Boundedness of log terminal Fano pairs of bounded index
James McKernan
我们证明了一个猜想的Batryev指出，家庭的所有Fano品种与kawamata日志终端奇异性和固定的指标，形成一个有界的家庭。
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.
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We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $ε$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $ε$-Fano type.
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。
Some boundedness results for Fano-like Moishezon manifolds
我们证明了Fano流形上光滑blow-down个数的有限性和非射影Fano流形几何的有界性结果。我们的证明使用适当的希尔伯特计划和森理论。
Boundedness of log Fano pairs with certain K-stability
本文证明具有特定K稳定性的对数Fano对的界性结果，特别是Maeda型K半稳定对数Fano对构成对数有界族，并计算了若干例子的K半稳定域。
Boundedness of log Calabi-Yau pairs of Fano type
证明了klt对$(X,B)$的有界性结果，使得$K_X+B\equiv 0$和$B$是大的.因此，我们得到了一个积极的答案，有效Iitaka纤维化猜想的klt对大边界。
We show boundedness of $3$-folds of $ε$-Fano type with Mori fibration structures. The proof is based on the birational boundedness result in our previous work arXiv:1509.08722 combining with arguments in Kawamata \cite{K} and Kollár--Miyaoka--Mori--Takagi \cite{KMMT}.
An optimal boundedness on weak $\bQ$-Fano threefolds
设$X$是一个终端弱$\bQ$ -Fano三重图.我们证明了$P_{-6}(X)>0$和$P_{-8}(X)>1$ .我们还证明了反正则体积有一个普适下界$-K_X^3 \geq 1/330$ .这个下界是最优的。
We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $ε$-klt log Fano pair of dimension three is bounded from above.
Boundedness of complements for fibered Fano threefolds in positive characteristic
本文在正特征大特征情形下证明了Fano型纤维化的典范丛公式，并解决了具有纤维化结构的Fano型三维配对的Shokurov补集有界性猜想，特别覆盖了$-(K_X+B)$为nef且非big的情形。
Deformations of weak Fano 3-folds with only terminal singularities
本文证明了任何只有终端奇点的$\Bbb Q$ -阶乘弱Fano 3-重都是光滑的.
Terminal Fano threefolds and their smoothings
我们研究了法诺的三重平滑。我们证明了皮卡德数保持不变的情况下，终端Gorenstein奇点。
We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.
Given $ε>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ is bounded from above.
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $Φ(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $α(X,B)^{d-1}\mathrm{vol}(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.
This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive.
We show the existence of $(ε,n)$-complements for $(ε,\Rr)$-complementary projective generalized pairs of Fano type $(X,B+M)$ when either the coefficients of $B$ and $μ_j$ belong to a finite set or the coefficients of $B$ belong to a DCC set and $M'\equiv 0$, where $M'=\sumμ_jM_j'$ and $M_j'$ are b-Cartier nef divisors.
We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
We completely classify the Q-factorial terminal toric Fano three-folds such that the sum of the squared torus invariant prime divisors is non-negative.
We continue the classification of terminal Fano threefolds with an effective two-torus action. In earlier work we settled the Q-factorial case with Picard number one. Here we treat the larger class of varieties that do not admit any contraction of a prime divisor; these are called combinatorially minimal.
