Jungkai A. Chen, Meng Chen
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\leq 24$. A direct consequence is the birationality of the pluricanonical map $φ_m$ for all $m\geq 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $ν(3)\geq \frac{1}{63\cdot 126^2}$.
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We determine birational superrigidity for a quasi-smooth prime Fano 3-fold of codimension 4 with no projection centers. In particular we prove birational superrigidity for Fano 3-folds of codimension 4 with no projection centers which are recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano 3-folds.
K-stability and birational rigidity of Fano 3-fold weighted complete intersections
本研究证明了双有理刚性的良构拟光滑法诺三维加权完全交具有K-稳定性。
We give some birationality criteria for $ϕ_m$ (m=4,5,6,7) on general type 3-folds with $p_g\geq 2$ by means of an intensive classification.
We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.
Birational geometry of terminal quartic 3-folds. I
这是该论文的未删节网络版本，将发表在《美国数学杂志》上。本文研究了某些轻度奇异四次3重的例子的双有理几何。一个四次3重是Fano簇的一个例子，即一个具有充足反正则层$Ø_X(-K_X)$的簇$X$。从双有理几何的观点来看，它们基本上属于两类：要么$X$ "接近有理“，然后它有非常多的双正则不同的双有理模型作为Fano 3-fold，或者，在另一个极端，$X$有一个唯一的模型，而且$X$的每一个双有理自映射通常都是双正则的。在本文中，我们构造的例子奇异四次3-folds正好有两个双有理模型Fano 3-folds。
This is a continuation of a series of papers studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consists of birationally rigid varieties and $14$ families consists of birationally birigid varieties. The aim of this paper is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.
In this paper we study 14 families among 85 families of anticanonically embedded Q-Fano threefolds weighted complete intersections of codimension 2 and show that every quasismooth member is birationally birigid, that is, it is birational to another Q-Fano threefold but not birational to any other Mori fiber space.
Birational Calabi-Yau 3-folds and BPS state counting
本文包括三角范畴上的Bridgeland-Douglas稳定性条件和Joyce关于计算半稳定对象不变量的工作在双有理几何研究中的一些应用。我们引入了motivic Gopakumar-Vafa不变量作为D2-膜的计数不变量的概念，并证明了它们在Calabi-Yau 3-folds之间的双有理变换下是不变的。其结果类似于双有理Calabi-Yau 3-folds具有相同的贝蒂数或Hodge数。
Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.
For nonsingular projective 3-folds $X$ (of general type) whose geometric genus $p_g=h^0(X, K_X)$ is $\geq 5$, the birationality of the fourth canonical map $φ_{4,X}=Φ_{|4K_X|}$ was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of $φ_{4,X}$ for those $X$ with $p_g=4$.
Toroidalization of birational morphisms of 3-folds
我们证明了在特征为零的域上，通过对域和目标上的点和非奇异曲线进行一系列的爆破，可以使射影3-折叠的双有理态射成为环面。
Birational geometry of terminal quartic 3-folds II: the importance of beings Q-factorial
本文研究了终四次三重的双有理几何。在这样做，我开发了一种新的方法来研究具有正维中心的极大奇点。这使得我们可以确定一个Q-阶乘四次二重点的可柔性，它显示了Q-阶乘性在双有理几何中的重要性
In this paper we consider Q-Fano 3-fold weighted complete intersections of codimension 2 in the 85 families listed in the Iano-Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to an another explicitly described Mori fiber space. As a consequence, 19 families are proved to be birationally rigid and the remaining 66 families are proved to be birationally nonrigid.
We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.
There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.
