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. Our main theorem is that there exists e>0 such that a_{mn} is contained in (a_{m-e})^n for all m and n. This can be viewed as a greatly strengthened form of Izumi's Theorem for Abhyankar valuations centered on smooth complex varieties.
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.
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.
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…
Abhyankar-Moh Semigroups for arbitrary hypersurfaces
Instituto de Matemáticas, Unidad Oaxaca, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所瓦哈卡分部）；Dipartimento Matematica Ed Applicazioni “R. Caccioppoli” Università Degli Studi Di Napoli “Federico II”（那不勒斯费德里科二世大学应用数学系）
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…
In a valuation domain $(V,M)$ every nonzero finitely generated ideal $J$ is principal and so, in particular, $J=J^t$, hence the maximal ideal $M$ is a $t$-ideal. Therefore, the $t$-local domains (i.e., the local domains, with maximal ideal being a $t$-ideal) are "cousins" of valuation domains, but, as we will see in detail, not so close. Indeed, for instance, a localization of a $t$-local domain is not necessarily $t$-local, but of course a localization of a valuation domain is a valuation domain. So it is natural to ask under what conditions is a $t$-local domain a valuation domain? The main …
Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$ we associate weights to the coordinates of the projective space, respectively, the product of projective spaces. Let $w_{\mathfrak v}$ be the vector whose entries are these weights. Our main result is that the value semi-group of $\mathfrak v$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak v}$ is prime. We further show that…
We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinato…
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…
Extending a valuation centered in a local domain to the formal completion
设（R; m; k）是局部Noether整环，R_v是赋值环，控制R（不一定是双有理的）.设v| K是v对K的限制;根据定义，v| K以R为中心。令\hat{R}表示R的m-adic完备化。在赋值论在交换代数和奇点研究中的应用中，人们经常被诱导用它的m-adic完备化\hat{R}代替R，用一个合适的扩张\hat{v}代替v，对于一个合适的素理想P，使得P \cap R =（0），它是\hat{R} /P的适当扩张。本文的目的是给，假设R是优秀的，一个系统的描述所有这样的扩展\hat{v}，并确定某些类别的扩展是特别感兴趣的应用程序。
