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Local monomialization of transcendental extensions. (English) Zbl 1081.14020

Let \(R\subset S\) be local domains essentially of finite type over a field \(k\) with char \(k=0\) and \(V\) a valuation ring of the quotient field \(K\) of \(S\). Then there exist sequences of monomial transforms \(R\rightarrow R'\), \(S\rightarrow S'\) along \(V\) such that \(R'\), \(S'\) are regular local rings, \(S'\) dominates \(R'\) and there exist regular system of parameters \((y_1,\ldots,y_n)\) in \(S'\), \((x_1,\ldots,x_m)\) in \(R'\), units \(\beta_1,\ldots,\beta_n\in S'\) and a \(m\times n\) matrix \((c_{ij})\) of non-negative integers such that Rank\((c_{ij})=m\) and \(x_i=\Pi_{j=1}^ny_j^{c_{ij}}\beta_i\), \(1\leq i\leq m\).
This is the most general possible relative “Local Uniformization Theorem for mappings”, the case \(R=k\) being given by Zariski. When \(K\) is a finite extension of the quotient field of \(R\) the result was already stated by the author in [“Local monomialization and factorization of morphisms”, Astérisque 260 (1999; Zbl 0941.14001)]. The above result is used to the construction of a monomialization by quasi-complete varieties, which proves a local version of the toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki and J. Wlodarczyk [J. Am. Math. Soc. 15, 531–572 (2002; Zbl 1032.14003)].

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13A18 Valuations and their generalizations for commutative rings
14B05 Singularities in algebraic geometry
14E05 Rational and birational maps
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References:

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