Comparison of divisorial valuations. (Comparaison des valuations divisorielles.) (French. English summary) Zbl 1190.13013

Cano, Felipe (ed.) et al., Équations différentielles et singularités. En l’honneur de J. M. Aroca. Paris: Société Mathématique de France (ISBN 978-2-85629-263-1/pbk). Astérisque 323, 17-31 (2009).
Summary: Using the notion of connexity in codimension one, we give in this paper a new geometric proof of Izumi’s theorem in two special cases. We also propose the following conjecture: let \((R,\mathfrak m)\) be a complete, normal local domain and \(v_1, v_2\) two divisorial valuations centered in \(\mathfrak m\). Then there exists an \(\mathfrak m\)-primary ideal \(I\) of \(R\) such that the centers of \(v_1\) and \(v_2\) in the normalised blowing up of Spec\(R\) along \(I\) are linked in codimension 1. At the end of the paper, we make some comments about this conjecture.
For the entire collection see [Zbl 1187.00035].


13F30 Valuation rings
13G05 Integral domains
14E05 Rational and birational maps
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