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

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.

Logarithmic deformations of normal crossing Enriques surfaces in characteristic two

Working in characteristic two, I classify nonsmooth Enriques surfaces with normal crossing singularities. Using Kato's theory of logarithmic structures, I show that such surfaces are smoothable and lift to characteristic zero, provided they are d-semistable.

Calabi-Yau广义费马流形的商与对数Enriques簇

Quotients of Calabi-Yau Generalized Fermat manifolds and Log Enriques varieties

本文通过刻画Calabi-Yau型广义费马流形的无不动点自同构商，构造了对数Enriques簇与Enriques流形，证明其为已知两类Calabi-Yau型Enriques流形族中一类的子族，且与另一类的第二Betti数不同。

A 1-dimensional family of Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1

We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (-2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.

On the complete classification of extremal log Enriques surfaces

We show that there are exactly, up to isomorphisms, seven extremal log Enriques surfaces Z and construct all of them; among them types D_{19} and A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X_3 or X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface with Picard number 20 and discriminant 3 (resp. 4), which are called the most algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by Shioda-Inose and Vinberg).

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or Enriques surface to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface to be birationally k-very ample and birationally k-spanned, and relate these concepts to the Clifford index and gonality of smooth curves in |L|.

A note on stability of syzygy bundles on Enriques and bielliptic surfaces

In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable with respect to $L$ for any ample and globally generated line bundle $L$ on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex Enriques surfaces improves a result of Torres-López and Zamora by removing a condition on Clifford index. Together with the results of Camere and Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample and globally generated line bundle $L$ on any smooth minimal co…

Exceptional linear systems on curves on Enriques surfaces

The main purpose in this paper is to study the gonality, the Clifford index and the Clifford dimension on linearly equivalent smooth curves on Enriques surfaces. The method is similar to techniques of M.Green $\&$ R.Lazarsfeld and G.Pareschi.

基变换型Enriques曲面及有理椭圆曲面上的Severi簇

Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces

本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。

Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds

We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational connectedness conjecture in [KoMiMo] which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group (now a Theorem of S. Takayama).

Analytic torsion for log-Enriques surfaces and Borcherds product

We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac{1}{4}(1,1)$. The moduli space of such log-Enriques surfaces with $k$ singular points is a modular variety of orthogonal type %of dimension $10-k$ associated with a unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed as a function on the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$. As a consequence, th…

Log-canonical threshold for curves on a smooth surface

It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.

Automorphisms of K3 surfaces and their applications

This paper is a survey about $K3$ surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at "Mathematical structures of integrable systems and their applications" held at Research Institute for Mathematical Sciences in September 2018.

Quotients of K3 Surfaces Modulo Involutions

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each of the three extremal cases where X^g contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2. Our approach is more geometrical rather than lattice-theoretical (see Nikulin's paper for the latter approach).

Algebraic surfaces with quotient singularities - including some discussion on automorphisms and fundamental groups

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 o…

Log D-modules and Index theorems

We study log D-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah's generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

K-stability of log del Pezzo hypersurfaces with index 2

We completely classify K-stability of log del Pezzo hypersurfaces with index 2.

Some Results about the Index Conjecture for log Calabi-Yau Pairs

We show some inductive statements for the index conjecture for log canonical Calabi-Yau pairs. Using it, we show that boundedness of log canonical index for log canonical Calabi Yau pairs with rational DCC coefficients in dimension 3. We also show boundedness of log canonical index for klt log Calabi-Yau pairs in dimension 4 with $B\neq 0$.

Boundedness of log terminal Fano pairs of bounded index

We prove a conjecture of Batryev which states that the family of all Fano varieties with kawamata log terminal singularities and fixed index, forms a bounded family.

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.

Index 1 covers of log terminal surface sigularities

We shall investigate index 1 covers of 2-dimensional log terminal singularities. The main result is that the index 1 cover is canonical if the characteristic of the base field is different from 2 or 3. We also give some counterexamples in the case of characteristic 2 or 3. By using this result, we correct an error in a previous paper.

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.

