Crepant Resolutions of Gorenstein Toric Singularities and Upper Bound Theorem
Dimitrios I. Dais
利用对单形球有效的经典上界定理的一个变形，导出了Gorenstein环面奇点的环面等变收缩分解存在的一个必要条件.
A necessary condition for the existence of torus-equivariant crepant resolutions of Gorenstein toric singularities can be derived by making use of a variant of the classical Upper Bound Theorem which is valid for simplicial balls.
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In this paper, we show a condition for two-parameter Gorenstein cyclic quotient singularities to have a crepant resolution by using the remainder polynomial in any dimension.
We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.
On the existence of crepant resolutions of Gorenstein Abelian quotient singularities in dimensions \geq 4
SL（r，C），r \geq 4的哪些有限子群G存在商空间C^r/G的可crepent去奇异化？这个问题的一个完整答案（也被称为“存在问题”）将对所有那些麦凯对应的高维版本有效的组进行分类。在本文中，我们考虑了SL（r，C）的交换有限子群的情况下，通过使用技术从环面和离散几何。我们给出了两个必要的存在性条件，涉及锥的Hilbert基元支持初级单形，和一个上界定理，分别。此外，一系列已知的Gorenstein阿贝尔商奇点承认投影，crepant决议（这是简要概括），我们添加了一系列新的非CI。具有这种性质的循环商奇点。
All toric l.c.i.-singularities admit projective crepant resolutions
众所周知，所有可嵌入仿射空间中的超曲面的完全相交的阿贝尔商奇点的底层空间都可以通过所有维度上的射影环面等变的可折双有理态射来整体求解。在本文中，我们将这一结果推广到整个复曲面l.c.i.-奇点我们的证明利用中岛的分类定理和一些特殊的技术，从复曲面和离散几何。
Gorenstein braid cones and crepant resolutions
本研究将偏序集对应到辫锥及扇集，证明扇集定义的光滑环面簇是锥对应奇异环面簇的 crepant 消解的充要条件，揭示环面簇的 Gorenstein 性质与偏序集结构的关联并验证相关猜想。
On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities
Gorenstein商空间$\Bbb{C}^{r}/G$在$r\geq 4$维上的经典McKay对应的直接推广主要要求存在投射的、可割的、完全的去奇异化。由于这并不总是可能的，自然会问满足上述性质的这种商空间的特殊类。本文给出了Gorenstein循环商奇点的2-参数级数在所有维上具有这种特殊环面等变分解的明确的充要条件。
We compute the motivic Donaldson-Thomas theory of small crepant resolutions of toric Calabi-Yau 3-folds.
We give a characterizaion of Gorenstein toric Fano $n$-folds with index $n-1$, which is called Gorenstein toric Del Pezzo $n$-folds, among toric varieties. In practice, we obtain a condition for a lattice $n$-polytope to be a Gorenstein Fano polytope. In our proof, we do not use Batyrev-Juny's classification of Gorenstein Fano polytopes.
Towards non-commutative crepant resolutions of affine toric Gorenstein varieties
本文推广了Špenko-Van den Bergh和Iyama-Wemyss的结果，利用Novaković的方法研究具有内部点的多面体锥的仿射环面Gorenstein簇，并证明了在自反多面体顶点数$\le \dim P+2$时NCCRs的存在性。
Non-commutative crepant resolutions for (almost) simplicial toric algebras
本文通过非交换crepant解析方法，证明了单纯形和几乎单纯形典范Gorenstein代数的典范解析存在性，并提供了新的简短证明。
Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.
We give a characterization of Gorenstein toric Fano varieties of dimension $n$ with index $n$ among toric varieties. As an application, we give a strong version of Fujita's freeness conjecture and also give a simple proof of Fujita's very ampleness conjecture on Gorenstein toric varieties.
We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR.
Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of all its commutative crepant resolutions. We also explain how the McKay correspondence in dimension 3 can be interpreted using brane tilings.
