Steven Dale Cutkosky
We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.
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We prove that every place $P$ of an algebraic function field $F|K$ of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field $FP$ over $K$ is equal to the transcendence degree of $F|K$, and the extension $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, assuming that the valued field $(K,v_P)$ is defectless, the factor group $v_P F/v_P K$ is torsion-free and the extension of resi…
We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the valuation is finitely generated (which implies that the valuation is Abhyankar) the valuation can be uniformized in an embedded way by a birational map which is monomial with respect to a suitable system of generators of the maximal ideal. We prove that conversely if a valuation is Abhyankar after a birational modification and localization at the point picked b…
We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar-Moh (semigroup) Theorem for it.
Using the theory of asymptotic test ideals, we prove the prime characteristic analogue of a characteristic $0$ result of Ein, Lazarsfeld and Smith (arXiv:math/0202303) on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties over perfect fields.
In this paper we use the theory of multiplier ideals to show that the valuation ideals of a rank one Abhyankar valuation centered at a smooth point of a complex algebraic variety are approximated, in a quite strong sense, by sequences of powers of fixed ideals. Fix a rank one valuation v centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that v is Abhyankar, that is, that its rational rank plus its transcendence degree equal the dimension of the variety. Let a_m denote the ideal of elements in the local ring of x whose valuations are at least m. Ou…
Suppose that R\rightarrow S is an extension of local domains and ν^* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation gr_{ν^*}(R)\rightarrow gr_{ν^*}(S). We give examples showing that in general, this extension does not share good properties of the extension $\rightarrow S, but after enough blow ups above the valuations, good properties of the extension R\rightarrow S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positiv…
In this paper, we give a valuation formula for rational top differential forms of function fields in characteristic zero for arbitrary Abhyankar places generalizing the classical valuation at prime divisors. This enables us to define log discrepancies for log pairs for arbitrary Abhyankar places. If the Abhyankar place has dimension greater than zero we restrict rational top differential forms with valuation zero to the residue field of the Abhyankar place, generalizing the classical restriction of a top differential form with a simple pole along a smooth divisor. This opens up the door to gen…
Abhyankar valuations, Prüfer-Manis valuations, and perfectoid Tate algebras
本文研究了完美Tate代数的商域，证明了其为特定估值的半直接扩展，并讨论了Abhyankar估值的性质。
Frobenius splitting of valuation rings revisited
针对特征p>0的F有限域上函数域的赋值环，证明其Frobenius分裂等价于对应赋值为Abhyankar赋值，还构造反例说明基域非F有限时该等价性不成立，方法不依赖局部一致化结果。
We give a simple algorithm showing that the reduction of the multiplicity of a characteristic p>0 hypersurface singularity along a valuation is possible if there is a finite linear projection which is defectless. The method begins with the algorithm of Zariski to reduce multiplicity of hypersuface singularities in characteristic 0 along a valuation. This gives a simple demonstration that the only obstruction to local uniformization in positive characteristic is from defect arising in finite projections of singularities.
We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension E of F. We show that E|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension EP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation…
A special form of Zariski's Uniformization Theorem in positive characteristic
本文简要阐述正特征下赋值环作为光滑代数滤过并的扩张，涉及Zariski均匀化定理的特殊形式。
It is known since the works of Zariski in early 40ies that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, i.e. any valuation can be uniformized by a purely inseparable alteration.
We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theorem for monic polynomials whose discriminant is weighted homogeneous.
Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to [arXiv:1507.06009], this paper emphasizes the role of centers in controlling Frobenius properties of valuations rings in prime characteri…
Essential finite generation of extensions of valuation rings
本文在完全一般性下证明了 Knaf 猜想，即用相关赋值扩张的经典不变量来刻画赋值环局部扩张何时本原有限型，推进了赋值局部一致化问题。
Height reduction for local uniformization of varieties and non-archimedean spaces
本文证明概形与非阿基米德解析空间的局部一致化可严格归结为高度为一的赋值情形，从而将非阿基米德空间的问题转化为研究光滑Berkovich空间的局部结构。
Abeloid varieties over higher rank valued fields: Raynaud uniformization and Néron models
本文研究高秩赋值环上的阿贝尔型簇，推广 Raynaud 均匀化，并证明代数与形式 Néron 模型的存在性，揭示两者联系。
A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension. After extension of the valuation to a suitable completion (whose existence is not established in the paper) the relation becomes much more precise, and suggests a possible way to establish local uniformization for an excellent equicharacteristic local domain with an algebraically closed residue field, by deforming a partial toric …
Abhyankar-Moh Semigroups for arbitrary hypersurfaces
该研究针对任意超曲面奇点构造了与代数闭域相关的半群，扩展了阿比扬卡尔和莫研究的平面曲线值半群，建立其与根选择无关的条件，还指出其他相关扩展是此构造族的特殊情况。
Let k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings R_i with quotient field K which are dominated by V, and such that the direct union of the R_i's is V. We investigate the ring T, which is the direct union of the completions of the R_i's. We give necessary and sufficient conditions for T to be a valuation ring. We then focus on the case in which the valuation has rank one.
