The 5-canonical system on 3-folds of general type
Jungkai A. Chen, Meng Chen, De-Qi Zhang
设$X$是具有典型奇点的一般型射影极小Gorenstein 3-折叠.证明了5-正则映射在其象上是双有理的。
Let $X$ be a projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.
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Addendum to "Termination of 4-fold canonical flips"
《四重典型翻转的终结》中的加权难度定义存在一些错误。在本文中，我们描述了这些错误以及如何纠正它们。无论如何，任何四重典型翻转序列都终止。
Termination of 4-fold canonical flips
我们证明了4重典型翻转的终止性。
We introduce K3 transitions as a geometric approach to studying canonical 3-folds. These transitions link different deformation classes of canonical 3-folds via a combination of birational contractions and smoothings. As applications, we investigate some basic properties of the web of canonical 3-folds in small codimension and give an interesting example of a singularity in codimension 4 with obstructed smoothings, similar to the famous example of the affine cone over a degree 6 del Pezzo surface.
We obtain that the nonzero accumulation points of the set of 3-fold canonical thresholds $\ct(X,S)$ are precisely $1/k$ where $k\ge 2$ is an integer and $S$ is an effective integral divisor of a projective 3-fold $X$ with only terminal singularities. Moreover, we generalize the ascending chain condition for the set of 3-fold canonical thresholds to pair.
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.
Let $X$ be a Gorenstein minimal projective $n$-fold with at worst locally factorial terminal singularities, and suppose that the canonical map of $X$ is generically finite onto its image. When $n<4$, the canonical degree is universally bounded. While the possibility of obtaining a universal bound on the canonical degree of $X$ for $n \geqslant 4$ may be inaccessible, we give a uniform upper bound for the degrees of certain abelian covers. In particular, we show that if the canonical divisor $K_X$ defines an abelian cover over $\mathbb{P}^n$, i.e., when $X$ is an \emph{abelian canonical $n$-fol…
We study canonically polarized Gorenstein $3$-folds with at most terminal singularities and satisfying $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$. We characterize the canonical maps of such $3$-folds, describe a structure theorem for the locally factorial ones and completely classify the smooth ones. New examples of canonically polarized smooth $3$-folds with $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$ are constructed. These examples are natural extensions of those constructed by M.~Kobayashi.
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.
We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T 0$, where $n$ is an uniform natural number. This means that every $3$-fold of log Calabi-Yau type can be turned into a log Calabi-Yau pair in an effective way.
We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $κ=2$.
We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.
A sharp lower bound for the canonical volume of 3-folds of general type
设V是一般类型的光滑射影3重。用$K^3$表示，一个有理数，V的任何极小模型的正则层的自相交。定义$K^3$为$V$的正则体积。假设为$p_g\ge 2$。我们发现$K^3\ge {1/3}$，这是一个尖锐的下界。然后，我们将那些具有小$K^3$的V分类到显式结构。我们也给出了一些新的例子与$p_g=2$，具有最大的典型稳定性指标。最后给出了某些代数4-折叠的应用。
Fibrations on four-folds with trivial canonical bundles
根据和乐群的性质，将平凡典范丛的四重分为六类。我们考虑的例子是由阿贝尔曲面包围的射影平面。我们在六个类中的五个类中构造了这样的纤维化，并证明了在第六类中不存在这样的纤维化。我们对类属纤维是亏格为2的曲线的雅可比矩阵的纤维化进行了分类。
For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $φ_{-m}$ is birational onto its image for all $m\geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $φ_{-m}$ is birational onto its image for all $m\geq 97$.
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 $p_g=4$.
On Log Canonical MMP for Kähler 3-Fold
该数学研究证明对数极小模型程序适用于仅含klt奇点的紧Kähler三维簇对应的$\u211a$-阶乘lc对，推进了复几何领域相关理论发展。
Canonical stability of 3-folds of general type with $p_g\geq 3$
研究了一般型光滑投影3重$V$的正则稳定性.证明了：（1）$|5K_V|$在几何亏格为$p_g\geq 4$的条件下给出到其象上的双有理映射;（2）$|6K_V|$在几何亏格为$p_g=3$的条件下给出到其象上的双有理映射.已知的例子表明，两者都是最优的。这一事实可以被看作是平行的表面情况下，虽然人们知道很少的一般类型的3倍与$p_g\leq 1$。
We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.
