On the number of minimal models of a log smooth threefold

A smooth, complete threefold with no nontrivial nef line bundles

We present a smooth, complete toric threefold with no nontrivial nef line bundles. This is a counterexample to a recent conjecture of Fujino.

On Castelnuovo theory and non-existence of smooth isolated curves in quintic threefolds

We give some necessary conditions for a smooth irreducible curve $C\subset \mathbb{P}^4$ to be isolated in a smooth quintic threefold, and also find a lower bound for $h^1(\mathcal{N}_{C/{\mathbb{P}^4}})$. Combining these with beautiful results in Castelnuovo theory, we prove certain non-existence results on smooth curves in smooth quintic threefolds. As an application, we can prove Knutsen's list of examples of smooth isolated curves in general quintic threefolds is complete up to degree 9.

On Pseudo-Effectivity of the Second Chern Classes for Smooth Threefolds

We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata's Effective Non-vanishing Conjecture holds for such threefolds.

关于具有零共正则性的光滑Fano三维流形

On smooth Fano threefolds with coregularity zero

本文构造了次数为2、4、6、8的光滑三维Fano完全交的零共正则性实例，并结合已有定理证明所有光滑Fano三维流形族均含零共正则性元素。

On the Chern number inequalities satisfied by all smooth complete intersection threefolds with ample canonical class

We obtain all linear Chern number inequalities satisfied by any smooth complete intersection threefold with ample canonical bundle.

平滑素数Fano三维流形的另一种观点：关于度数为22且具有无限自同构群的分类

Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups

本文提供了一个自包含的替代证明，用于重新证明Kuznetsov、Prokhorov和Shramov关于度数为22且具有无限自同构群的平滑素数Fano三维流形的分类。

Smooth rationally connected threefolds contain all smooth curves

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.

光滑三维簇上的极小对数差异

Minimal log discrepancies on smooth threefolds

本文证明了光滑三维簇上极小对数差异的ACC，并推导出a-lc阈值ACC、一致m-adic半连续性及计算除子有界性等推论。

Abel-Jacobi maps associated to smooth cubic threefolds

We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by analyzing vector bundles on cubic threefolds. Our proofs use residuation rather than vector bundles.

皮卡秩为二的K-稳定光滑法诺三维流形

K-stable smooth Fano threefolds of Picard rank two

本文证明了皮卡秩为二的多个光滑法诺三维流形族具有K-稳定性，并给出了族2.5中满足明确一般性条件的流形也具有K-稳定性的结论。

$K$-polystability of smooth Fano $\text{SL}_2$-threefolds

We prove the $K$-polystability of all smooth complex Fano threefolds admitting an effective action of $\text{SL}_2$ but not of a 2-torus or 3-torus. In particular, the existence of Kähler-Einstein metrics on varieties in the families (1.10), (1.15), (1.16), (1.17), (2.21), (2.27), (2.32), (3.13), (3.17), (3.25) and (4.6) of the Mori-Mukai classification of smooth Fano threefolds is proved.

On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds

Building on results of Clemens and Kley, we find criteria for a continuous family of curves in a nodal $K$-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersections threefolds.

On smooth and isolated curves in general complete intersection Calabi-Yau threefolds

Recently Knutsen found criteria for the curves in a complete linear system $|\mathcal{L}|$ on a smooth surface $X$ in a nodal K-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. In this article we develop new methods to check whether the set of nodes of $Y_0$ imposes independent conditions on $|\mathcal{L}|$. As an application, we find new smooth isolated curves in complete intersection Calabi-Yau threefolds.

光滑 Fano 三维簇的余正则性

Coregularity of smooth Fano threefolds

本文研究了光滑 Fano 三维簇的余正则性，证明了在105族中100族的一般成员余正则性为0，其中92族任意成员均为0，并发现存在一般元素具有正余正则性的族。

Non cancellation for smooth contractible affine threefolds

We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X and Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic affine 3-spaces.

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

We prove a generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macrì and Toda for the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions.

Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with $c_1\leq 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on $Q$ with $c_1=3$ is an odd instanton up to twist.

关于K-极稳定极化的光滑Fano三维流形的Matsushima定理

A Matsushima theorem for K-polystable polarised smooth Fano threefolds

本文证明了在光滑Fano三维流形上，若极化L满足K-极稳定性，则自同构群为可约群，验证了Yau-Tian-Donaldson猜想在任意极化下的可约性预测。

光滑法诺三维簇的皮卡群上的作用

Actions on the Picard group of smooth Fano threefolds

该研究对光滑法诺三维簇自同构群在其皮卡群上作用的可能像分类，同时探究了这类簇族皮卡群的第一群上同调，属代数几何领域的相关工作。

On higher rank globally generated vector bundles over a smooth quadric threefold

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with $c_1\leq 2$ and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

光滑Fano三维簇的环面模型

Toric models of smooth Fano threefolds

本研究证明一般有理光滑Fano三维簇存在环面模型，其双有理复杂度为零，并论证了有理性、一般性、光滑性三个条件的必要性。

Tangent scrolls in prime Fano threefolds

In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2 (1999), 221-244) this implies that any morphism from a smooth Fano threefold of index 2 to a smooth Fano threefold of index 1 must be constant, which gives an answer in dimension 3 to a question stated by Peternell.

