Local monomialization and factorization of morphisms
By Steven Dale Cutkosky. Astérisque, 2018
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— Suppose that R C S are regular local rings of a common dimension, which are essentially of finite type over a field k of characteristic zéro, such that the quotient field K of S is finite over the quotient field of R. If V is a valuation ring of K which dominâtes S, then we show that there are séquences of monoidal transforms (blowups of regular primes) R -¥ R± and S » Si along V such that Ri —> S\ is a monomial mapping. It follows that a generically finite morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blowups of nonsingular subvarieties. We give applications to factorization of birational morphisms and simultaneous resolution of singularities. Résumé (Monomialisation et factorisation locales des morphismes) Soient R C S deux anneaux locaux réguliers de même dimension, essentiellement de type fini sur un corps k de caractéristique zéro, et tels que le corps des fractions K de S est fini sur celui de R. Si V est un anneau de valuation de K dominant S, nous montrons qu'il existe des suites de transformés monoïdaux (éclatements d'idéaux premiers réguliers) R —)> Ri et S —• Si le long de V tels que Ri —> Si est une application monomiale. Il s'ensuit qu'un morphisme génériquement fini de variétés non singulières peut être rendu monomial le long d'une valuation après éclatement de sous-variétés non singulières. Nous donnons des applications à la factorisation des morphismes birationnels et à la résolution simultanée des singularités. © Astérisque 260, SMF 1999
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/0b15b8b424a98bc210e0f7de9cc02c91baecf0c9
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In Algebra and Number Theory
In Geometry and Topology
Citations in the corpus
- Resolution of singularities and modular Galois theory (2000, methodology): “Recently, in [Cu1] and [ Cu2 ], Dale Cutkosky settled the Local Conjecture for any dimension in characteristic zero.”