Complements on log surfaces

Lectures on complements on log surfaces

The aim of these notes is to explain main ideas of the theory of complements. Basically we will follow Shokurov's work alg-geom/9711024.

On effective log Iitaka fibrations and existence of complements

We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_X$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a gene…

Approximating delta invariants in the sense of complements of plt type

In this note, we will show that delta invariant of a log Fano pair can be approximated by lc places of complements of plt type if it is no greater than one. Give a log Fano pair with delta invariant no greater than one, under the assumption that delta invariant of the log Fano pair can be approximated by lc places of bounded complements of plt type, we show the existence of the divisorial valuation computing delta invariant of this log Fano pair.

对数 Calabi-Yau 三维簇的补集有界性

Boundedness of complements for log Calabi-Yau threefolds

本研究针对系数集为[0,1]的对数Calabi-Yau型簇，证明三维簇对若存在局部klt的ℝ-补集，则对应有限正整数集中的某个n有n-补集，同时证得ℝ-互补曲面对的补集有界性。

ACC for minimal log discrepancies of exceptional singularities

We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Γ_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.

带有受限分量类的光滑曲面上可约曲线补集的基本群

Fundamental groups of the complements to reducible curves on smooth surfaces with restricted classes of components

本文分类光滑单连通射影曲面上满足对偶簇次数≤6等奇异曲线束族，进而描述不可约分量为超平面截影类且基本群可满射到秩>6自由群的曲面上曲线补集的基本群。

关于曲面上补集的注记

Remark on complements on surfaces

本文给出任意维数例外对奇点的显式刻画，证明例外 Fano 曲面是 1/42-lc，并由此首次给出曲面的 n-补集和 Tian alpha 不变量的显式上界。

Complements on surfaces

The main result of the paper is a boundedness for $n$-complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.

Strong $(δ,n)$-complements for semi-stable morphisms

We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fibe…

非一般型补集的无理直纹曲面上的曲线

Curves on irrational ruled surfaces whose complements are of non-general type

研究无理直纹曲面上曲线补集的对数Kodaira维数，给出其等于K_X+B的Iitaka维数的证明，并分类维数小于2时的曲线配置，以及维数为1时对数多重典范系统的结构。

Decomposition of log crepant birational morphisms between log terminal surfaces

We prove that every log crepant birational morphism between log terminal surfaces is decomposed into log-flopping type divisorial contraction morphisms and log blow-downs. Repeating these two kinds of contractions we reach a minimal log minimal surface from any log minimal surface.

On homological mirror symmetry for the complement of a smooth ample divisor in a K3 surface

We introduce a conjecture on homological mirror symmetry relating the symplectic topology of the complement of a smooth ample divisor in a K3 surface to algebraic geometry of type III degenerations, and prove it when the degree of the divisor is either 2 or 4.

曲面上 Fano 纤维化的 klt 补的有界性

Boundedness of klt complements on Fano fibrations over surfaces

本文证明 Birkar–Shokurov 关于 ε-lc 对局部有效 Cartier 除子补的猜想等价于 Shokurov 的 Fano 纤维化 klt 补有界性猜想，并在二维情形加以证实，从而推出曲面上 Fano 纤维化的 klt 补有界性。

Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space

In this article we prove that every entire curve in the complement of a generic hypersurface of degree $d\geq 586$ in $\mathbb{P}_{\mathbb{C}}^{3}$ is algebraically degenerate i.e there exists a proper subvariety which contains the entire curve.

The fundamental group of the complement of the branch curve of T times T in C^2

This paper is the second in a series of three papers concerning the surface T times T, where T is a complex torus. We compute the fundamental group of the branch curve of the surface in C^2, using the van Kampen Theorem and the braid monodromy factorization of the curve.

The Sarkisov program on log surfaces

We show that the Sarkisov program holds for $\mathbb{Q}$-factorial log surfaces and log canonical surfaces over any algebraically closed field.

Minimal model theory for log surfaces

We discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

Elliptic Gorenstein singularities, log canonical thresholds and log Enriques surfaces

We point out an interesting relation between hypersurface elliptic singularities and log Enriques surfaces: with a few exceptions, every hypersurface elliptic singularity define some klt log Enriques surface $(S,Diff)$. In many cases, the log canonical cover of $(S,Diff)$ is a Du Val K3 surface with ``large'' nonsymplectic automorphism group.

Classification of log Enriques surfaces with delta=1

Log Enriques surfaces with delta=1 are classified.

Log-plurigenera in stable families of surfaces

We study the flatness of log-pluricanonical sheaves on stable families of surfaces.

对数典范Fano簇的弱特殊测试构型

Weakly special test configurations of log canonical Fano varieties

该研究针对严格对数典范Fano簇，证明补的所有lc位置均为dreamy型，并建立了(X,-K_X)的弱特殊测试构型与补的lc位置的对应关系，丰富了对数典范Fano簇的相关理论。

The classification of log Enriques surfaces of rank 18

Log Enriques surface is a generalization of K3 and Enriques surface. We will classify all the rational log Enriques surfaces of rank 18 by giving concrete models for the realizable types of these surfaces.

Classification of exceptional log Del Pezzo surfaces with delta=1

The exceptional log Del Pezzo surfaces with delta=1 are classified.

带不可约边界的对数K3曲面

Log K3 surfaces with irreducible boundary

该研究针对带不可约边界的开对数K3曲面，完成了其自同构群的确定工作，为对数K3曲面的自同构性质研究提供了明确结果。

Ideal-adic semi-continuity of minimal log discrepancies on surfaces

We prove the ideal-adic semi-continuity of minimal log discrepancies on surfaces.