For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of lack of defect.
We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.
We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential $1$-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.
We prove that for any compact quasi-smooth strictly $k$-analytic space $X$ there exist a finite extension $l/k$ and a quasi-étale covering $X'\to X\otimes_kl$ such that $X'$ possesses a strictly semistable formal model. This extends a theorem of U. Hartl to the case of the ground field with a non-discrete valuation.
On Kaplansky's Embedding Theorem
本文证明完备等特征诺特局部整环可保赋值嵌入Hahn级数环，方法基于环面局部均匀化与半赋值逼近。
The reduction of singularities of codimension one foliations is known in the case of ambient dimension 2 (Seidenberg, A. (1968). Reduction of singularities of the differential equation Ady= Bdx. American Journal of Mathematics, 90(1), 248-269) and 3 (Cano, F. (2004). Reduction of the singularities of codimension one singular foliations in dimension three. Annals of mathematics, 160(3), 907-1011). However, in greater ambient dimension it is still an open problem. In this work we prove the existence of local uniformization for codimension one foliations in arbitrary ambient dimension, in the cas…
An extension (K(X)|K, v) of valued fields is said to be valuation transcendental if we have equality in the Abhyankar inequality. Minimal pairs of definition are fundamental objects in the investigation of valuation transcendental extensions. In this article, we associate a uniquely determined positive integer with a valuation transcendental extension. This integer is defined via a chosen minimal pair of definition, but it is later shown to be independent of the choice. Further, we show that this integer encodes important information regarding the implicit constant field of the extension (K(X)…
Let $R$ be a complete equicharacteristic noetherian local domain and $ν$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $ν$ on $R\setminus \{0\}$ is not finitely generated in general. We produce equations in an appropriate generalized power series ring for the algebra encoding the degeneration of $R$ to the toric graded algebra ${\rm gr}_νR$ associated to the filtration defined by $ν$. We apply this to represent $ν$ as the limit of a sequence of Abhyankar semivaluations (valuations on quotients) of $R$ w…
We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then alternatively, F' can be obtained from F by at most two Galois extensions; if in addition P is zero-dimensional, then we only need one Galois extension. Certain rational places of rank 1 can be uniformized already on F. We introduce the notion of "relative uniformization" for arbitrary finitely generated extensions of valued fields. Our proofs are based solely on…
We prove an embedded local uniformization theroem for a valuation centered on a point of a quasi-excellent scheme of characteristic zero. The proof reduces to valuations of rank 1 and consists in desingularizing the ideal formed by the elements of infinite value and monomializing the key polynomials. We then prove a monomialization theorem valid in all characteristic under certain conditions, including that of non-existence of limit key polynomials, a condition that is always satified in characteristic zero.
Extending valuations of local domains to complete local domains without changing the value group
研究在不改变值群的情况下，将局部域的赋值扩展到完备局部域的问题，通过给出泰西耶猜想的证明，为正特征下局部一致化的两种方法提供重要步骤。
Suppose that (K, $ν$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $ω$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ν$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $ω$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $ω$ as an extension of the associated graded ring of A for the filtration defined by $ν$. In parti…
Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $ν^*$ is a valuation of $K^*$ such that $ν^*$ dominates $S$. Let $ν$ be the restriction of $ν^*$ to $K$. The associated graded ring ${\rm gr}_ν(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,ν)\rightarrow (K^*,ν^*)$ of valued fields is without defect if and only if there exist regular l…
Key polynomials for simple extensions of valued fields
本文提出改进的关键多项式理论，用于描述简单域扩张中的关键多项式及其序型上界。
K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This generalizes the work of M. Kerz, S. Saito and U. Jannsen for finite fields to the case of all higher local fields. For that purpose a $p$-alteration theorem for the local uniformization of schemes over valuation rings of arbitrary finite rank and a corresponding Bertini theorem is developed extending the results of O. Gabber, J. deJong, L. Illusie, M. Temkin, S. S…
Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathrm{char}\ k > 2g+1$. Let $S = \mathrm{Spec}\ R$. Let $X$ be the minimal proper regular model of $C$ over $S$. Let $\mathrm{Art}\ (C/K)$ denote the Artin conductor of the $S$-scheme $X$ and let $ν(Δ_C)$ denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}\ (C/K) \leq ν(Δ_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent…
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