Valery Alexeev, Alexander Borisov
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.
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The main purpose of this notes is to supplement the paper reid, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. As an application, we obtain a generalization of Fujita's conjecture for singular toric varieties. We also prove that every toric variety has a small projective toric $\bQ$-factorialization.
We can run the MMP for any divisor on any $\mathbb{Q}$-factorial projective toric variety. We show that two Mori fiber spaces, which are outputs of the above MMP, are connected by finitely many elementary transforms.
The main purpose of this paper is to give a simple and non-combinatorial proof of the toric Mori theory. Here, the toric Mori theory means the (log) Minimal Model Program (MMP, for short) for toric varieties. We minimize the arguments on fans and their decompositions. We recommend this paper to the following people: (A) those who are uncomfortable with manipulating fans and their decompositions, (B) those who are familiar with toric geometry but not with the MMP. People in the category (A) will be relieved from tedious combinatorial arguments in several problems. Those in the category (B) will…
We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces. For some of these toric surfaces, the question whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of Gonzalez and Karu. We give new examples and non-examples of Mori Dream Spaces among these blow-ups.
We describe the set of minimal log discrepancies of toric log varities, and study its accumulation points.
We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.
We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb Q$-factorial toric varieties. So, our theory seems to be quite different from Reid's original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb Q$-factorial contractions, which imply that the $\mathbb Q$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Prog…
Given a smooth Mori dream space $X$ we construct a model dominating all the small $\mathbb{Q}$-factorial modifications via tropicalization. This construction allows us to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on $X$ and hence the movable cone of $X$. The existence of such basis allows us to prove the polyhedrality of the global Newton-Okounkov body and the existence of toric degenerations.
Constructing geometric realizations of birational maps between Mori Dream Spaces
本文为Mori Dream Spaces间的双有理映射构造几何实现，证明其仍为MDS且诱导墙穿越分解，并提供SageMath工具及环面情形下的矩多胞体构造。
We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.
This is the major revision. The main purpose of this paper is to prove that minimal discrepancies of $n$-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less than $n$. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.
In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $B$-model on toric varieties to semi-orthogonal decompositions on the $A$-model on the mirror Landau-Ginzburg models. As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $¶^2$ at three infinitesimally near points.
Cox rings of projectivized toric vector bundles and toric flag Bundles
本文研究射影化环面向量丛在直和运算下何时保持Mori dream空间性质，利用全旗丛关系给出判定条件，并计算了射影空间切丛的全旗丛Cox环表示。
The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties
本文将伪加权射影空间的重数概念推广到Mori梦想空间，引入完备toric簇的权群$G_Q$和权模$|Q|$，为Fano类toric簇分类提供框架，并给出Batyrev极对偶性的代数解释。
We prove that for each positive integer $n$ there exists a positive number $ε_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,ε_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.
We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.
The geometry of Frobenius on toric varieties
本文通过定义Frobenius支撑及除子与1-闭链的F-有效性，揭示了F-有效曲线锥与Mori锥的相互作用如何刻画环面簇的极值Mori收缩，并证明Frobenius迹核丰沛当且仅当Picard秩为1。
We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.
The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric surfaces of Picard number one into an interpolation problem involving points in the projective plane. An instance of such an interpolation problem is the Gonzalez-Karu theorem that gives new examples of weighted projective planes whose blow-up at a general point is not a Mori Dream Space.
On semi-continuity problems for minimal log discrepancies
本文证明了具有Crepan解析的簇的极小对数差异的半连续性，并据此解决了环面对的理想-adic半连续性问题。
In 1988 S. Mori, D. Morrison, and I. Morrison gave a computer-based conjectural classification of four-dimensional cyclic quotient singularities of prime index. It was partially proven in 1990 by G. Sankaran. In 1991 Jim Lawrence basically proved this and much more, been unaware of algebro-geometric meaning of what he did. In this short note I just bring this all together to prove that all n-dimensional toric singularities with log-discrepancy greater than (or equal to) $ε$ form finitely many "series".
We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other …
With the help of Mori theory for projective toric manifolds due to M. Reid, we study non projective toric manifolds which become projective after a single blow up along an invariant curve.
Bounding minimal log discrepancies of general arrangement varieties
本文证明了在一般排列簇上，最小对数歧变可由簇的维数上界限制，扩展了关于该猜想的先前结果。
We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric varieties are given in terms of fans over the facets of reflexive lattice polytopes. The program performs crepant star triangulations of reflexive polytopes and determines the Mori cones of the resulting toric varieties. Furthermore, it computes the intersection rings and characteristic classes of hypersurfaces.
Minimal log discrepancies of hypersurface mirrors
本文研究加权射影空间中拟光滑 Calabi-Yau 超曲面商的最小对数差异，证明其与 BHK 镜像的权和次数相关，并构造出已知最小对数差异的 klt Calabi-Yau 簇。
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。
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