On the birational geometry of toric Fano 4-folds
我们宣布一个因式分解结果的等变双有理态射之间的环面4倍，其源是法诺：这样的态射总是一个复合材料的爆破沿光滑不变中心。此外，我们用一个反例表明，与三维情况不同，即使源和目标都是Fano，中间品种也不能选择Fano。
Birational Geometry of 3-fold Mori Fibre Spaces
我们研究了P^2上的3重锥丛和P^1上的三次del Pezzo纤维化的地理学和双有理几何。我们讨论了许多明确的例子，并提出了一些开放的问题。该文件已提交2002年10月在都灵举行的“法诺会议”。
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $ε$-CY type form a birationally bounded family for $ε>0$. Moreover, we show that the set of $ε$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
Strong Toroidalization of Birational Morphisms of 3-Folds
本文证明了3重双有理态射的强超环面化。设f：X \to Y是非奇异完全3重的双有理态射，D_Y，D_X是Y和X上的单正规交叉因子，使得f^{-1}（D_Y）=D_X，且D_X包含态射f的奇异轨迹.本文证明了存在态射Φ：X_1 \to X和Φ：Y_1 \to Y，它们是D_Y的原像所支持的点和非奇异曲线的爆破的乘积，并且与原像作简单的正规交叉，使得f_1= \circ \circ Φ_1是环面态射。这个定理推广了我们在“3-folds的双有理态射的Toroidization”中证明的Toroidization定理。
Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {\it pluricanonical section index} $δ(V):=\text{min}\{m|P_m\geq 2\}$ since $1\leq δ(V)\leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $δ(V)\geq 13$ and an intensive geometrical investigation to those with $δ(V)\leq 12$, we prove that $\text{Vol}(V) \geq \frac{1}{1680}$ and that the pluricanonical map $Φ_{m}$ is birational for all $m \geq 61$, which greatly improves known results. An optimal birationality of $Φ_m$ for the case $δ(V)=2$…
The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…
Classification of elliptic and K3 fibrations birational to some Q-Fano 3-folds
本文给出了椭圆和K3纤维化对某些轻度奇异复Fano 3-folds的双有理的完全分类。对于一个例子给出了详细的证明，即加权P^4中30次的一般超曲面X的例子，权值为1，4，5，6，15;但是我们的方法适用于更一般的情况。为了构造从X到椭圆和K3纤维化的双有理映射，我们使用Kawamata blowup和Mori理论来计算反典范环;为了排除其他可能的纤维化，我们仔细检查（X，（1/n）H）的严格典范奇异性，其中H是与假定的双有理映射相关的线性系，n是其反典范度。
The Curve Exclusion Theorem for elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces
在标题中提到的定理是一个技术性的结果，需要分类的椭圆和K3纤维化双有理Fano 3重超曲面在加权射影空间。我们提出了曲线排除定理的完整证明，该定理首次出现在作者未发表的博士论文[Ry02]中，并且此后一直依赖于纤维化分类问题的许多情况的解决方案。我们给出这些解决方案的例子，并简要讨论它们。
Explicit birational geometry of 3-folds, Foreword
这是A. Corti和M. Reid，CUP Jun 2000，ISBN：0 521 63641 8，with papers by K. Altmann，A. Corti，A.R. Iano-Fletcher，J. Kollár，A.V. Pukhlikov and M.里德其中一个主要成就的代数几何在过去20年的工作森和其他扩展极小模型和恩里克-小平分类3倍。这本书是一个综合套件的文件围绕应用森理论双有理几何。四个文件（那些由Pukhlikov，弗莱彻，科尔蒂，和长期联合文件科尔蒂，Pukhlikov和里德）工作的详细理论的双有理刚性的法诺3倍，这些贡献的工作首次与一个代表性的一类法诺品种，3倍超曲面在加权射影空间，并包括一个有吸引力的介绍性治疗和丰富的详细计算的特殊情况。
Birational rigidity of quartic three-folds with a double point of rank 3
本文证明复射影空间中仅含秩3二重点的一般四次三重流形具有双有理刚性，其双有理自映射群为25个2阶循环群的自由积，并由此推出刚性阶乘四次流形补集的余维数至少为3。
In this note we present two equivalent definitions for the genus of a birational map X --> Y between smooth complex projective 3-folds. The first one is the definition introduced in 1973 by M. A. Frumkin, the second one was recently suggested to me by S. Cantat. By focusing first on proving that these two definitions are equivalent, one can obtain all the results of the paper of Frumkin in a much shorter way. In particular, the genus of an automorphism of $\mathbb{C}^3$, view as a birational self-map of the projective space, will easily be proved to be 0.
Factoriality and birational rigidity of two families of singular quartic three-folds
本文研究两个三维四次折面家族的阶乘性和双射刚性，分析其双射自映射群。
Birational complexity of log Calabi-Yau 3-folds
本文研究log Calabi-Yau 3-折叠的双有理复杂度，证明其取值于{0,2,3}，并构造具有到射影空间收缩的crepant双有理模型，通过标准$\mathbb{P}^1$-链的几何刻画实现。
The relative pluricanonical stability for 3-folds of general type
本文改进了JanosKollar的一个定理。对给定的一般型复射影三重映射X，设p_k（X）为多亏格\ge 2，Kollar证明了（11 k +5）-标准映射是双有理的.这里我们证明了（7 k +3）-正则映射或（7 k +5）-正则映射是双有理的，并且对于m \ge 13 k +6，m-正则映射是稳定双有理的.若P_k（X）\ge 3，则m-正则映射对m \ge 10 k +8是稳定双有理的.特别地，当p_g（X）\ge 2时，12-正则映射是双有理的，当p_g（X）\ge 3时，11-正则映射是双有理的。
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementar…
Explicit birational geometry of determinantal quartic 3-folds
研究$\mathbb{P}^4$中一般线性行列式四次三维流形的双有理几何，发现其族包含有理$\mathbb{Q}$-factorial成员并可特化为有理非$\mathbb{Q}$-factorial双二次曲面，且几何由$\mathbb{P}^3$中10次11亏格曲线控制。
Invariants in equivariant birational geometry
本文研究了等变理性几何中不变量的性质与应用。
The Grassmannian Gr(2,5) is embedded in $\Bbb{P}^9$ via the Plücker embedding. The intersection of two general PGL(10)-translates of Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y, deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain "birational" Torelli statement for…
Bernardi and Tirabassi show the existence of full strong exceptional collections consisting of line bundles on smooth toric Fano $3$-folds under assuming Bondal's conjecture, which states that the Frobenius push-forward of the structure sheaf $\mc O_X$ generates the derived category $D^b(X)$ for smooth projective toric varieties $X$. In this article, we show Bondal's conjecture for smooth toric Fano $3$-folds and also improve their result, using birational geometry.