We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss …
Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension
一个$n$维Gorenstein环面Fano簇$X$称为Del Pezzo簇，如果反正则类$-K_X$是一个Cartier因子的$(n-1)$ -倍数.我们的目的是给出任意维数$n \geq 2$上Gorenstein环面Del Pezzo簇的完全双正则分类.我们表明，同构存在37 Gorenstein环面德尔佩佐品种的尺寸$n$不是锥以上$(n-1)$维Gorenstein环面德尔佩佐品种。我们的结果与Emiris和Tsigaridas对自反多边形的Minkowski和分解的分类以及Jahnke和Peternell对$n$维几乎Del Pezzo流形的变形的分类密切相关。
Gorenstein toric Fano varieties
我们调查Gorenstein环面法诺品种组合的方法使用的概念，出现在连接镜像对称的自反多面体。该文件包含概括的工具和以前已知的结果，非奇异复曲面法诺品种。作为应用，我们得到了新的分类结果，不变量的界和自反多面体的组合和几何性质的公式。
Gorenstein toric Schubert varieties in Grassmannians
本文研究格拉斯曼流形中的环面Schubert簇，给出了Gorenstein环面Schubert簇的扇的显式描述，并证明其均为Fano簇。
Crepant Transformation Correspondence For Toric Stack Bundles
本文通过构造辛变换和 Fourier-Mukai 同构，研究了簇堆叠包在 crepant 墙交叉下的 Gromov-Witten 理论对应关系。
Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples
在本文中，我们分析了六个例子的双有理复曲面orbifolds之间的转换：三个crepant决议，两个crepant部分决议，和翻转。我们研究了这些变换对零格Gromov-Witten不变量的影响，证明了每种情况下的Coates-Corti-Iritani-Tseng/Ruan形式的Crepant归结猜想。我们的研究结果表明，这种形式的Crepant解决猜想也可能持有更一般的crepant双有理变换。他们还建议，阮的原始Crepant分辨率猜想应修改，包括适当的“量子修正”，并没有阮的原始猜想或上同调Crepant分辨率猜想的crepant部分决议的情况下直接推广。我们的方法是基于复曲面orbifolds镜像对称。
The Open Crepant Transformation Conjecture for Toric Calabi-Yau 3-Orbifolds
本文证明环面 Calabi–Yau 3-轨形的开 Ruan Crepant 变换猜想，借助镜像定理、GKZ 次级扇穿墙与镜像曲线解析延拓，等同 K-等价轨形的圆盘不变量。
Q-factorial Gorenstein toric Fano varieties with large Picard number
具有Picard数rho（X）的维数为d的Q-阶乘Gorenstein环面Fano簇X对应于具有rho（X）+d个顶点的单纯自反d-多面体。Casagrande证明了任何单纯自反d-多面体最多有3d个顶点，如果d是偶数，3d-1，如果d是奇数。此外，已知d的相等甚至意味着直到幺模等价的唯一性。本文对所有顶点数为3d-1的单纯自反d-多面体进行了完全分类，对应于Picard数为2d-1的d维Q-阶乘Gorenstein环面Fano簇.对于d偶数，存在三个这样的簇，其中两个是奇异的，而对于d奇数（d > 1），恰好存在两个，都是投射线上的非奇异复曲面纤维丛。这概括了第二作者最近的工作。
Crepant resolutions, mutations, and the space of potentials
本文通过开发网络应用程序，利用带势箭图的变异与 crepant 三角剖分研究 Gorenstein 环面商奇点的 crepant 消解之间的 flop，并发现箭图变异可分类这些消解，同时实现了 McKay 对应的关键构造。
On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution
本文证明了\frac 11（1，…，1，l-（r-1））与$l\geq r\geq 2$，有一个\textit{unique}环面-等变投影，收缩，部分分辨率，它是“全”的当且仅当$l\equiv 0$ mod $% (r-1) $或$l\equiv 1$ mod $(r-1) $。事实证明，如果满足这两个条件之一，则完全去奇异化的例外轨迹由$\lfloor \frac{l}{r-1} \rfloor$素因子组成，其中$\lfloor \frac{l}{r-1}\rfloor - 1$同构于$\Bbb{P}_{\Bbb{C}%}^{r-2}$上的$\Bbb{P}_{\Bbb{C}}^1$ -丛的全空间。此外，它示出的交叉数是可计算的明确和分辨率态射可以被看作是一个连续的（归一化）爆破的复合物。显然，上述类型的单参数化奇异级数包含（作为其“第一成员”）由位于$\Bbb{P}_{\Bbb{C}}^{r-1}$的$r$元组Veronese嵌入之上的仿射锥的原点定义的著名Gorenstein奇异性。
Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties
该研究推广了Eikelberg仿射依赖理论，证明维数不小于3的极化Gorenstein环面Fano簇对的Picard秩之和的上界，且该上界在极化对为单-单纯对时取到。
Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier-Mukai transformation between the K-groups of X and Y, via an equivariant version of the …
Crepant Resolutions of C^n/A_1(n) and Flops of n-Folds for n = 4,5
本文确定了$\hl^{A_1(n)}(\CZ^n)$对$n=4,5$的显式环面簇结构，其中$A_1(n)$是所有阶为2的元素的特殊对角群.通过对$\hl^{A_1(n)}(\CZ^n)$的复曲面数据的分析，得到了$\CZ^n/A_1(n)$的某些复曲面可分辨性，并通过$n=4,5$的$n$ -folds的触发器将不同的可分辨性联系起来.
