Cylinders in Fano varieties

Fano簇的刚性性质

Rigidity properties of Fano varieties

我们概述了最近的一些结果Fano品种的证据，他们的刚性性质下的小变形。

一类Fano簇的半单量子上同调

Semisimple quantum cohomology of some Fano varieties

给出了Picard秩为1的Fano簇的量子上同调半单性的充分条件。我们应用这些技术来证明新的半简单性结果的一些Fano品种的皮卡德秩1和大指数。我们也给出了Fano簇具有非半单小量子上同调但具有半单大量子上同调的例子。

半球Fano簇的伪指数

The pseudo-index of horospherical Fano varieties

证明了L.Bonavero，C.卡萨格兰德岛Debarre和S. Druel，关于光滑Fano品种的伪指数，在horospheric品种的特殊情况下。

低秩Fano对称簇

Fano symmetric varieties with low rank

根据Akhiezer的一个经典结果，秩为1的对称射影簇都是光滑的和Fano的。我们分类的局部阶乘（分别光滑）的射影对称$G$ -品种的秩2是Fano。当$G$是半单的，我们也分类的局部阶乘（分别顺利）射影对称$G$ -品种秩2，这只是准法诺。此外，我们分类的Fano对称$G$ -品种的秩3可从一个奇妙的品种的爆破沿$G$ -稳定品种的序列。最后，我们对任意秩的Fano对称簇进行了分类，这些簇可以通过沿闭轨道的一系列爆破从一个奇妙簇得到。

K-stability of Fano varieties: an algebro-geometric approach

We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.

Fano型变种的特征化

A characterization of Fano type varieties

本文通过证明Fano型变种的特征化，揭示了其几何结构的核心属性。

具有大度自同态的Fano簇

Fano Varieties with Large Degree Endomorphisms

我们给出了Picard数为1的Fano簇$X$的例子，这些簇具有终端奇点，并且允许度大于1的自同态。

Picard数1的Q-Fano簇的有界性

Boundedness of Q-Fano varieties of Picard number one

证明了对于每个$n$，Picard数为1且维数为$n$的Q-Fano簇的双有理有界性.我们不使用森理论。

奇异Fano簇的双有理几何

Birational geometry of singular Fano varieties

证明了具有初等奇点的一般位置的Fano超曲面和对偶空间的除子标准性。

曲线具有整除收缩的Toric Fano簇

Toric Fano varieties with divisorial contractions to curves

在本文中，我们得到了一个完整的分类光滑复曲面Fano品种配备极值收缩因子曲线的任何维。作为应用，我们得到了沿曲线等变爆破到Fano的光滑投射环面簇的一个完全分类。

$\mathbb{Q}$ -Fano品种一致K稳定性的评价准则

A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties

给出了$\mathbb{Q}$ -Fano簇一致K-稳定的一个简单的充要条件.

Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields

In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields.

三次四重Fano簇的极化辛自同构的分类

Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds

我们对光滑三次四重Fano簇的极化辛自同构进行了分类（配备Plücker极化），并研究了其不动轨迹.

Fano复曲面簇中Calabi-Yau完全交的镜像对称性

Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties

推广Batyrev和Borisov的自反多面体和nef划分的概念，我们提出了一个镜像对称的Fano复曲面簇中的Calabi-Yau完全相交。

Computation and data in the classification of Fano varieties

Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called Q-Fano threefolds. The classification of Q-Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.

关于有效因子在数值上最终自由的Fano簇

On Fano varieties whose effective divisors are numerically eventually free

本文对具有最大Picard数且其有效因子在数值上最终自由的轻度奇异Fano簇进行了分类。

关于Fano簇的Landau-Ginzburg模型

On Landau--Ginzburg models for Fano varieties

我们观察到一种方法找到弱朗道-金斯堡模型的法诺品种，并找到他们光滑法诺三倍的属9，10和12。

On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{φ_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.

Topics on Fano varieties of K3 type

This is a survey paper about a selection of recent results on the geometry of a special class of Fano varieties, which are called of K3 type. The focus is mostly Hodge-theoretical, with an eye towards the multiple connections between Fano varieties of K3 type and hyperkähler geometry.

族中Fano簇的不变量

Invariants of Fano varieties in families

我们证明了Picard秩在Fano簇族中是常数（任意特征），并进一步研究了指数的恒定性。

1-Fano品种的周期

1-Cycles on Fano varieties

我们证明结果1-循环的某些Fano品种使用的技术，依赖于有理曲线。首先，我们证明了指数大于维数一半的Fano加权完全交具有平凡的第一Griffiths群。其次，证明了指数足够高的$\mathbb{P}^n$（$3$除外）中的大多数$2$-Fano加权完全交和$2$-Fano锥连通簇的第一Chow群都是由直线生成的.此外，如果线的Fano簇是不可约的，则第一Chow群同构于$\mathbb{Z}$。

Fano varieties of K3 type and IHS manifolds

We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.