Rational curves on Fano threefolds with Gorenstein terminal singularities
本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。
Boundedness of weak Fano threefolds with fixed Gorenstein index in positive characteristic
该研究在特征p>5的不可数代数闭域上，通过引入Gorenstein指数、补与Kodaira型零化条件，部分证明了三维簇的BAB猜想，同时给出了射影三维簇上nef大可逆层Seshadri常数的一致下界。
Kawamata boundedness for Fano threefolds and the Graded Ring Database
本研究实现了法诺三维簇的川又有界性，给出39550个候选希尔伯特级数列表，为法诺三维簇分类提供初步近似，明确了相关数据的含义及与分类计划的关联。
Boundedness of Fano threefolds with log-terminal singularities of given index
证明了具有对数终端奇点的Fano三重方程的有界性定理。这是我们早期结果的改进，在那里我们还要求簇是Q阶乘的，Picard数为1。本文的创新之处在于：1.利用Alexeev极小模型程序，在适当的边界条件下，求出水平极值收缩。2.使用Kollár的有效基点自由度定理。3.利用Kawamata关于极值曲线长度的结果，在适当的边界条件下，避免了某些情况下曲线的粘连。
In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singulari…
Terminalizations of quotients of Fano varieties of lines on cubic fourfolds
本文研究了立方四维流形上线的Fano簇商的终端化，计算了相关拓扑不变量，并识别出新的四维不可约解析辛流形变形类。
Boundedness theorem for Fano log-threefolds
本文的主要目的是证明所有具有有界指标的对数终端奇点的Fano三重性族是有界的。
Kawamata-Miyaoka-type inequality for $\mathbb Q$-Fano varieties with canonical singularities II: Terminal $\mathbb Q$-Fano threefolds
本文证明了 Fano 指数至少为 $3$ 的终端 $\mathbb Q$-Fano 三维簇的最优 Kawamata-Miyaoka 型不等式，并应用得到所有终端 $\mathbb Q$-Fano 三维簇满足 $c_1(X)^3 < 3c_2(X)c_1(X)$。
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。
The defect of Fano 3-folds
本文研究了终末Gorenstein Fano 3褶皱的缺陷。我确定了皮卡德秩1的终端Gorenstein Fano 3折叠缺陷的界限，该折叠不包含平面。给出了四次3-折叠的一般界，并指出了如何研究Picard秩为1的含有平面的终端Gorenstein Fano 3-折叠的亏损。
On terminal Fano 3-folds with 2-torus action
本文通过反正则复形推广环面法诺簇与格多胞体的对应关系，分类了具有二维环面作用且皮卡数为一的终端法诺三维流形，并给出终端性与典范性的简单刻画。
Deformations of Q-Calabi-Yau 3-folds and Q-Fano 3-folds of Fano index 1
本文证明了任何只具有普通终端奇点的Q-Calabi-Yau 3-fold和任何只具有终端奇点的Fano指数为1的Q-Fano 3-fold都有Q-smoothing。
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.
We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…
Boundedness of spherical Fano varieties
我们证明了对任意e>0，只存在1000个固定维数的e-log终球Fano簇.我们还引入了一个不变量的球形子群H在一个约化群G的措施如何好的等变Fano紧化G/H存在。
Terminal Fano four folds in low codimension
本文构造并分类了低余维数下含至多孤立轨形点的指标为1的良态拟光滑终端 Fano 4-folds，利用反典范嵌入将其分为余维2至4子簇，共分类出95族第一类型与32族第二类型，并强调了计算机代数的关键作用。
Characterizing terminal Fano threefolds with the smallest anti-canonical volume
本文刻画了反典范体积达到最小下界 1/330 的非有理 Q-阶乘终端法诺三维簇，证明其为 P(1,5,6,22,33) 中次数 66 的加权超曲面。
Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.
On finite generation and boundedness of adjoint foliated structures
本文证明了一般型klt代数可积伴随叶状结构存在良好极小模型、特定Fano代数可积伴随叶状结构构成有界族，并给出环境簇为Fano型的应用，分别类比典范环有限生成性与Fano簇有界性结果。
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.
We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of singular points of certain Fano threefolds with~terminal factorial singularities.
On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$
本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。
We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.