We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relat…
All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions
对于Gorenstein商空间$C^d/G$，在维数$d\geq 4$上的经典McKay对应的直接推广将主要要求存在投射的、crepant的去奇异化。由于这原来并不总是可能的，里德问特殊类的商空间，这将满足上述财产。我们证明了所有Gorenstein阿贝尔商奇点的基础空间，这是嵌入作为仿射空间中的超曲面的完全交叉，有环面等变的投影crepant决议在所有维度。我们使用复曲面和离散几何的技术。
We describe the non-Gorenstein loci of normal toric varieties. In the case of Hibi rings a combinatorial description is provided in terms of the underlying partially ordered set. As a non-toric application we compute the dimensions of the non-Gorenstein loci of the first secant variety of Segre varieties.
A central question in liaison theory asks whether every Cohen-Macaulay, graded ideal of a standard graded K-algebra belongs to the same G-liaison class of a complete intersection. In this paper we answer this question positively for toric ideals defining edge subrings of bipartite graphs.
Log Crepant Birational Maps and Derived Categories
我们将关于导出等价和K-等价的猜想推广到对数情形，并在复曲面情形下证明了这一猜想。
Gorenstein Quotient Singularities of Monomial Type in Dimension Three
本文利用SL（3，C）的单项式型有限子群构造了商奇点的一个一致分解，并证明了该分解的Euler数等于共轭类的个数.这一结果是猜想II的一部分，在以前的文件“Crepant分辨率的三面体奇点”（alg-geom 9404008）。这些奇点与三面体奇点不同，但证明的主要思想是基于三面体情形的方法。
On Gorenstein Fano toric complete intersections
本文分类了假加权射影空间中的Q-因子Gorenstein Fano非退化完全交三维流形。
Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples
设X是Gorenstein orbifold，Y是X的临界分解.本文提出了一个关于X的零亏格Gromov-Witten不变量与Y的零亏格Gromov-Witten不变量的猜想，它与Ruan和Bryan-Graber的Crepant归结猜想不同，并在X = P（1，1，2）和X = P（1，1，1，3）时证明了我们的猜想.因此，我们看到Bryan-Graber猜想的原始形式对P（1，1，2）成立，但对P（1，1，1，3）可能是假的。我们的方法是基于复曲面orbifolds镜像对称。
Let $R$ be a Cohen--Macaulay normal domain with a canonical module $ω_R$. It is proved that if $R$ admits a noncommutative crepant resolution (NCCR), then necessarily it is $\mathbb{Q}$-Gorenstein. Writing $S$ for a Zariski local canonical cover of $R$, then a tight relationship between the existence of noncommutative (crepant) resolutions on $R$ and $S$ is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of…
In this paper, we study splitting (or toric) non-commutative crepant resolutions (= NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group $\mathbb{Z}^2$ have a splitting NCCR. In the appendix, we also discuss Gorenstein toric rings with class group $\mathbb{Z}$, in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange gr…
We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.
Rigid Gorenstein toric Fano varieties arising from directed graphs
本文利用 Totaro 的刚性判据，分类所有使对应 Gorenstein 环面 Fano 簇满足余维数 2 光滑、余维数 3 $\mathbb{Q}$-factorial 的有向图。
Complete toric varieties with reductive automorphism group
我们给出了一个完整的复曲面簇的自同构群，分别Gorenstein复曲面Fano簇，是还原的等价和充分的标准。特别是，我们证明了自同构群的Gorenstein环面Fano品种是约化的，如果重心相关联的自反多面体是零。通过对Demazure根集的研究，证明了完备复曲面簇的约化自同构群维数的一个严格界。
The aim of this paper is to study an analog of non-commutative Donaldson-Thomas theory corresponding to the refined topological vertex for small crepant resolutions of toric Calabi-Yau 3-folds. We define the invariants using dimer models and provide "wall-crossing" formulas. In particular, we get normalized generating functions which are unchanged under "wall-crossing".
