Jaroslaw Wlodarczyk
Building upon work of Villamayor and Bierstone-Milman we give a proof of the canonical Hironaka principalization and desingularization. The idea of "homogenized ideals" introduced in the paper gives {\it a priori} the canonicity of algorithm and radically simplifies the proof.
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Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk, we give an a priori estimate for the complexity of the simplified Hironaka algorithm. As a consequence of this result, we show that there exists canonical Hironaka embedded desingularization and principalization over fields of large characteristic (relative to the degrees of generating polynomials).
We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.
Hironaka's concept of characteristic polyhedron of a singularity has been one of the most powerful and fruitful ideas of the last decades in singularity theory. In fact, since then combinatorics have become a major tool in many important results. However, this seminal concept is still not enough to cope with some effective problems: for instance, giving a bound on the maximum number of blowing--ups to be performed on a surface before its multiplicity decreases. This short note shows why such a bounding is not possible, at least with the original definitions.
Algorithms for resolution of singularities in characteristic zero are based on Hironaka's idea of reducing the problem to a simpler question of desingularization of an "idealistic exponent" (or "marked ideal"). How can we determine whether two marked ideals are equisingular in the sense that they can be resolved by the same blowing-up sequences? We show there is a desingularization functor defined on the category of equivalence classes of marked ideals and smooth morphisms, where marked ideals are "equivalent" if they have the same sequences of "test transformations". Functoriality in this sen…
We consider a finite analytic morphism $ϕ= (f,g) : (X,p)\to (\C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $π: (Y,E_{Y})\to (X,p)$ be a good resolution of $ϕ$ with exceptional divisor $E_{Y}=π^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $π$. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected compone…
When $X$ is a $d$-dimensional variety defined over a field $k$ of characteristic zero, a constructive resolution of singularities can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers. This is done by stratifying the maximum multiplicity locus of $X$ by means of the so called {\em resolution functions}. The most important of these functions is what we know as {\em Hironaka's order function in dimension $d$}. Actually, this function can be defined for varieties when the base field is perfect; however if the characteristic of $k$ is positiv…
We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.
Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by ideals in a simpler form. In the special Cohen-Macaulay case we obtain a singular analog of the inverse function theorem. The singular implicit function theorem is closely related to a (proven here) extended version of the Weierstrass-Hironaka-Malgrange division and preparation …
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…
These expository notes, addressed to non-experts, are intended to present some of Hironaka's ideas on his theorem of resolution of singularities. We focus particularly on those aspects which have played a central role in the constructive proof of this theorem. In fact, algorithmic proofs of the theorem of resolution grow, to a large extend, from the so called Hironaka's fundamental invariant. Here we underline the influence of this invariant in the proofs of the natural properties of constructive resolution, such as: equivariance, compatibility with open restrictions, with pull-backs by smooth…
The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by restriction to smooth hypersurfaces (hypersurfaces of maximal contact). Our alternative approach, introduced here, replaces restrictions to smooth sub-schemes by generic projections on smooth schemes of smaller dimension. We also introduce a generalization of the discriminant, and our result makes use of the elimination theory. In the case of fields of char…
We prove an equivariant version of Hironaka's theorem on elimination of points of indeterminacy. Our arguments rely on canonical resolution of singularities.
We show that a version of the desingularization theorem of Hironaka holds for certain classes of infinitely differentiable functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, Lojasiewicz inequalities, division properti…
These are introductional notes to resolution of singularities and Log principalization of ideals over fields of characteristic zero. We refer to `A simplified proof of desingularization and applications', A. Bravo, S. Encinas and O. Villamayor, (to appear in {\em Revista Matemática Iberoamericana}), for more details. The notes follow the algorithmic proof in `A new proof of desingularization over fields of characteristic zero', S. Encinas and O. Villamayor U., {\em Revista Matemática Iberoamericana}, 19 (2) (2003), which provides an important conceptual simplification over the original proofs …
We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.
We present a concise proof for the existence and construction of a {\it strong resolution of excellent schemes} of finite type over a field of characteristic zero. Our proof is based on earlier work of Villamayor, Encinas-Villamayor and Bierstone-Milman. It apports some substantial simplifications which may be helpful for a better understanding of how to prove Hironaka's famous theorem.
This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a…
Hironaka's characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition $ (\mathrm{Pol}) $ holds, or a mild condition $(*) $ on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.
Functorial flatification of proper morphisms
本研究针对正常态射的平坦化问题，借鉴Hironaka算法思路提出通过吹胀的函子平坦化算法，特征零下可通过光滑中心吹胀实现，还给出了特征零下基于Kummer吹胀的函子平展化算法。
Updated version. Includes comments on the advances in the field from the Kyoto workshop on Resolution of Singularities in Positive Characteristic, December 2008. The article surveys the theory of kangaroo points as they appear in the resolution of singularities in positive characteristic. They represent one of the main obstructions for transcribing the characteristic zero proof of resolution to positive characteristic. Kangaroo points are classified through the concept of oblique polynomials. The results of the article are used in Hironaka's recent program towards the resolution of singulariti…
Logarithmic resolution via weighted toroidal blow-ups
本文提出一种基于加权环面爆破的快速函数性对数分辨率程序，适用于一般对数光滑的 fs 对数概形，并恢复特征零下 Hironaka 奇点消解的更简单方法。
We study Hironaka's idealistic exponents over $ \operatorname{Spec} ( \mathbb{Z} ) $. We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.