On the pluricanonical map and the canonical volume of projective 4-folds of general type
本文研究了一般型非奇异射影4-重的多重典范映射双有理性的有效界及典范体积下界，证明了当m≥76m0+77时φ_m双有理，并给出Vol(V)的下界，改进了Meng Chen的早期结果。
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 $…
Let $V$ be a nonsingular projective 3-fold of general type. When the pluricanonical section index $δ(V)>12$, Chen-Chen \cite{Chen3} has a complete list of the possibility for the weighted basket ${\mathbb B}(V)$. However the possibility of ${\mathbb B}(V)$ could be infinite in the situation $δ(V)\leq 12$, which is the main challenge to the classification. In this paper we mainly study the case with $δ(V)=12$ and show that $Vol(V)\geq \frac{31}{48048}$, which improves the corresponding result of Chen--Chen \cite[Prop. 4.9(5)]{Chen3}.
The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.
Rational points on 3-folds with nef anti-canonical class over finite fields
本文利用极小模型程序在完美域上建立了广义log Calabi-Yau 3-折叠对的结构结果，并证明了在特征p>5且基数q>19的有限域上，具有丰反典范类与负Kodaira维数或具有平凡典范类且非零第一Betti数的几何整光滑3-折叠存在有理点。
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。
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.
Families of stable 3-folds in positive characteristic
本文研究了正特征下稳定3折族的模空间问题，发现其无法形成适当模空间，并得到了log canonical 4折对的性质。
The Deformation Space of Calabi-Yau n-folds with canonical singularities can be obstructed
本文给出了一个简单的例子，证明了任意维的Calabi-Yaus族具有一维的标准奇点，其Kuranishi空间是奇异的。因此，Bogomolov-Tian-Todorov无障碍定理对具有正则奇点的Calabi-Yaus不成立。
Arithmetic and geometric deformations of 3-folds
该研究证明特征$p>5$完美剩余域下三维典范/终端奇点的混合与等特征小形变保持奇点类型，探讨其在三维Fano簇与极小模型族的应用，结果依赖四维簇对数解消存在性。
Motivated by previous computations of Y. Cao, M. Kool and the author, we propose square roots and sign rules for the vertex and edge terms that compute Donaldson-Thomas invariants of a toric Calabi-Yau 4-fold, and prove that they are canonical, exploiting the combinatorics of plane and solid partitions.
We construct some new deformation families of four-dimensional Fano manifolds of index $1$ in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anti-canonical embedding in some weighted projective space. The constructed families have relatively smaller anti-canonical degrees than most other known families of smooth Fano 4-folds.
Quartic 3-fold: Pfaffians, instantons and half-canonical curves
一般的四次3重X允许一个7维表示族作为线性型的8 × 8反对称矩阵的Pfiran。这提供了X上秩为2的向量丛的7维模空间M。给出了X中亏格为15的14维半标准曲线族C的精确几何描述，使得上述向量丛由C通过Serre构造得到.证明了这类因子通过M的Abel-Jacobi映射以及由此得到的从M到中间Jacobi的映射是拟有限的。特别地，M的每个分量都具有非负的Kodaira维数。讨论了小陈类秩2向量丛的其他构造，证明了X上瞬子的最小可能电荷为4。
The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to th…
We show that there exists a positive real number $δ>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-δ$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.
In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general type, namely so called Kanev surfaces and Todorov surfaces. We show how properties of our examples of surfaces could be derived directly from properties of some polytopes and we compute the singularities of their canonical models.
We find an explicit upper bound for the anticanonical volume of Fano 4-folds with canonical singularities.
A canonical Fano threefold has Fano index $\leq 66$
该研究证明典范弱法诺三维簇的ℚ-法诺指数≤66，正面回答程熙的三维猜想，用到新黎曼-罗赫公式、川又-宫冈型不等式及秩2叶状结构几何等，明确最优上界。
The index of a 3-fold canonical singularity at a crepant centre is at most 6.
The sharp lower bound for the volume of 3-folds of general type with χ(\Co{X})=1
设$V$是一般型光滑射影3重。用一个有理数$K^{3}$表示$V$的任何极小模型的规范层的自交。将$K^{3}$定义为$V$的规范体积。本文证明了$K^{3}\ge {1/420}$的一个精确的下界，并给出了一个例子：$X_{46}\subseteq \mathbb{P}(4,5,6,7,23)$ .