On logarithmic reduction of cohomologically tame elliptic surface

We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth morphisms.

Classification of log Enriques surfaces with delta=2

Log Enriques surfaces with delta=2 are classified.

Classification of log del Pezzo surfaces of index three

A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all of the log del Pezzo surfaces of index three. The technique for the classification based on the argument of Nakayama.

Log del Pezzo Surfaces of Rank 2 and Cartier Index 3 with a Unique Singularity

Log del Pezzo surfaces play the role of the opposite of surfaces of general type. We will completely classify all the log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity.

Log del Pezzo surfaces with simple automorphism groups

In this paper we consider del Pezzo surfaces with only log terminal singularities admitting an action of a finite simple group.

四次曲面补集中的代数环面

Algebraic tori in the complement of quartic surfaces

本文研究四次曲面补集中的代数环面，证明其存在等价于核心正则度为零，并分类此类簇型对，给出环面簇上log Calabi-Yau对的簇型判据。

对数曲面的丰沛定理

Abundance theorem for log surfaces

该论文在不假设Q-因子性或对数典范性的条件下，证明了对数曲面的丰沛定理，推广了已有结果。

$\mathbb{A}^1$ curves on log K3 surfaces

In this paper, we study $\mathbb{A}^1$ curves on log K3 surfaces. We classify all genuine log K3 surfaces of type II which admits countably infinite $\mathbb{A}^1$ curves.

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.

Minimal model theory for log surfaces in Fujiki's class $\mathcal C$

We establish the minimal model theory for $\mathbb Q$-factorial log surfaces and log canonical surfaces in Fujiki's class $\mathcal C$.

Effective basepoint-free theorem for semi-log canonical surfaces

This paper proposes a Fujita-type freeness conjecture for semi-log canonical pairs. We prove it for curves and surfaces by using the theory of quasi-log schemes and give some effective very ampleness results for stable surfaces and semi-log canonical Fano surfaces. We also prove an effective freeness for log surfaces.

具有四个log终端奇点的Del Pezzo表面

Del Pezzo surfaces with four log terminal singularities

本文研究了Picard数为一且具有四个log终端奇点的Del Pezzo表面的分类问题。

Log del Pezzo surfaces with large volumes

We classify all of the log del Pezzo surfaces $S$ of index $a$ such that the volume $(-K_S^2)$ is larger than or equal to $2a$.

K3 surfaces and log del Pezzo surfaces of index three

We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give. Our methods include the definition of right resolutions of quotient singularities of index three and some analysis of automorphism-stable elliptic fibrations on K3 surfaces. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three.

On log surfaces

This paper is an announcement of the minimal model theory for log surfaces in all characteristics and contains some related results including a simplified proof of the Artin-Keel contraction theorem in the surface case.

On minimal model theory for algebraic log surfaces

We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.

Log Enriques surfaces of index 7 and type $A_{15}$

We show that there is only one log Enriques surface of index 7 and type $A_{15}$.

Finite generation of adjoint ring for log surfaces

We prove the finite generation of the adjoint ring for $\mathbb{Q}$-factorial log surfaces over any algebraically closed field.

关于解析环境下log曲面的最小模型理论的评论

Remarks on the minimal model theory for log surfaces in the analytic setting

本文研究了复曲面在解析环境下log最小模型理论，证明了极小模型程序、充足性定理及log canonical环的有限生成性。

The maximal number of singular points on log del Pezzo surfaces

We prove that a del Pezzo surface with Picard number one has at most four singular points.

Log rationally connected surfaces

In this paper, combining the works of Miyanishi-Tsunoda and Keel-McKernan, we prove the log Castelnuovo's rationality criterion for smooth quasiprojective surfaces over complex numbers.

Braid Group, Algebraic Surfaces and Fundamental Groups of complements of Branch Curves

This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely $\tilde B_n$ which plays an important role in the description of fundamental groups of complements of branch curves.It turns out that all suc…

Log del Pezzo surfaces with not small fractional indices

For a log del Pezzo surface $S$, the fractional index $r(S)\in\mathbb{Q}_{>0}$ is the maximum of $r$ with which $-K_S$ can be written as $r$ times some Cartier divisor. We classify all the log del Pezzo surfaces $S$ with $r(S)\geq 1/2$, after the technique of Nakayama.

Bounding the volumes of singular weak log del Pezzo surfaces

We give an optimal upper bound for the anti-canonical volume of an $ε$-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

Abundance theorem for semi log canonical surfaces in positive characteristic

We prove the abundance theorem for semi log canonical surfaces in positive characteristic.

Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.

热带曲面上中等位置热带曲线的补集

The complement of tropical curves in moderate position on tropical surfaces

本文针对紧致热带流形上的热带 Cartier 除数，引入中等位置概念，提出 Riemann-Roch 数等于补集拓扑欧拉示性数的猜想，并在二维且具有 Delzant 面结构的情形下给出证明与推广。

紧曲面上半正反正则除子补集的$\overline{\partial}$上同调

$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface

本文研究具有半正曲率反正则线丛的紧复曲面上，其反正则除子补集的$\overline{\partial}$上同调群$H^1(M,\mathcal{O}_M)$的结构与性质。

On Gorenstein log del Pezzo Surfaces

In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two Gorenstein log del Pezzo surfaces. Finally we present the complete list of the singularity type of the relatively minimal Gorenstein log del Pezzo surface and the number of the isomorphism classes with t…