Universal valued fields and lifting points in local tropical varieties

Stepanov, D. A.

doi:10.1080/00927872.2016.1175447

Cited by 3 publications

(1 citation statement)

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“…A recent result for valuations of rank one on complete regular local rings, including the mixed characteristic case and making significant use of key polynomials, is due to San Saturnino in [73]. 17 See [74] for motivation and results.…”

Section: Abhyankar Valuations and Quasi Monomial Valuationsmentioning

confidence: 99%

Overweight deformations of affine toric varieties and local uniformization

Teissier

2014

EMS Series of Congress Reports

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Given an equicharacteristic complete noetherian local domain R with algebraically closed residue field k, we first present a combinatorial proof of embedded local uniformization for zero-dimensional valuations of R whose associated graded ring gr ν R with respect to the filtration defined by the valuation is a finitely generated k-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to gr ν R also provide local uniformizations for ν on R. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is Z r with r = dimR). In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain R with algebraically closed residue field, if the zero-dimensional valuation ν of R is Abhyankar, there are local domains R ′ which are essentially of finite type over R and dominated by the valuation ring Rν (ν-modifications of R) such that the semigroup of values of ν on R ′ is finitely generated, and therefore so is the k-algebra gr ν R ′ . Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations.As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by N of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring k[[x 1 , . . . , xr]] to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding. 1

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Section: Abhyankar Valuations and Quasi Monomial Valuationsmentioning

confidence: 99%

Overweight deformations of affine toric varieties and local uniformization

Teissier

2014

EMS Series of Congress Reports

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Resolving singularities of curves with one toric morphism

2022

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We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity (C, O) contained in a non singular surface S such a reembedding may be defined in terms of a sequence of maximal contact curves of the minimal embedded resolution of C. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of C. We use properties of the semivaluation space of S at O to describe how the dual graph of the minimal embedded resolution of C may be seen on the local tropicalization of S associated to this reembedding.

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