The analogue of Izumi's Theorem for Abhyankar valuations

Rond, Guillaume; Spivakovsky, Mark

doi:10.1112/jlms/jdu045

Cited by 7 publications

(6 citation statements)

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“…Our main results are that u L is an ultrametric on any such set if and only if X is arborescent (see Theorem 1. 46) and that even when X is not arborescent, it is still an ultrametric in restriction to arbitrarily large sets of branches, which may be characterized topologically in terms of their total transform on any good resolution of their sum (see Theorem 1. 42).…”

Section: Ultrametric Distances On Finite Sets Of Branchesmentioning

confidence: 99%

“…Namely, we prove that if u L is an ultrametric for every branch L on X, then X is arborescent (see Theorem 1. 46).…”

Section: 5mentioning

confidence: 99%

“…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%

“…We prove a converse of one of the main theorems of [19], which stated that u L is an ultrametric whenever X is arborescent (see Theorem 1. 46):…”

mentioning

confidence: 99%

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Ultrametric properties for valuation spaces of normal surface singularities

Barroso

Pérez

Popescu-Pampu

et al. 2019

Trans. Amer. Math. Soc.

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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.

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Section: Ultrametric Distances On Finite Sets Of Branchesmentioning

confidence: 99%

“…Namely, we prove that if u L is an ultrametric for every branch L on X, then X is arborescent (see Theorem 1. 46).…”

Section: 5mentioning

confidence: 99%

Section: B-divisors On Normal Surface Singularitiesmentioning

confidence: 99%

“…We prove a converse of one of the main theorems of [19], which stated that u L is an ultrametric whenever X is arborescent (see Theorem 1. 46):…”

mentioning

confidence: 99%

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Ultrametric properties for valuation spaces of normal surface singularities

Barroso

Pérez

Popescu-Pampu

et al. 2019

Trans. Amer. Math. Soc.

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“…Finally, as in [ELS03], Theorem B also gives a new proof of a prime characteristic version of Izumi's theorem for arbitrary real-valued Abhyankar valuations with a common regular center (see also the more general work of [RS14]).…”

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confidence: 96%

Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic

Datta¹

2019

Trans. Amer. Math. Soc.

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We prove the prime characteristic analogue of a characteristic 0 result of Ein, Lazarsfeld and Smith [ELS03] on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties.Let X be a variety over a perfect field k of prime characteristic, with function field K. Suppose v is a real-valued valuation of K/k centered on X. Then for all m ∈ R, we have the valuation idealsThe goal of this paper is to use the theory of asymptotic test ideals in positive characteristic to prove the following uniform approximation result for Abhyankar valuation ideals established in the characteristic 0 setting by Ein, Lazarsfeld and Smith [ELS03].Theorem A. Let X be a regular variety over a perfect field k of prime characteristic with function field K. For any non-trivial, real-valued Abhyankar valuation v of K/k centered on X, there exists e ≥ 0, such that for all m ∈ R ≥0 and ℓ ∈ N,In this paper, we employ an asymptotic version of the test ideal of a pair to prove Theorem A, drawing inspiration from the asymptotic multiplier ideal techniques in [ELS03]. However, instead of utilizing tight closure machinery, our approach to asymptotic test ideals is based on Schwede's dual and simpler reformulation of test ideals using p −e -linear maps, which are like maps inverse to Frobenius [Sch10, Sch11] (see also [Smi95,LS01]).Asymptotic test ideals are associated to graded families of ideals (Definition 4.2.1), an example of the latter being the family of valuation ideals a • := {a m (A)} m∈R ≥0 . For each m ≥ 0, one constructs the m-th asymptotic test ideal τ m (A, a • ) of the family a • , and then Theorem A is deduced using Theorem B. Let v be a non-trivial real-valued Abhyankar valuation of K/k, centered on a regular local ring (A, m), where A is essentially of finite type over the perfect field k of prime characteristic with fraction field K. Then there exists r ∈ A − {0} such that for all m ∈ R ≥0 ,In other words, m∈R ≥0 (a m : τ m (A, a • )) = (0). Finally, as in [ELS03], Theorem B also gives a new proof of a prime characteristic version ofIzumi's theorem for arbitrary real-valued Abhyankar valuations with a common regular center (see also the more general work of [RS14]). Corollary C (Izumi's Theorem for Abhyankar valuations in prime characteristic).Let v and w be non-trivial real-valued Abhyankar valuations of K/k, centered on a regular local ring (A, m), as in Theorem B. Then there exists a real number C > 0 such that for allThus, Corollary C implies that the valuation topologies on A induced by two non-trivial real-valued Abhyankar valuations are equivalent.Hara defined and used asymptotic test ideals to give a prime characteristic proof of uniform bounds on symbolic power ideals [Har05], which is independent of Hochster and Huneke's earlier proof [HH02] and similar to the multiplier ideal approach of [ELS01]. This paper continues the efforts of Hara and other researchers to use test ideals to prove prime characteristic analogues of statements in characteristic 0 that were esta...

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Uniform Symbolic Topologies and Hypersurfaces

Huneke,

Katz

2024

Acta Math Vietnam

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We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.

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The analogue of Izumi's Theorem for Abhyankar valuations

Cited by 7 publications

References 15 publications

Ultrametric properties for valuation spaces of normal surface singularities

Ultrametric properties for valuation spaces of normal surface singularities

Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic

Uniform Symbolic Topologies and Hypersurfaces

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