On Izumi's theorem on comparison of valuations

Moghaddam, Mohammadreza Sadeghi

doi:10.2996/kmj/1301576758

Cited by 3 publications

(3 citation statements)

References 12 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Its study has been the focus of several works, see e.g. [45,10,37,46,6]. The b-divisor interpretation given by Favre and Jonsson is more recent, and it has been used to study several properties of valuation spaces for smooth and singular surfaces (see e.g.…”

Section: B-divisors On Normal Surface Singularitiesmentioning

confidence: 99%

Ultrametric properties for valuation spaces of normal surface singularities

Barroso

Pérez

Popescu-Pampu

et al. 2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let L be a fixed branch -that is, an irreducible germ of curve -on a normal surface singularity X. If A, B are two other branches, define uL(A, B) :where A · B denotes the intersection number of A and B. Call X arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P loski by proving that whenever X is arborescent, the function uL is an ultrametric on the set of branches on X different from L. In the present paper we prove that, conversely, if uL is an ultrametric, then X is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on X, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which uL is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing L to be an arbitrary semivaluation on X and by defining uL on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if X is arborescent, and without any restriction on X we exhibit special subspaces of the space of semivaluations in restriction to which uL is still an ultrametric.

show abstract

Section: B-divisors On Normal Surface Singularitiesmentioning

confidence: 99%

Ultrametric properties for valuation spaces of normal surface singularities

Barroso

Pérez

Popescu-Pampu

et al. 2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Other approaches, which also construct key polynomials step by step (in a transfinite sense) are found in the work of M. Spivakovsky and his collaborators, who have developed (see [36]) a general theory of key polynomials from the viewpoint of generating sequences for valuations of rank one on complete regular local rings, generalized to higher rank by W. Mahboub (see [56]) and in the work of M. Moghaddam (see [58], [59]) who has generalized the constructions of Favre and Jonsson (see [24]) for C{x, y} 20 . The relationship of the first approach with Vaquié's is explained in [55] but again there, the key polynomials are sought in K 0 [y] and not R 0 [y] although Mahboub corrects a lapsus in the definition of key polynomials in [36] where one seeks generators of the graded ring with respect to the valuation which is associated to K 0 (y) and not…”

Section: Key Polynomials and The Valuative Cohen Theoremmentioning

confidence: 99%

Overweight deformations of affine toric varieties and local uniformization

Teissier

2014

Valuation Theory in Interaction

View full text Add to dashboard Cite

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

show abstract

“…His argument has been generalized by Rees [Ree89], and alternative proofs given by Hübl and Swanson [HS01], and Beddani [Bed09]. Another approach, based on the notion of key polynomials, was recently developed by Moghaddam [Mog11], see [FJ04] in the two-dimensional case. For a connection between Izumi's Theorem and the Artin-Rees Lemma, see [Ron06].…”

Section: Introductionmentioning

confidence: 99%

A refinement of Izumi's Theorem

Boucksom

Favre

Jönsson

2014

Valuation Theory in Interaction

View full text Add to dashboard Cite

We improve Izumi's inequality, which states that any divisorial valuation v centered at a closed point 0 on an algebraic variety Y is controlled by the order of vanishing at 0. More precisely, as v ranges through valuations that are monomial with respect to coordinates in a fixed birational model X dominating Y , we show that for any regular function f on Y at 0, the function v → v(f )/ ord0(f ) is uniformly Lipschitz continuous as a function of the weight defining v. As a consequence, the volume of v is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.

show abstract

On Izumi's theorem on comparison of valuations

Cited by 3 publications

References 12 publications

Ultrametric properties for valuation spaces of normal surface singularities

Ultrametric properties for valuation spaces of normal surface singularities

Overweight deformations of affine toric varieties and local uniformization

A refinement of Izumi's Theorem

Contact Info

Product

Resources

About