Three-dimensional terminal toric flips

Normality of general elephants on 3-fold terminal flips

We prove that the general elephant $E_{X^+}\in |-K_{X^+}|$ is normal where $X--> X^+$ is a 3-fold terminal flip. Hence $E_{X^+}$ has at worst Du Val singularities. As a corollary, there exists no non-Gorenstein singularity of type $cE/2$, $cD/3$, nor $cAx/4$ on the flipped curve $C^+$.

Toric Fano 3-folds with terminal singularities

This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.

On the Log Discrepancies in Toric Mori Contractions

It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.

Purely log terminal blow-ups of three-dimensional terminal singularities

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.

环面爆裂与翻转下的纤维积

Fiber products under toric flops and flips

研究两个扇细化诱导的双有理映射的图闭包和概形纤维积，给出图闭包正规性的组合判据，并证明在三维情形下当有至多终端奇点时纤维积等于其约化且正规。

Flips from semi-stable 4-folds whose degenerate fibers are unions of Cartier divisors which are terminal factorial 3-folds

We shall investigate flipping contractions from a semi-stable 4-fold $X$ whose degenerate fiber is a union of Cartier divisors which are terminal factorial 3-folds. Especially we shall prove that $X$ is smooth along the flipping locus, and that the flip exists for such contractions.

Combinatorics of the toric Hilbert scheme

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P^4 whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of t…

Toric $G$-Fano Threefolds

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

Toric G-solid Fano threefolds

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

三维情形下的一些终端序

Some terminal orders on 3-folds

给出三维非平凡终端局部序的首个已知例子，通过两种构造方法，一种利用给定环面分歧数据生成极大序得到环面例子，另一种在具克莱因奇点的曲面上分歧产生极大序。

关于非射影完全环面簇

On non-projective complete toric varieties

本文证明每个Picard数至多为五的光滑非射影完全环面三重簇在经历有限次flop或anti-flip后可变为射影簇。

关于具有二维环面作用的终端法诺三维流形

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。

三维除子收缩中的托里纤维Landau-Ginzburg模型

Toric Landau-Ginzburg models in threefold divisorial contractions

本文研究了终端Fano三维流形的除子收缩下的量子周期和托里纤维Landau-Ginzburg模型，证明了正则化周期恒等式，为计算三维中心模型的Sarkisov链接和高阶syzygies提供了镜像方法。

射影空间在环面不变点处爆破的反翻转

Anti-flips of the blow-ups of the projective spaces at torus invariant points

本文通过反翻转显式构造光滑环面法诺簇，使其同构于射影空间在余维一环面不变点处的爆破。

典范性判据与典范四单纯形的分类

A canonicity criterion for toric varieties and the classification of canonical 4-simplices

本文基于Reid-Shepherd-Barron-Tai判据，利用局部类群作用特征化代数簇的典范性和终端性，并在Picard数为一的情况下，提出对任意维数的典范和终端假加权射影空间的分类算法，四维情况下得到710450个典范假加权射影空间。

Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint

It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (except the vertices). There have been a lot work, especially in the last 20 years or so on classification of these objects. The main goal of this paper is to bring together these and related results, that are…

Factoring 3-fold flips and divisorial contractions to curves

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a point.

非退化环面完全交的反典范复形

The anticanonical complex for non-degenerate toric complete intersections

本文将反典范复形推广至由非退化Laurent多项式系统定义的环面簇完全交情形，并据此分类了嵌入伪加权射影空间的终端Fano三重簇。

Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes

We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at w…

A Minimal Model Program for $\mathbb{Q}$-Gorenstein varieties

The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal mode…

终端三维簇的陈数

Chern numbers of terminal threefolds

本文证明在光滑三维簇的K_X-MMP中，每次翻转引起的第一陈数三次方之差仅由第二贝蒂数界定，给出终端三维簇陈数的有界性。

On Gorenstein Fano Threefolds with an Action of a Two-Dimensional Torus

We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.

Birational maps of 3-folds

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.

Stringy Chern classes of singular toric varieties and their applications

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional…

Divisorial contractions of 3-folds

In this paper the three-dimensional divisorial contractions $f\colon Y\to (X\ni P)$ are classified provided that $\Exc f=E$ is an irreducible divisor, $f(E)=P$, the variety $Y$ has canonical singularities and $(X\ni P)$ is a toric terminal singularity.

On the three dimensional minimal model program in positive characteristic

Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the flip of $f$ exists. As a consequence, we prove the existence of minimal models for any projective $\Q$-factorial terminal variety $X$ with pseudo-effective canonical divisor $K_X$.

Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For every such a polytope, one naturally obtains a family of Fano hypersurfaces X with at worst conifold singularities. A generic 3-dimensional Fano hypersurface X can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano varieties X…

非交换沙漏 II 1/2：具有良构奇点的双有理联络的导出范畴

Noncommutative Hourglasses II 1/2: Derived Category of Birational Linkage with Well-Formed Singularities

本文针对奇点为环面上奇点内完全交的良构簇，将有界凝聚层导出范畴的经典公式推广到奇异情形，并讨论终复三维簇及其栈间态射的Bondal-Orlov局部化问题。

On Chern number inequality in dimension 3

We prove that if $X---> X^+$ is a threefold terminal flip, then $c_1(X).c_2(X)\leq c_1(X^+).c_2(X^+)$ where $c_1(X)$ and $c_2(X)$ denote the Chern classes. This gives the affirmative answer to a Question by Xie \cite{Xie2}. We obtain the similar but weaker result in the case of divisorial contraction to curves.

Fano 索引为十三的有理系数三重曲面

$\mathbb{Q}$-Fano threefolds of Fano index thirteen

本研究证明了 Picard 等级为 1 且 Fano 索引为 13 的 Fano 三重曲面为特定加权超曲面

皮卡数为三的环面椭圆对

Toric Elliptic Pairs with Picard Number Three

本文研究皮卡数为三的环面椭圆对，通过分析 $K_X+C$ 的 Zariski 分解中的曲线，证明了不存在 Lang-Trotter 四边形，从而否定回答了已有文献提出的公开问题。