Closure operations induced via resolutions of singularities in characteristic zero
在零特征中，通过解析奇点引入KH闭包操作，用于检测理性奇点并改进紧闭操作，同时探索相关闭包操作及其与正特征紧闭操作的联系。
Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope
本文为代数超曲面引入 Newton-Hironaka 多面体的扰动变体，由实数参数化，在参数值合适时接近原多面体，并提供不依赖参数选择的额外信息。
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. The results extend to more general varieties defined locally by binomial equations.
Using results of Hironaka-Matsumura and Faltings, we prove a strong version of the well known Fulton-Hansen connectivity theorem for weighted projective spaces. As a consequence we get the following result. If $Y$ is an irreducible subvariety of the $n$-dimensional projective space (over a field of arbitrary characteristic), then the diagonal embedding $Δ_Y$ is $G_3$ in $Y\times Y$. This fact implies a generalized version (with a characteristic-free proof) of a result of Ogus (in char. zero) and Speiser (in positive characteristic).
We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness. This proof, already sketched in [A course on constructive desingularization and equivariance. In {\em Resolution of singularities (Obergurgl, 1997)}, vol. 181 {\em Progr. Math.}, Birkhäuser, 2000.] page 224, is done by showing that desingularization of a closed subscheme $X$, in a smooth sheme W, is achieved by taking an algorithmic principalization for the ideal $I(X)$, associated to the embedded scheme $X$.
Thresholds of singularities in characteristic zero
该研究在特征零情形下构造了Frobenius阈值的两个类似物Hironaka阈值与Koszul-Hironaka阈值，证明其对参数理想一致且为有理数，揭示了其与乘子理想跳跃数、模p约化Frobenius阈值极限的关联。
The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective r…
Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not modify the subset $ V $ and transforms $X$ into a toroidal embedding $Y$, with singularities extending those on $V$. Moreover, the exceptional divisor has simple normal crossings on $Y$. The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus $V$ and transforms $X$ into a nonsingular variety $Y$. Th…
A dynamical version of Silverman-Tate's height inequality
本文以吹胀替代 Hironaka 奇点消解，将 Silverman-Tate 高度不等式推广到一般动力系统及任意可定义高度的乘积公式域，并导出高维基底下的 Silverman 特殊化定理变体。
We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to automorphisms or etale or Zariski localizations. We treat the embedded case as well as the non-embedded case, with or without a boundary, and we relate the diferent versions. In the non-embedded case, a boundary is a collection of locally principal closed subschemes. Our main tools are the stratifications by Hilbert-Samuel functions and the characteristic polyhedra in…
Based on an idea in Hironaka's proof of resolution of singularities, we present an algorithmic smoothness test for algebraic varieties. The test is inherently parallel and does not involve the calculation of codimension-sized minors of the Jacobian matrix of the variety. We also describe a hybrid method which combines the new method with the Jacobian criterion, thus making use of the strengths of both approaches. We have implemented all algorithms in the computer algebra system Singular, and compare the different approaches with respect to timings and memory usage. The test examples originate …
Logarithmic resolution via multi-weighted blow-ups
本文引入多重加权爆破概念，并系统构造了特征零域上Hironaka意义下的函子性对数分辨的显式高效算法。
Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic
本文通过进化搜索方法发现正特征下标准吹胀过程的候选排名函数，并提出两个猜想作为主要贡献。
Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a category of complexes. Roughly speaking, the criterion shows that if such a functor satisfies the standard exact sequence of a blowing-up, then the functor can be extended to all separated k-schemes of finite type. The result is applied to the Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an analytic space, and to the rational homotopy of k-schemes in algebraic De Rham theory.
This article contains an elementary constructive proof of resolution of singularities in characteristic zero. Our proof applies in particular to schemes of finite type and to analytic spaces (so we recover the great theorems of Hironaka). We introduce a discrete local invariant $\inv_X (a)$ whose maximum locus determines a smooth centre of blowing up, leading to desingularization. To define $\inv_X$, we need only to work with a category of local-ringed spaces $X=(|X|,{\cal O}_X)$ satisfying certain natural conditions. If $a\in |X|$, then $\inv_X(a)$ depends only on $\widehat{\cal O}_{X,a}$. Mo…
Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions
本文通过构造普通爆破算法将二元$C^{\infty}$函数零点簇化为几乎法向交叉形式，并借助van der Corput型引理，用零点簇的重数刻画了相应局部zeta函数的亚纯延拓区域。
In characteristic zero, if a quotient variety has a crepant resolution, the Euler characteristic of the crepant resolution is equal to the number of conjugacy classes of the acting group, by Batyrev's theorem. This is one of the McKay correspondence. It is natural to consider the analogue statement in the positive characteristic. In this paper, we present sequences of crepant resolutions of quotient varieties in the positive characteristic and show that one of the sequences gives a counterexample to the analogue statement of Batyrev's theorem.