Smooth complete toric threefolds with no nontrivial nef line bundles

We describe all of the smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.

光滑三维簇的MMP序列数量的有效界

Effective bounds for the number of MMP-series of a smooth threefold

本文证明了光滑射影三维簇在正小平维数且Picard数为三时，其MMP序列数量不超过二。

On equivalent conjectures for minimal log discrepancies on smooth threefolds

On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.

The cone of moving curves of a smooth Fano-threefold

In this short note we show that the closed cone of moving curves of a smooth Fano-threefold is polyhedral. The proof relies on a famous result of Bucksom, Demailly, Paun and Peternell which says that the strongly movable cone is dual to the cone of pseudoeffective divisors. Finally, we relate the extremal rays of the cone of moving curves on a threefold with extremal rays in the cone of strongly movable curves on a birational model obtained by a Mori contraction.

Halphen pencils on quartic threefolds

For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

We prove that smooth Fano threefolds have toric Landau--Ginzburg models. More precise, we prove that their Landau--Ginzburg models, presented as Laurent polynomials, admit compactifications to families of K3 surfaces, and we describe their fibers over infinity. We also give an explicit construction of Landau--Ginzburg models for del Pezzo surfaces and any divisors on them.

Type III contractions and quintic threefolds

We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this contraction and smoothing deformation. A generalization of a fomula for calculating Hodge numbers of hypersurfaces in $\mathbb{P}^4$ with ordinary double and triple points is presented. We use these results to construct new Calabi-Yau threefolds of Picard rank two arising from a family of quintic threefolds containing a cone.

复光滑三次三维簇首次稳定的非理性

Irrationality of the first stabilization of a complex smooth cubic threefold

本文证明所有光滑复三次三维簇与射影直线乘积的非理性，并构造一个稳定有理但非理性的 Fano 簇，其高次幂与对称平方均为有理。

固定三维簇上的最小对数差异

Minimal log discrepancies on a fixed threefold

本文证明任意固定三维簇上最小对数差异满足ACC，为三维簇奇点不变量的有界性提供结论。

Smooth contractible threefolds with hyperbolic $\mathbb{G}_{m}$-actions via ps-divisors

The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group $\mathbb{G}_{m}\simeq\mathbb{C}^{\text{*}}$, using the language of polyhedral divisors developed by Altmann and Hausen as generalization of $\mathbb{Q}$-divisors.

具有 Artin-Mumford 有理性障碍的光滑二次三重覆盖的双重覆盖

Double covers of smooth quadric threefolds with Artin-Mumford obstructions to rationality

本文分类了作为光滑二次三重流形双重覆盖且具有 Artin-Mumford 有理性障碍的节点 Fano 三重流形，证明其属于三个明确描述的族，其中两个族为首次发现。

On stability conditions for the quintic threefold

We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macrì, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.

光滑环面法诺曲面与三维簇的线丛的特殊集合

Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds

该研究利用Bayer-Popescu-Sturmfels的对角胞腔分解，得到单模射影环面曲面上满的强例外线丛集合，又通过Hanlon-Hicks-Lazarev的对角分解，为18个光滑环面法诺三维簇中的16个构造了此类线丛集合。

Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

本文研究Picard秩为2和3的光滑射影环面三维体中曲面的代数双曲性，利用组合描述与Haase和Ilten的方法，在这些簇中确定了众多代数双曲性曲面。

Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture

We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli spac…

Fujita-type freeness for quasi-log canonical threefolds

In this paper, we show that Fujita's basepoint-freeness conjecture for projective quasi-log canonical singularities holds true in dimension three. Immediately, we prove Fujita-type basepoint-freeness for projective semi-log canonical threefolds.

Normality of minimal log canonical centers of threefolds in mixed and positive characteristic

We prove the normality of minimal log canonical centers on threefold pairs which residue fields are perfect of residue characteristics $p\neq 2,3 $ and $5$. We also show that the union of all log canonical centers on threefold pairs with standard coefficients are seminormal provided that the residue characteristic is large enough. We provide an example of a non-seminormal log canonical center on a threefold in characteristic $3$, and give sufficient conditions to construct similar examples.