On deformations of toric Fano varieties

In this note we collect some results on the deformation theory of toric Fano varieties.

具有$\mathrm{Nef}(X)=\mathrm{Psef}(X)$和$ρ(X)=\mathrm{dim}\, X-1$的Fano品种

Fano varieties with $\mathrm{Nef}(X)=\mathrm{Psef}(X)$ and $ρ(X)=\mathrm{dim}\, X-1$

我们对轻度奇异Fano簇$X$进行分类，使得$\mathrm{Nef}(X)=\mathrm{Psef}(X)$和$X$的Picard数等于$X$减去$1$的维数.

具有大Seshadri常数的Fano簇

Fano varieties with large Seshadri constants

我们表明，一组Fano品种（任意奇点）的反正则因子有大的Seshadri常数满足一定的弱和双有理有界性。我们还分类奇异Fano品种的尺寸$n$的anticanonical因子的Seshadri常数至少$n$，推广了刘和作者的早期结果。

一类三次函数的Fano簇上的代数圈

Algebraic cycles on Fano varieties of some cubics

本文研究光滑三次五重线上的Fano簇的圈论性质。这些论点是基于这样一个事实，即这种法诺品种有限维动机。我们还提出了一些结果，关于周群的Fano品种线的某些立方在其他方面。

Toric Sarkisov links of toric Fano varieties

We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.

On the Chow ring of Fano varieties on the Fatighenti-Mongardi list

Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.

具有许多自映射的Fano簇

Fano varieties with many selfmaps

我们研究具有端商奇异性的反正态嵌入准光滑加权Fano三重超曲面上的全局对数正则阈值，以证明其中大多数超曲面上存在卡勒-爱因斯坦度量，并给出具有双有理自同构无限离散群的Fano簇的例子。

来自奇异Fano簇的K3结构

K3 structures from singular Fano varieties

本文综述了寻找K3型Fano簇的成果，并探讨了奇异簇对发现新K3结构的意义。

反标准环与Fano型簇

On the anti-canonical ring and varieties of Fano type

本文研究了投射簇的反正则环，并根据这些模型的奇异性研究了对数Fano型的投射簇。

Gorenstein球面Fano簇

Gorenstein spherical Fano varieties

我们得到的Gorenstein球面Fano品种的组合描述的某些多面体，推广Gorenstein环面Fano品种的组合描述的自反多面体和它的扩展Gorenstein horosepherical Fano品种由于Pasquier。利用这种描述，我们证明了任意$d$ -维$\mathbb{Q}$ -阶乘Gorenstein球面Fano簇的Picard群的秩由$2d$有界.本文还包含一个概述的自然代表的anticanonical因子类的球形品种由于Brion的描述。

具有一束Fano双覆盖的有理刚性簇。III

Birationally rigid varieties with a pencil of Fano double covers. III

研究了Fano纤维空间$π\colon V\to {\mathbb P}^1$的双有理几何，其中纤维是指数为1的Fano双超曲面。对于这些品种的每一个家庭，我们要么证明双有理刚性或生产明确的非平凡结构的Fano纤维空间。本文发展了一种研究Fano纤维空间$V/{\mathbb P}^1$上可动系统的新的线性方法。

高维复曲面Fano簇的分类

Toward the classification of higher-dimensional toric Fano varieties

本文的目的是通过Batyrev的原始集合和原始关系的概念，给出非奇异环面Fano簇的分类的基本工具。利用它们，我们可以很容易地处理等变的blow-up和blow-down问题，并得到一个简单的判定给定的非奇异复曲面簇是否为Fano簇的准则。作为这些结果的应用，我们得到了Mori定理的一个环面形式，并且原则上可以对由一个非奇异环面Fano簇通过非奇异环面Fano簇的等变blow-up和blow-down的有限序列得到的所有非奇异环面Fano簇进行分类.特别地，我们得到了一个新的分类方法的非奇异环面Fano簇的维数最多为4。这些方法被扩展到Gorenstein环面Fano品种赋予自然决议的奇异性的情况下。特别地，我们很容易地得到一个新的方法，Gorenstein复曲面Fano曲面的分类。

关于完全交的Fano簇的注记

Notes on Fano varieties of complete intersections

Fano簇是格拉斯曼簇的子簇，其点参数化包含在给定投射簇中的线性子空间。这些简要说明给出了Fano各种完全交叉的结果，以期在机器学习中应用。为了使广大受众能够了解这些成果，将先决条件保持在最低限度。

具有小最小对数差异的例外Fano簇

Exceptional Fano varieties with small minimal log discrepancy

本文构造了加权射影空间中的良构超曲面作为例外Fano簇，其最小对数差异随维度双指数衰减，并在二维达到最优值。

On toric geometry and K-stability of Fano varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.

