On Zariski decomposition problem

光滑射影曲面上的整Zariski分解

Integral Zariski decompositions on smooth projective surfaces

该研究针对代数几何中的除子分解问题，刻画了满足所有整伪有效除子都存在整Zariski分解的光滑射影曲面的特征。

Weak Zariski decompositions and log terminal models for generalized polarized pairs

We show that the existence of a birational weak Zariski decomposition for a pseudo-effective generalized polarized lc pair is equivalent to the existence of a generalized polarized log terminal model.

Weyl and Zariski chambers on K3 surfaces

The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study the mutual inclusions of Zariski and simple Weyl chambers. Finally, we establish the fact that -- even though the decompositions themselves may differ -- the number of Zariski chambers always equals the number of simple Weyl chambers.

On log minimal models and Zariski decompositions II

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.

A simple proof for the existence of Zariski decompositions on surfaces

In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.

On the Fujita-Zariski decomposition on threefolds

We prove that, on a smooth threefold, pseudoeffective divisors with closed and one-dimensional diminished base locus have birationally a Fujita-Zariski decomposition.

Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle

We study the divisorial Zariski decomposition on varieties whose first Chern class is zero. We first prove that any exceptional divisor is contractible (up to a birational map that is an isomorphism in codimension one). We then characterize prime exceptional divisors on holomorphic symplectic manifolds.

On Zariski Decomposition with and without support

In this paper we study Zariski Decomposition with support in a negative definite cycle, a variation introduced by Y. Miyaoka. We provide two extensions of the original statement, which was originally meant for effective $\Q$-divisors: we can either state it for any $\Q$-divisor, or we can take the support to be in any cycle. Ultimately, we present a new approach to Zariski Decomposition of pseudo-effective $\Q$-divisors, which consists in iterating Zariski Decomposition with support.

On the boundedness of the denominators in the Zariski decomposition on surfaces

Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a 'universal denominator' that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded N…

具有整数Zariski分解的射影K3曲面

Integral Zariski decompositions on smooth projective surfaces II

本文通过一个显式的有限步终止算法，刻画了所有使得每个整数伪有效除子都具有整数Zariski分解的射影K3曲面。

On the semiampleness of the positive part of CKM Zariski decompositions

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.

Nakayama-Zariski 分解与翻转的终止性

Nakayama-Zariski decomposition and the termination of flips

本文证明在伪有效射影对中，假设关于Nakayama-Zariski分解在极小模型程序操作下行为的自然猜想，一次翻转序列的终止性蕴含所有翻转的终止性。

Okounkov bodies and Zariski decompositions on surfaces

The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Secondly, we study how the shapes of Okounkov bodies change as we vary the divisors in the big cone.

某些乘积三维簇上的Fujita-Zariski分解

Fujita-Zariski decompositions on some product threefolds

本文通过显式爆破和双有理Fujita-Zariski分解，计算三类乘积三维簇上盒积极化的Newton-Okounkov体及大但非nef除子的体积。

An integral version of Zariski decompositions on normal surfaces

We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and P…

反典范极小模型与 Zariski 分解

Anticanonical minimal models and Zariski decomposition

本文将 Birkar-Hu 关于 lc 配对极小模型存在性的 Zariski 分解判据平行推广到 pklt 配对，证明了反典范除子容许双有理 Zariski 分解时反典范极小模型存在。

关于不可约辛流形的Boucksom-Zariski分解与负数有界性

On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

本文研究了不可约辛流形上Boucksom-Zariski分解的性质，并证明了在光滑情况下负数有界猜想的有效版本，从而得出关于辛流形上大线包的有效应性结论。

Weyl and Zariski chambers on projective surfaces

Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

Explicit computations of Zariski decompositions on P_Z^1

In this note, we observe several properties of arithmetic divisors on the projective line over Z and give their Zariski decompositions.

Divisorial Zariski decomposition and algebraic Morse inequalities

In this note we use the divisorial Zariski decomposition to give a more intrinsic version of the algebraic Morse inequalities.