The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields.
Logarithmic resolution of singularities in characteristic 0 using weighted blow-ups
本文研究特征零域上的奇点消解问题，核心方法是使用加权爆破，构造出例外因子呈简单正常交叉的对数消解。
We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution via blowing ups in regular centers. We illustrate this for generic determinantal varieties. The article is partially expository and is addressed to non-experts who aim to construct resolutions for other special classes of singularities in positive or mixed characteristic.
Trim resolutions, stringy and Mather classes, and IC characteristic cycles
本文提出trim resolutions的概念，并证明具有trim resolutions的簇的交变层特征循环不可约，且stringy和Chern-Mather类相等。
This is Part II of the series of our papers under the title "Toward resolution of singularities over a field of positive characteristic (The Idealistic Filtration Program)". See http://arxiv.org/abs/math/0607009 for Part I.
A new proof for the embedded resolution of surface singularities in a three-dimensional smooth ambient space over algebraically closed fields of arbitrary characteristic. The proof makes use of an upper semicontinuous resolution invariant which prescribes the center in each step of the resolution algorithm. The resolution invariant strictly decreases under each blowup. The definition of the resolution invariant is inspired by the proofs of resolution of singularities in arbitrary dimension over fields of characteristic zero. It was developed in an attempt to generalize these proofs to the sett…
Crepant resolution of $\mathbb{A}^4/A_4$ in characteristic 2
本文在特征 2 下构造了 $\mathbb{A}^4/A_4$ 的 crepant 消解，并通过计算其 Euler 数给出正特征 McKay 对应类似命题的新反例。
This paper represents the main portion of the Ph.D. Thesis of the author, and is the first of the series of four papers, which is a joint work with K. Matsuki as a whole. We present a program toward constructing an algorithm for resolution of singularities in positive characteristic. We would like to emphasize, however, that the program is actually valid in all characteristics. As such, it leads in characteristic zero to a complete algorithm, which is slightly different from the existing ones. In positive characteristic, almost all the ingredients work nicely forming a perfect parallel to the …
A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.
This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial transformations and based on Weierstrass preparation theorem. This approach entails such new methods as canonical reduction and synthesis of monomial transformations as well as latency and revival of primary variables. The orders of primary variables serve as the decreasing singularity invariants for the algorithm albeit with some temporary increases. A finite parti…
We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It i…
In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations in centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transformations with a normal crossing, Weierstrass representations, reduction sequences, and so forth.
Introduction to Universal Equations and Characteristic-free Resolution
本文是无特征奇点分解证明的综述，阐述该证明的方法动机、新思想并介绍相关证明内容。
We unify results of Artin, Brieskorn, Slodowy and others by showing that, in all characteristics, the Artin component of the deformation space of a rational surface singularity has a ramified cover where simultaneous resolution exists and the Galois group of this cover is the Weyl group determined by the configuration of $(-2)$-curves in the minimal resolution. This verifies a conjecture made by Burns and Rapoport. We use an integral version of this to show that certain actions of Weyl groups on polynomial rings over the integers give rings of invariants that are also polynomial, and deduce th…
This paper is devoted to give all the technical constructions and definitions that will lead to the construction of an algorithm of resolution of singularities for binomial ideals. We construct a resolution function that will provide a resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic of the base field. This resolution function is based in a modified order function, called $E$-order. The …
This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number o…
We describe the embedded resolution of a quasi-ordinary surface singularity (V,p) which results from applying the canonical resolution of Bierstone-Milman to (V,p). We show that this process depends solely on the characteristic pairs of (V,p), as predicted by Lipman. We describe the process explicitly enough that a resolution graph for (V,p) could in principle be obtained by computer using only the characteristic pairs.
We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the previous structures. In the case of characteristic zero one may use these structures to obtain a resolution of singularities.
Let $X$ be an integral affine or projective scheme of finite presentation over a perfect field. We prove that $X$ admits a resolution, that is, there exists a smooth scheme $\widetilde X$ and a projective birational morphism from $\widetilde X$ onto $X$.
In this paper a concise, complete proof of resolution of singularities of 3-folds in positive characteristic (>5) is given. The first proof of this theorem was given by Abhyankar in 1966. The resolution morphism in our proof is an isomorphism over the nonsingular locus.
For a smooth variety $X$ over an algebraically closed field of characteristic $p$, to a differential 1-form $α$ on the Frobenius twist $X^{(1)}$ one can associate an Azumaya algebra $\mathcal D_{X,α}$, defined as a certain central reduction of the algebra $\mathcal D_X$ of "crystalline differential operators" on $X$. For a resolution of singularities $π:X\to Y$ of an affine variety $Y$, we study for which $α$ does the class $[\mathcal D_{X,α}]$ in the Brauer group $\mathrm{Br}(X^{(1)})$ descend to $Y^{(1)}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations …
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