textit{W. Heinzer} et al., J. Algebra 474, 213--239 (2017; Zbl 1365.13036)] that if \(\{(R_n, \mathbf{m}_n)\}_{n \geq 0}\) is an infinite sequence of local quadratic transforms of regular local rings, then the order valuations converges.