秩2且次数为28的K-稳定Fano三维流形

K-stable Fano threefolds of rank 2 and degree 28

本文证明了除明确例外情况外，所有Picard秩2且次数为28的光滑Fano三维流形均为K-多稳定，并对光滑二次三维流形中有理正规四次曲线的法丛进行了分类。

Smooth, isolated curves in families of Calabi-Yau threefolds in homogeneous spaces

We show the existence of smooth isolated curves of different degrees and genera in Calabi-Yau threefolds that are complete intersections in homogeneous spaces. Along the way, we classify all degrees and genera of smooth curves on BN general K3 surfaces of low genera. By results of Mukai, these are the K3 surfaces that can be realised as complete intersections in certain homogeneous spaces.

强$\mathbb{Q}$-factorial紧Kähler三维簇的对数Sarkisov纲领

Log Sarkisov Program for strongly $\mathbb{Q}$-factorial compact Kähler threefolds

该研究为强$\mathbb{Q}$-factorial紧Kähler三维簇建立对数Sarkisov纲领，解决了该类簇的双有理几何核心问题。

On the existence of certain weak Fano threefolds of Picard number two

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.

关于具有二维消没循环空间的射影三维簇

On projective threefolds with two-dimensional space of vanishing cycles

本文完整分类了复数域上消没循环空间维数为2的光滑射影三维簇，以及光滑射影曲面上第二陈类为3的秩2非常充足向量束。

Weak Fano threefolds with del Pezzo fibration

This article treats smooth weak Fano 3-folds having an extremal ray of type D. Smooth weak Fano 3-folds with an extremal ray of type D except of degree 6 are classified into 47 deformation types.

K-polystability of two smooth Fano threefolds

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+ωx_1^2+ω^2x_2^2+\big(x_3^2+ωx_4^2+ω^2x_5^2\big)+\big(x_0x_3+ωx_1x_4+ω^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=x_1=x_2=0$, where $ω$ is a~primitive cube root of unity.

关于三次超曲面的Kuznetsov分量的Fourier-Mukai型自等价的探讨

On Fourier-Mukai type autoequivalences of Kuznetsov components of cubic threefolds

本文确定了光滑复三次超曲面的Kuznetsov分量的所有Fourier-Mukai型自等价群，并为该范畴的Torelli定理的Fourier-Mukai版本提供了新证明。

On smooth rational threefolds of P^5 with rational non-special hyperplane section

It is known that the smooth rational threefolds of P^5 having a rational non-special surface of P^4 as general hyperplane section have degree d=3,... ,7. We study such threefolds X from the point of view of linear systems of surfaces in P^3, looking in each case fosr an explicit description of a birational map from P^3 to X. For d=3,..., 6 we prove that there exists a line L on X such that the projection map of X centered at L is birational; we completely describe the base loci B of the linear systems found in this way and give a description of any such threefold X as a suitable blowing-down o…

允许具有Picard数一的Fano三维簇的$\mathbb Q$-Gorenstein光滑化的加权射影空间

Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to Fano threefolds with Picard number one

研究加权射影三维簇允许Q-戈伦斯坦光滑化为Picard数一的光滑Fano三维簇，推导必要数值与局部条件，计算机搜索确定数值候选，构造若干光滑化并证明一个非光滑化例子，给出消失环准则保持Picard数一。

ACC for minimal log discrepancies of terminal threefolds

We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-δ,+\infty)$, where $δ>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's conjecture on the existence of $(ε,n)$-complements for threefolds for any $ε\geq 1$. As a key important step, we prove the uniform boundedness of divisors computing minimal log discrepancies for terminal threefolds. We show the ACC for threefold canonical thresholds, and that the set of accumulation points of threefold canonical thresholds is equal to $\{0\}\cup\{\frac{1}{n}\…

Artin-Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge-Witt decompositions and Chow groups of quasi-$F$-split threefolds

In this article we prove a fundamental inequality between Artin-Mazur heights and Yobuko heights of certain proper log smooth schemes of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field $κ$ of characteristic $p>0$. We also prove that the cohomologies of Witt-sheaves of them are finitely generated ${\cal W}(κ)$-modules if the Yobuko heights of them are finite. As an application, we prove that the $p$-primary torsion parts of the Chow groups of codimension $2$ of proper smooth threefolds over $κ$ is of finite cotype if the Yobuko heights of them are …

Vector Fields on Smooth Threefolds Vanishing on Complete Intersections

The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducib…

On birational maps from cubic threefolds

We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two were previously left as open numerical possibilities, namely $(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold wi…

An optimal gap of minimal log discrepancies of threefold non-canonical singularities

We show that the minimal log discrepancy of any $\mathbb Q$-Gorenstein non-canonical threefold is $\leq\frac{12}{13}$, which is an optimal bound.