关于势三元组的极小模型程序与Zariski分解

On minimal model program and Zariski decomposition of potential triples

本文研究了由对与伪有效除子构成的势三元组的性质，证明若除子允许双有理Zariski分解则可关联广义对结构并运行广义MMP，同时证明了pklt对在特定分解条件下极小模型的存在性。

Higher dimensional Zariski decompositions

Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $α$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of $α$. The negative part of $α$ is then defined as the real effective divisor $N(α)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $α$ along $D$, and we get in that way a divisorial Zariski decomposition of $α$ into the sum of a class $Z(α)$ which is nef in codimension 1 and the class of its negative part $N(α)$, which is exceptio…

Orthogonality of divisorial Zariski decompositions for classes with volume zero

We show that the orthogonality conjecture for divisorial Zariski decompositions on compact Kahler manifolds holds for pseudoeffective (1,1) classes with volume zero.

Zariski decompositions on arithmetic surfaces

In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.

Boucksom-Zariski 与 Weyl 腔室在不可约全纯辛流形上的研究

Boucksom-Zariski and Weyl chambers on irreducible holomorphic symplectic manifolds

受 Bauer 等人启发，本文为射影不可约全纯辛流形的大锥建立腔室分解，并研究其与 Weyl 腔室及体积函数的关系，最终确定伪有效锥结构。

When are Zariski chambers numerically determined?

The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer-Funke on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on Enriques surfaces.

Zariski decomposition: a new (old) chapter of linear algebra

In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential concept is purely within the realm of linear algebra. In this paper, we formulate Zariski decomposition as a theorem in linear algebra and present a linear algebraic proof. We also sketch the geometric context in which Zariski first introduced his decomposition.

Pseudo-effective cones of projective bundles and weak Zariski decomposition

In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition…

$P$-平凡MMP、Zariski分解与广义配对的极小模型

$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

本研究建立$P$-平凡MMP理论，证明满足特定Nakayama-Zariski分解及数值有效条件的广义klt配对存在极小模型，并推广至三维广义lc配对，拓展了相关领域的经典结论。

非阿基米德Green函数与Zariski分解

Non-Archimedean Green's functions and Zariski decompositions

本文研究了非阿基米德Monge-Ampère方程，通过几何构造给出了反例以否定Burgos Gil等人的问题，并提出利用该理论高效编码伪有效数值类的Zariski分解。

Divisorial Zariski Decomposition and some properties of full mass currents

Let $α$ be a big class on a compact Kähler manifold. We prove that a decomposition $α=α_1+α_2$ into the sum of a modified nef class $α_1$ and a pseudoeffective class $α_2$ is the divisorial Zariski decomposition of $α$ if and only if $\operatorname{vol}(α)=\operatorname{vol}(α_1)$. We deduce from this result some properties of full mass currents.

Minimal singular metrics of a line bundle admitting no Zariski-decomposition

We give a concrete expression of a minimal singular metric of a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski-decomposition even after any proper modifications. As an application, we discuss the Zariski-closedness of non-nef loci and the openness conjecture of Demailly and Kollár in this class.

Zariski F-decomposition and Lagrangian fibration on hyperkähler manifolds

For a compact hyperkähler manifold X, we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a (holomorphic) Lagrangian fibration. We also prove that any sequence of D-flops between projective hyperkähler manifolds terminates after finite steps.

Higher Dimensional Elliptic Fibrations and Zariski Decompositions

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian.

Restricted volumes and divisorial Zariski decompositions

We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in the line of Boucksom's work. It enables us to define the restricted volume of a transcendental class on a compact Kähler manifold in natural way. The relation can be extended to a transcendental class.

Counting Zariski chambers on Del Pezzo surfaces

Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. In the present paper we present an algorithm that allows to effectively determine Zariski chambers when the negative curves on the surface are known. We show how the algorithm can be used to compute the number of chambers on Del Pezzo surfaces.

相对Nakayama-Zariski分解与广义对的最小模型

Relative Nakayama-Zariski decomposition and minimal models of generalized pairs

本文证明了相对Nakayama-Zariski分解的基本性质，并将其应用于研究lc广义对，证明了由丰富除子极化的lc广义对的log极小模型或Mori纤维空间的存在性。

Mobile Product and Zariski decomposition

We explain the relationship between $α_{1}...α_{q}$ (standard cohomology product) and $(α_{1}...α_{q})$ (mobile intersection product) of pseudo-effective classes $α_{1},...,α_{q}$ on a compact K$\ddot{a}$hler manifold. We also show how to use this relationship for proving some holomorphic Morse inequalities. Then we prove a result concerning the direct image of Lelong numbers under a modification in dimension 3, deriving a continuity property for the Lelong numbers of the wedge of $(1,1)-$currents.

