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Toroidalization of generating sequences in dimension two function fields. (English) Zbl 1170.14003

Let \({\Bbbk}\) be an algebraically closed field of characteristic 0, and let \(K^* / K\) be a finite extension of algebraic function fields of transcendence degree 2 over \({\Bbbk}\). Let \(\nu^*\) be a \({\Bbbk}\)-valuation of \(K^*\) with valuation ring \(V^*\) and value group \(\Gamma^*\). Let \(\nu\) be the restriction of \(\nu^*\) to \(K\) with valuation ring \(V\) and value group \(\Gamma \). \(K^* / K\) is the birational model of an extension of local rings \(R\rightarrow S\), that is \(K\) is the fraction field of \(R\), \(K^*\) is the fraction field of \(S\) and \(S\) is dominated by \(V^*\). Valuations in dimension two are completely described by a compact set of data called generating sequence. Let recall the definition of generating sequences: Let \(\Gamma _+=\nu (R\setminus \{0\})\) be a semigroup of \(\Gamma \). For \(\gamma \in \Gamma _+\), let \(I_\gamma =\{ f\in R\mid \nu(f)\geq \gamma\}\). A (possibly infinite) sequence \(\{Q_i\}\) of elements of \(R\) is a generating sequence of \(\nu\) if for every \(\gamma \in \Gamma _+\) the ideal \(I_\gamma \) is generated by the set \[ \left\{\prod Q_i^{a_i}\mid a_i\in \mathbb N_0, \sum_i a_i\nu(Q_i)\geq \gamma \right\}. \] The aim of this paper is to find structure theorems for generating sequences of \(\nu\) and \(\nu^*\). In fact after quadratic transforms the map between generating sequences of \(\nu\) and \(\nu^*\) has a toroidal structure. This work extends the Strong Monomialization theorem of S. D. Cutkosky and O. Piltant [Adv. Math. 183, No. 1, 1–79 (2004; Zbl 1105.14015)].

MSC:

14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
13A18 Valuations and their generalizations for commutative rings
14E05 Rational and birational maps
14B05 Singularities in algebraic geometry
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References:

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