We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
Birational geometry of Deligne-Mumford stacks
本文研究 Deligne-Mumford 栈的双有理几何，并定义了新的双有理不变量。
Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.
We study linear actions of finite groups in small dimensions, up to equivariant birationality.
Birational Calabi--Yau n-folds have equal Betti numbers
设X和Y是C上具有平凡标准线丛的n维光滑射影代数簇X和Y。本文利用局部数域上代数簇的p-adic分析方法证明了：当X和Y是双有理的时，它们具有相同的Betti数。
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
Exercises in the birational geometry of algebraic varieties
这是一个大约100个练习的集合。它可以作为一个补充书Kollár-森：双有理几何的代数簇。
Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces.
We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have nonvanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.
Derived Categories and Birational Geometry
2005年在西雅图的AMS夏季研究所的调查文章。
Birational geometry of actions on del Pezzo surfaces
本文完成有限群在del Pezzo表面的正则一般自由作用的分类，解决了若干等变理查德几何中的开放问题。
Birational geometry of Fano double covers
证明了一般位置的Fano双超曲面的除子正规性。
This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.
Birational geometry of defective varieties
在这里，我们研究的双有理几何的任意维数的射影簇有缺陷的高割线品种。我们应用经典的工具切向投影，我们确定自然条件uniruledness，合理的连接，合理性。
Birational geometry
这篇文章包含双有理几何研究生课程的笔记，重点是极小模型程序。涵盖的主题包括奇点，消失，非消失，锥和收缩，基点自由，有限生成，翻转，终止，极小模型和森纤维空间。
Witt rings, Pfister forms, and equivariant birational geometry
该研究针对带有限群作用的二次型的等变双有理几何，发展了Witt群与Pfister理论的等变类似物，拓展了双有理几何的研究范畴。
We determine the birational geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification, by looking at the locus of non-locally-free sheaves on the relevant moduli space.
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。
We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.
Motivic measures and stable birational geometry
本文从稳定双有理几何的角度研究了代数簇的运动Grothendieck群。特别地，我们得到了M的一个猜想的反例。Kapranov论动机zeta函数的合理性。
Explicit birational geometry of threefolds of general type
设$V$是一般型的复非奇异射影3重。本文证明了$P_{12}(V)>0$和$P_{24}(V)>1$（它回答了J.Kollar和S. Mori）。我们还证明了正则体积有一个普适下界$\text{Vol}(V) \geq 1/2660$，并且对于所有$m\geq 77$，多正则映射$Φ_m$在其象上是双有理的.作为该方法的一个应用，我们证明了关于具有终端商奇点的加权超曲面3重的Fletcher猜想。另一个特征性结果是在所有具有$χ({\mathcal O}_V)\leq 1$的3倍$V$中的最佳下界$\text{Vol}(V)\geq {1/420}$。
Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $m\geq 73$ and that the canonical volume $\text{Vol}(V)\geq {1/2660}$. When $χ(\mathcal{O}_V)\leq 1$, our result is $\text{Vol}(V)\geq {1/420}$, which is optimal. Other effective results are also included in the paper.
In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.
Linearizability notions in equivariant birational geometry
本文讨论了有限群作用在代数簇上的双有理性质、线性化、torsors和versality。
Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.
Birational Geometry of Rationally Connected Manifolds via Quasi-lines
这主要是一个调查的结果有关的双有理几何的合理连接的流形，使用有理曲线类似的线在${\mathbb P}^n$（{ \it quasi-lines}）。得到了${\mathbb P}^n$中直线的Zapriki邻域的各种特征，其中有些是新的.此外，形式几何的方法适用于推导结果的双有理性质。
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