Mori fibre spaces for $3$-folds over imperfect fields
研究特征\(p>5\)的函数域上射影三维空间，证明典范除子非伪有效时LMMP有效，过程中表明此情形下三维空间对数典范阈值的ACC完全成立。
We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…
On minimal varieties growing from quasismooth weighted hypersurfaces
本文研究小典范体积极小簇的构造，通过建立加权超曲面加权爆破的有效nef准则构造了多族极小3-折叠，并证明了一般型极小n-折叠典范体积的有效下界，证明了其在低维最优、高维近似最优。
We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 5$ and all $P\in \text{Pic}^0(X)$. These results are also optimal.
We show that, for nonsingular projective 4-folds V of general type with geometric genus $p_g\geq 2$, the 33-canonical map is birational onto the image and the canonical volume has the lower bound $1/520$, which improves a previous theorem by Chen and Chen.
We prove that, for any nonsingular projective irregular 3-fold of general type, the 6-canonical map is birational onto its image.
Three-dimensional toric morphisms with anti-nef canonical divisors
本文将Gorenstein环面3-折叠的环面双有理态射分类到具有较充分反典范除子的三维仿射空间上。
We classify the irreducible components of the space of foliations on Fano 3-folds with rank one Picard group. As a corollary we obtain a classification of holomorphic Poisson structures on the same class of 3-folds.
Termination of pseudo-effective 4-fold flips
本文证明特征零代数闭域上对数典范四维簇的任意伪有效翻转移除序列必然终止。
On explicit birational geometry for minimal n-folds of canonical dimension n-1
本文研究了典范纤维化极小射影n-叠的典范体积下界与稳定性指数上界，证明了$i=1$时的具体最优值公式，并指出该方法适用于所有典范维数。
The Noether inequality for threefolds and three moduli spaces with minimal volumes
本文建立了几何亏格5≤p_g≤10的一般型射影三维簇的诺特不等式，解决三维簇诺特不等式所有剩余情形，并证明三个特定极小体积典范三维簇的粗模空间是不可约单有理簇且成员单连通。
A lifting principle for canonical stability indices of varieties of general type
本文证明了一般型光滑射影簇典范稳定性指标序列的提升原理，将 $r_n$ 与高维、大典范体积簇的指标联系起来，并给出多典范映射双有理性的等价体积判据。
On K-stability of singular hyperelliptic Fano 3-folds
本文研究了具有特定奇点和反canonical线性系性质的Fano 3折面的K稳定性问题。
Let $(X,Δ)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $Δ$ an effective $\mathbb{Q}$-divisor, and $K_X+Δ$ nef. We give a non-vanishing criterion for $K_X+Δ$ in dimension $n$ with $X$ uniruled, assuming various conjectures of LMMP in dimensions (up to) $n-1$ or $n$, and a semi-ampleness criterion in the irregular case. In particular, we obtain that if $X$ is a uniruled $4$-fold, then $κ(K_X+Δ)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+Δ$ is semi-ample.
Let $X$ be a nonsingular projective $n$-fold $(n\ge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $κ$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $κ$ is $0$, then the Chern ratios $(\frac{c_{2,1^{n-2}}}{c_{1^n}}, \frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \cdots, \frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for al…
A canonical Fano threefold has degree $\leq 72$
本文证明典范弱 Fano 三维簇的反典范体积上界为 72，且该上界最优。
We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold Kähler-Einstein metric and are K-stable. As an application, we give examples of super-rigid affine Fano 4-folds.
The canonical volume of threefolds of general type with $χ<1$
证明了一般型射影3-折叠的典范体积$K^3\geq {1/30}$与$χ(\mathcal{O})\leq 0$。这个边界很锋利.
An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds
本文证明了典范弱 $\mathbb{Q}$-Fano 三维流形的反典范体积上界为 $324$，给出了有效估计。
We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type over a curve, $c_1^3<18c_3$.
GW/PT-correspondence for CY 4-folds-I: Canonical Orientations
该文为非紧复解析Calabi-Yau 4-流形构建模叠的典范定向理论，得到所有PT对模空间的典范族定向，为证明GW/PT对应奠定基础。
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.
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