Unobstructedness of affine Gorenstein terminal toric fourfolds
该研究证明维数不超过4的仿射终态戈伦斯坦环面簇无阻碍，针对四维情形确定齐次阻碍次数，构造双参数形变消除潜在阻碍，完善了环面簇形变相关理论。
We establish a genus zero correspondence between the equivariant Gromov-Witten theory of the Deligne-Mumford stack $[\mathbb{C}^N/G]$ and its blowup at the origin. The relationship generalizes the crepant transformation conjecture of Coates-Iritani-Tseng and Coates-Ruan to the discrepant (non-crepant) setting using asymptotic expansion. Using this result together with quantum Serre duality and the MLK correspondence we prove LG/Fano and LG/general type correspondences for hypersurfaces.
Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite subgroup G of SL(3,C). However, in dimension 4, very few crepant resolutions are known. In this paper, we will show several examples of crepant resolutions in dimension 4 and show examples in which Hilb^G(C^4) is blow-up of certain crepant resolutions for C^4/G, or Hilb^G(C^4) has singularity.
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals fo…
Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities
本文研究有向复合Du Val奇点的二聚体模型，利用锯齿路径序列确定稳定性参数空间的墙-腔结构，实现跨墙时稳定表示变化的追踪，为三维Gorenstein有向奇点的射影平展解析提供新视角。
We prove the crepant resolution conjecture for Donaldson-Thomas invariants of toric Calabi-Yau 3-orbifolds with transverse A-singularities.
We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polyto…
We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.
We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.
Non-commutative crepant resolutions of toric singularities with divisor class group of rank one
本文证明并分类了除子类群秩一的Gorenstein环状奇点的环状非交换凝聚解消（NCCR），通过上集与Iyama-Wemyss突变建立导出等价，并在非金字塔情形下给出二聚体模型实现。
One-parameter families containing three-dimensional toric Gorenstein singularities
对于仿射复曲面簇，向量空间T1（包含无穷小变形）将通过定义多面体锥的横切的Minkowski和来解释。这一结果将被应用于研究（特别是非孤立的）三维Gorenstein奇点的变形理论。
Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties
我们提出了一个组合对偶的格多面体，这在理论上产生了对镜像对称家庭的Calabi-Yau完全相交的环面Fano品种Gorenstein奇点。我们的建设是一个推广的极对偶提出的Batyrev的情况下的超曲面。
Crepant resolutions of log-terminal singularities via Artin stacks
本文证明任意 log-terminal 奇点簇均可由光滑 Artin 叠实现 crepant 消解，并建立相应 McKay 对应，用光滑叠弧空间上的动机积分表达 Q-Gorenstein 簇的弦不变量。
Smoothing Gorenstein toric Fano 3-folds
该研究针对三维自反多面体提出可容许闵可夫斯基分解数据的组合概念，证明其可确定对应Gorenstein环面Fano 3-簇的光滑化，并给出计算该光滑化贝蒂数的有效方法。
Deformations of an affine Gorenstein toric pair
本文研究了仿射环面 Gorenstein 簇及其边界的形变，计算了形变函子的切空间与阻碍空间，构造了特定度数下的极小泛形变，并揭示了不可约分量与多面体 Minkowski 分解的一一对应关系。
The versal Deformation of an isolated toric Gorenstein Singularity
给定R^n中的一个格多面体Q，我们定义一个仿射方案M（Q），它反映了将Q分裂成Minkowski和的可能性。另一方面，Q诱导了复曲面Gorenstein奇点Y，我们构造了M（Q）上的一个平坦族，其中Y是特殊纤维。如果Y只有一个孤立的奇点，这个族就是π。(This修订版包含了现在的证明）。
Decomposition theorem for the cd-index of Gorenstein* posets
我们证明了一个分解定理的CD-指标Gorenstein* 偏序集类似的分解定理的交集上同调的环面品种。由此解决了Stanley的一个猜想：Gorenstein* 格的cd-指数在布尔代数上是极小的.
We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.
Equivariant noncommutative crepant resolutions and $G$-maximal modifying modules
本文针对含特征零域上有限群作用的Cohen-Macaulay正规环，研究等变非交换平展分解与极大修改代数的导出等价性，证明三维指数2终端奇点的MM模存在性及对应MMA的导出等价。
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