On the Chow ring of Fano varieties of type $S6$

Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.

Gorensteinness and iteration of Cox rings for Fano type varieties

We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.

Fano型簇的反正则映射的因子分解

Factorization of anticanonical maps of Fano type varieties

本文的目的是推广酒井的工作anticanonical模型的理性曲面品种的Fano型。我们首先证明了一个特征的Fano型品种使用的奇异性anticanonical模型。其次，利用$K_X$ -极小模型程序研究了反正则映射的分解。

Horospherical 2-Fano 簇

Horospherical 2-Fano varieties

本文分类了皮卡德数1的2-Fano horospherical簇，回顾已知2-Fano流形，并探讨2-Fano条件与稳定性概念的关系。

Coregularity of Fano varieties

The regularity of a Fano variety, denoted by ${\rm reg}(X)$, is the largest dimension of the dual complex of a log Calabi--Yau structure on $X$. The coregularity is defined to be \[ {\rm coreg}(X):= \dim X - {\rm reg}(X)-1. \] The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of $X$. In this note, we review the history of Fano varieties, give some examples, survey some important theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.

Derived categories of centrally-symmetric smooth toric Fano varieties

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.

Fano簇的双有理刚性与域扩张

Birational rigidity of Fano varieties and field extensions

本文的目的是解决关于双有理刚性在代数闭域扩张中的行为的一些基本问题。

Q-abelian and $\mathbb Q$-Fano finite quotients of abelian varieties

We study finite quotients of abelian varieties (fqav for short) i.e. quotients of abelian varieties by finite groups. We show that Q-abelian varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors) are characterized by the existence of quasiétale polarized (or int-amplified) endomorphisms. We show that every fqav has a finite quasiétale cover by the product of an abelian variety and a $\mathbb Q$-Fano fqav. Using such coverings, we give a characterization of $\mathbb Q$-Fano fqav's, and show that $\mathbb Q$-Fano fqav's and Q-abelian varieties are ``building blocks'' of gen…

log $Q$-Fano簇的有理连通性

Rational connectedness of log $Q$-Fano varieties

本文对极小模型程序中的一个猜想给予肯定的回答。证明了dim $n$的log $Q$-Fano簇是有理连通的.我们还研究了正则丛在射影态射下的行为。

Complements on log canonical Fano varieties

In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of sdlt log Calabi-Yau varieties in dimension 2.

计算某些Fano簇的Gromov-Witten不变量

Computing Gromov-Witten invariants of some Fano varieties

对于具有一般驯服的半单量子（和小量子）（p，p）-型上同调的Fano簇，其第一Chern类是有效积分基因子的严格正组合，我们给出了一个从有限个初始不变量计算所有亏格为零的Gromov-Witten不变量的递归算法.

高度Fano品种

Fano varieties with high degree

对任意正整数$k$和任意足够大的整数$n$，构造了一个具有Picard数$k$和维数$n$的Fano簇$X$，使得$((-K_X)^n)^{1/n}$像$n^k/(\log n)^{k-1}$一样增长.

所有完整的交叉品种是法诺游客

All complete intersection varieties are Fano visitors

证明了光滑完备交簇的导出范畴等价于光滑射影Fano簇的导出范畴的一个满子范畴。这使我们能够定义一些新的不变量的光滑射影簇，并提出了许多有趣的问题。

具有Picard数为1的divisor的奇异Fano簇

On singular Fano varieties with a divisor of Picard number one

本文研究了具有Picard数1的有效素除子的轻度奇异Fano簇的几何。之后，我们解决复曲面品种的情况下。最后，我们讨论了余维为1的万有覆叠空间的极值压缩的提升。

Higher index Fano varieties with finitely many birational automorphisms

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivarian…

余指标3 Fano簇上的有理曲线

Rational Curves on Coindex 3 Fano Varieties

本文研究光滑余指标3 Fano簇上有理曲线的模空间结构，证明高维情形下每个有效数值类对应单个不可约分支，并利用Fujita的$a$-不变量刻画四维情形，从而验证几何Manin猜想。

具有大伪指数的Fano簇

On Fano varieties with large pseudo-index

设$X$是一个在最坏情况下具有孤立商奇点的Fano簇.我们的结果断言，如果$C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$对每一条曲线$C \subset X$，则$ρ_X=1$。