Big arithmetic divisors on the projective spaces over Z

This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on P_Z^1". In this paper, we observe several properties of a certain kind of an arithmetic divisor D on the n-dimensional projective space over Z and give the exact form of the Zariski decomposition of D on the projective line over Z. Further, we show that, if n>=2 and D is big and non-nef, then, for any birational morphism f: X --> P^n_Z of projective, generically smooth and normal arithmetic varieties, we can not expect a suitable Zariski decomposition of f^*(D). We also give a concrete constru…

The Zariski Cancellation Problem and related problems in Affine Algebraic Geometry

In this article, we shall discuss the solution to the Zariski Cancellation Problem in positive characteristic, various approaches taken so far towards the possible solution in characteristic zero, and several other questions related to this problem.

A note on a smooth projective surface with Picard number 2

We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108].

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Ca…

Equimultiplicity of families of map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$

In 1971, Zariski proposed some questions in Theory of Singularities. One of such problems is the so-called, nowadays, Zariski's multiplicity conjecture. In this work, we consider the version of this conjecture for families. We answer positively Zariski's multiplicity conjecture for a special class of non-isolated singularities.

On Zariski's multiplicity conjecture

We discuss some features of the so-called Zariski's multiplicity problem especially the application of the work of A'Campo on the zeta function of a monodromy of an isolated singularity of a complex hypersurface to the problem.

Variations on the theme of Zariski's Cancellation Problem

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.

Multiplicity of complex hypersurface singularities, Rouche' satellites and Zariski's problem

Soient $f,g\colon (\hbox{åC}^n,0) \to (\hbox{åC},0)$ des germes de fonctions holomorphes réduits. Nous montrons que $f$ et $g$ ont la même multiplicité en 0 si et seulement s'il existe des germes réduits $f'$ et $g'$ analytiquement équivalents à $f$ et $g$, respectivement, tels que $f'$ et $g'$ satisfassent une inégalité du type de Rouché par rapport à un `petit' cercle générique autour de~0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicité et une réponse partielle positive à celle--ci.

On Zariski's multiplicity problem at infinity

We address a metric version of Zariski's multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the bi-Lipschitz homeomorphism has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In p…

A note on Jets of Entire Curves in Semi-Abelian Varieties

We prove a product decomposition of the Zariski closure of the jet lifts of a holomorphic map f from C into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of such a map f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of some articles which claimed that such a product decomposition is always true.

平面分支的Zariski模问题的构造性解

Constructive solution of Zariski's Moduli Problem for Plane Branches

该研究以算法方式计算不可约全纯平面曲线芽的Kähler微分集合，讨论其值半模的多种生成概念并给出对应基的几何性质与算法构造，得到平面分支Zariski模问题的显式构造性解。

Inseparable local uniformization

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.

The Analytic Classification of Plane Branches

In this paper we give a solution to Zariski's problem of analytic classification of plane branches.

关于阿诺尔德余秩、扎里斯基重数与切米尔诺纤维问题

On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems

本文研究阿诺尔德余秩等问题，证明余秩模2的拓扑不变性，得到三维阿诺尔德余秩问题的另一证明，还证明相关弱版本扎里斯基重数猜想与欧拉特征等价，给出广义半比雷斯科尔恩-法姆超曲面重数>2为右拓扑等价不变量。

The generalized Lipman-Zariski problem

We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x \in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$, the sheaf $Ω_X^{[p]}$ of reflexive differential $p$-forms is free. Does this imply that $(x \in X)$ is smooth? We give an example showing that the answer is no even for $p = 2$ and $X$ a terminal threefold. However, we prove that if $p = n - 1$, then there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. As an application, we show that if $X$ is a projecti…

Sextics with singular points in special position

In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The difference in the topology of their complements can only be detected via finer invariants or techniques. In our case the generic braid monodromies, the fundamental groups, the characteristic varieties and the existence of dihedral coverings of $\PP^2$ ramified along them can be used to distinguish the two sextics. Our intention is not only to use different methods a…