Equations of log del Pezzo surfaces of index 2

Del Pezzo surfaces over C with log terminal singularities of index \le 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to describe del Pezzo surfaces of index \le 2 by equations in some weighted projective space. Obtained results are natural completions of similar results of Du Val, and Hidaka and Watanabe, describing del Pezzo surfaces of index 1.

On isolated log canonical singularities with index one

We give a method to investigate isolated log canonical singularities with index one which are not log terminal. Our method depends on the minimal model program. One of the main purposes is to prove that our invariant coincides with Ishii's Hodge theoretic invariant.

复杂度为零且指数任意的对数 Calabi--Yau 对

Log Calabi--Yau pairs of complexity zero and arbitrary index

本文刻画了复杂度为零且指数任意的对数 Calabi--Yau 对，并证明双有理复杂度为零的对数 Calabi--Yau 对存在 crepant 双有理模型，即广义 Bott 塔。

关于固定 Gorenstein 指数簇上的极小对数差异

On minimal log discrepancies on varieties with fixed Gorenstein index

本文推广了对数典范阈值聚点的有理性定理，并将其应用于极小对数差异的 ACC 问题，证明了固定 Gorenstein 指数簇上具有固定系数的三维典范对的极小对数差异满足 ACC。

余正则性为零的对数卡拉比-丘对的指数

Index of coregularity zero log Calabi-Yau pairs

本文研究余正则性为0的对数卡拉比-丘对的指数，证明其Weil指数λ满足2λ(K_X+B)~0，给出精确上界，并推广到广义对数卡拉比-丘对、半对数典范对及孤立对数典范奇点等情形，还应用于Gross-Siebert纲领与带有限群作用的卡拉比-丘簇。

A supplement to Fujino's paper: On isolated log canonical singularities with index one

Let $E$ be the essential part of the exceptional locus of a good resolution of an isolated, log canonical singularity of index one. We describe the dimension of the dual complex of $E$ in terms of the Hodge type of $H^{n-1}(E, O_E)$, which is one of the main results of the paper [1] of Fujino. Our proof uses only an elementary classical method, while Fujino's argument depends on the recent development in minimal model theory.

Boundedness of Fano threefolds with log-terminal singularities of given index

We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The new ideas of the paper include the following. 1. Using Alexeev Minimal Model program with suitable boundary to find horizontal extremal contractions. 2. Using Kollár's effective Base Point Freeness theorem. 3. Using Kawamata's result on the length of extremal curves with suitable boundary to avoid gluing curves in some cases.

具有固定指数的有理连通对数 Calabi–Yau 对的双有理有界性

Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index

本文研究固定维数的有理连通 klt 对数 Calabi–Yau 对，证明在反对数典范类的固定 Cartier 指数与 nef 条件下，相应射影簇的集合模 flops 有界。

非klt情形下Shokurov指标猜想的一个反例

A non-klt counterexample to Shokurov's index conjecture

该研究针对非klt情形的Shokurov指标猜想，构造了反例，为相关代数几何问题提供了否定性依据。

Classification of Toric log Del Pezzo Surfaces having Picard Number 1 and Index $\le 3$

Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18 surfaces of this kind.

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 $2a$.

叶状曲面奇点的补集、指标定理与极小对数差异

Complements, index theorem, and minimal log discrepancies of foliated surface singularities

本文将配对与簇的若干结果推广到叶状曲面对，证明局部补集有界性、局部指标定理及mlds一致有界性，否定Cascini-Spice的两个问题，并恢复叶状曲面mlds升链条件猜想的证明。

On the combinatorial classification of toric log del Pezzo surfaces

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index l. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all l<17 is obtained.

Classification of log del Pezzo surfaces of index $\le 2$

This is an expanded version of our work [AN88], 1988, in Russian. We classify del Pezzo surfaces over C with log terminal singularities of index \le 2. By classification, we understand a description of the intersection graph of all exceptional curves on an appropriate (somewhat stronger than minimal) resolution of singularities together with the subgraph of the curves which are contracted to singular points. The final results are similar to classical results about classification of non-singular del Pezzo surfaces and use the usual finite root systems. However, the intermediate considerations u…