On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two
本文证明Picard秩为2的弱$\mathbb{Q}$-Fano terminal三维簇的反典范体积上界为72，并确定等号成立的唯一情形，为一般canonical Fano三维簇的最优上界问题提供了策略。
Fano-Mori contractions of high length on projective varieties with terminal singularities
设X是一个具有终端奇点的射影簇，L是X上的一个充分Cartier因子.证明了：如果f是一个与极值射线$ R \subset \bar {NE(X)}$相关联的双有理压缩，使得R.（K_X+（n-2）L）<0，则f是光滑点的加权blow-up。然后，我们分类与极值射线R相关联的除数收缩，使得R。（K_X+rL）<0，其中r是非负有理数，并且f的纤维具有小于或等于r+1的维数。
Sharp anticanonical volume bounds for Gorenstein Fano fourfold of Picard number one
本文证明终端和典范 Gorenstein Fano 四折叠的反典范体积上界，并给出尖锐界及特殊情形下的改进。
Boundedness of n-complements on surfaces
本文在无重数限制且不假设Fano型的广义情形下，证明了曲面对的n-补有界性。
Boundedness of $ε$-log Canonical Complements on Surfaces
证明了V. V. Shokurov关于曲面上$ε$ -log正则补的有界性的一个猜想.作为应用，我们给出了弱对数Fano曲面有界性的一个新的证明.
Toric Fano 3-folds with terminal singularities
本文对具有终端奇点的复曲面Fano 3-折叠进行了分类。这是通过解决等价的组合数学问题来实现的，即在GL（3，Z）的作用下，找到Z ^3中所有以原点为唯一非顶点格点的凸多胞体。
We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
On deformations of Q-Fano threefolds
本文研究了只具有终端奇点的Q-Fano 3-fold的形变理论。首先，我们证明了Q-Fano 3-fold的Kuranishi空间是光滑的。其次，我们证明了每个只具有“普通”终端奇点的Q-Fano 3-fold是Q-smoothable的，也就是说，它可以变形为只具有商奇点的Q-Fano 3-fold。最后，我们证明了Q-光滑的Q-Fano 3-倍假设存在的DuVal anticanonical元素。作为应用，我们得到了具有Du Val反正则元的准素Q-Fano 3-折叠的亏格界.
We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an…
Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II
本文证明具有最小反典范体积 1/330 且皮卡德数为 1 的 Q-阶乘终端法诺三维簇必为 P(1,5,6,22,33) 中的 66 次加权超曲面，并将同类刻画推广到 Iano-Fletcher 列表中的另外 11 个加权超曲面例子。
By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical weak $\mathbb{Q}$-Fano $3$-fold $V$, we show that there exists a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, being birational to $V$, such that the $m$-th anti-canonical map defined by $|-mK_{X}|$ is birational for all $m\geq 52$. As an intermediate result, we show that for any $K$-Mori fiber space $Y$ of a canonical weak $\mathbb{Q}$-Fano $3$-fold, the $…
On the Picard Number of Fano 3-Folds with Terminal Singularities
本文证明了如下主要结果：设X是一个具有终端Q-阶乘奇点的Fano 3-fold，且X的Mori多面体不存在小的极值射线和Kodaira维数为1或2的面。则Picard数ρ（X）< 8。
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.
We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.
On deformations of Q-Fano threefolds II
本文研究了$3$维终端奇点的余边界映射，这在研究奇异$3$-褶皱的形变中具有重要意义.我们证明了这个地图消失，只有商奇点和$A_{1,2}/4$-奇点，即终端奇点解析同构的$\mathbb{Z}_4$-商的奇点$ (x^2+y^2 +z^3+u^2=0)$。作为应用，我们证明了具有终端奇点的$\mathbb{Q}$-Fano $3$-折叠可以变形为只具有商奇点和$A_{1,2}/4$-奇点的折叠.我们也处理$\mathbb{Q}$-光滑性问题的$\mathbb{Q}$-Calabi-Yau $3$-折叠。
Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
利用Riemann-Roch定理讨论了定维最小对数偏差的有界性问题。在将其归结为Gorenstein终端奇点的情形后，首先证明了当重数或嵌入维数有界时，其最小对数偏差有界。其次，我们恢复的Gorenstein终端三重奇异里德的特性，其最小的日志差异的精确边界Markushevich，没有明确的分类。最后给出了一般超平面截面具有一个终端段的四重奇点的精确边界。
It is conjectured that the base varieties of the Iitaka fibrations are bounded when the Iitaka volumes are bounded above. We confirm this conjecture for Iitaka $ε$-lc Fano type fibrations.
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