On the Strong Factorization of Toric Birational Maps

构造Mori Dream Spaces之间双有理映射的几何实现

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。

Toric Birational Geometry and Applications to Lattice Polytopes

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco and Piene. Our approach consists in interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then applying techniques from Adjunction Theory and Mori Theory. In the opposite direction, we use the combinatorics of fans to descr…

量子刻面几何中的准合理映射

Birational morphisms in quantum toric geometry

本文研究量子刻面几何中由加权吹胀和边界关系诱导的合理映射，探讨其在刻面堆中的应用与特性。

Erratum to the paper "A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension"

This is an erratum to math.AG/9803126, Tohoku 51 (1999) 489-537. This erratum describes: 1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak factorization theorem for toroidal birational morphisms in [AMR], in the form utilized in [AKMR] for the proof of the weak factorization theorem for general birationla maps, avoiding the use of the above mentioned algorithm for the strong factorization, and 3. a list of corrections for a few other mistakes in [AMR], mostly pointed out by Laurent Bonavero.

Lectures on Factorization of Birational Maps

This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to give a detailed account of the proof for the (weak) factorization theorem of birational maps by Abramovich-Karu-Matsuki-Włodarczyk.

Toric non-equalized flips associated to $\mathbb{C}^*$-actions

Starting from $\mathbb{C}^*$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally described by toric flips, either of Atiyah type or so called non-equalized. We relate this notion of toric flip with the property of the action being non-equalized. Moreover, we find explicit examples of rational homogeneous varieties admitting a $\mathbb{C}^*$-action whose weighted blow-up at the extremal fixed point components gives a birational map among two proj…

Functorial factorization of birational maps for qe schemes in characteristic 0

We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces.

Relative quantum cohomology under birational transformations

We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.

Equivariant Burnside groups and toric varieties

We study $G$-equivariant birational geometry of toric varieties, where $G$ is a finite group.

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…

Log Crepant Birational Maps and Derived Categories

We extend the conjecture on the derived equivalence and K-equivalence to the logarithmic case and prove it in the toric case.

Three-dimensional toric morphisms with anti-nef canonical divisors

In this paper, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.

Gromov-Witten Theory of Toric Birational Transformations

We investigate the effect of a general toric wall crossing on genus zero Gromov-Witten theory. Given two complete toric orbifolds $X_+$ and $X_-$ related by wall crossing under variation of GIT, we prove that their respective $I$-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in $X_+$ and $X_-$. This extends the work of the previous authors in Acosta-Shoemaker to the case of complete intersections in toric varieties, and generalizes some of the results of Coates-Iritani-Jiang on th…

Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anti-canonical hypersurfaces in G/B. Recently Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev's in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of deg…

Some examples of Calabi-Yau pairs with maximal intersection and no toric model

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that admit no toric model.

Monomial Cremona transformations and toric polar maps

Given a birational map in the three dimensional projective space defined by monomials of degree $d$, we prove that its inverse is defined by monomials of degree at most $d^2-d+1$.

Essential bases and toric degenerations arising from birational sequences

We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair $(S,>)$ consisting of a birational sequence and a monomial order, we attach to the affine variety $G/\hskip -3.5pt/U$ a monoid $Γ=Γ(S,>)$. As a side effect we get a vector space basis $\mathbb B_Γ$ of $\mathbb C[G/\hskip -3.5pt/U]$, the elements being indexed by $Γ$. The basis $\mathbb B_Γ$ has multiplicative properties very similar to those o…

Local Strong Factorization of birational maps

The strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns. We prove a local version of the strong factorization conjecture for toric varieties. Combining this result with the monomialization theorem of S.D.Cutkosky, we obtain a strong factorization theorem for local rings dominated by a valuation.

Factorizations of birational extensions of local rings

We give a proof of local strong factorization of a birational extension of regular local rings (of equicharacteristic zero) along a valuation of rank 1 and maximal rational rank. This gives an alternate proof to the geometric proof of this Theorem by Karu. Our proof uses only linear algebra, and is in the spirit of Christensen's proof of this result in dimension 3.

Birational cobordisms and factorization of birational maps

In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in the considerations is played by K^*-actions where K is the base field. This paper serves as a basis for proving the weak factorization conjecture on factorization of birational maps in characteristic zero into blow-ups and blow-downs. This is carried out in two subsequent papers, o…

超越环面 Fano 三维流形的双有理变换与二维 (0,2) 箭袋规范理论

Birational Transformations and 2d (0,2) Quiver Gauge Theories beyond Toric Fano 3-folds

本文证明联系环面 Fano 三维流形的双有理变换对应二维 (0,2) 箭袋规范理论的质量形变，并推广至环面 Calabi-Yau 四维流形，为相关“极小模型纲领”奠定基础。

二维情形下的分解中心与格罗滕迪克簇环

Factorization centers in dimension two and the Grothendieck ring of varieties

本文研究双有理映射的分解中心，证明完美域上光滑射影曲面的双有理自同构的弱分解中胀开与胀缩中心同构，由此推出无法用曲面双有理自同构构造0维簇的非平凡L-等价，且有理曲面的有理性中心是良定义的。

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。

Birational sequences and the tropical Grassmannian

We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$, Speyer-Sturmfels' tropical Grassmannian. For $\text{Gr}(2,\mathbb C^n)$ we show that the associated valuations induce toric degenerations. We describe recursively the vertices of the corresponding Newton--Okounkov polytopes, which are particular vertices of a hypercube and hence integral. We show further that every toric degeneration of $\text{Gr}(2,\mathbb C^{n})$ constructed using the tropical Gra…

Simple Constructive Weak Factorization

We give a simplified algorithm of the functorial weak factorization of birational morphisms of nonsingular varieties over a field of characteristic zero into a composite of blow-ups and blow-downs with smooth centers.

Berglund-Hübsch-Krawitz Mirrors via Shioda Maps

We give an elementary approach to proving the birationality of multiple Berglund-Hübsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. Although a similar result has been obtained in recent months by Shoemaker, our proof is new in that it sidesteps using toric geometry and drops an unnecessary hypothesis. We give an explicit quotient of a Fermat variety to which the mirrors are birational.

Torification and Factorization of Birational Maps

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/990407…

A GIT proof of Włodarczyk's weighted factorization theorem

We give a short proof of W{\lodarczyk's theorem that any birational map between smooth projective varieties in characteristic zero is a composition of weighted blowups and blowdowns.

Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples

In this paper we analyze six examples of birational transformations between toric orbifolds: three crepant resolutions, two crepant partial resolutions, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appropria…

Factorization for stacks and boundary complexes

We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $Δ(X,D)$ depends only on $U$.

环面极映射与示性类

Toric polar maps and characteristic classes

该研究证明复射影空间中超曲面的环面极映射多重度与特定开集的Chern-Schwartz-MacPherson类系数符号一致，给出平面曲线映射次数的局部不变量公式，并构造了任意维数的双有理环面极映射的不可约超曲面族。

On Oda's Strong Factorization Conjecture

The Oda's Strong Factorization Conjecture states that a proper birational map between smooth toric varieties can be decomposed as a sequence of smooth toric blowups followed by a sequence of smooth toric blowdowns. This article describes an algorithm that conjecturally constructs such a decomposition. Several reductions and simplifications of the algorithm are presented and some special cases of the conjecture are proved.

二聚体可积系统上的双有理变换

Birational Transformations on Dimer Integrable Systems

本文证明环面卡拉比-丘三维流形间的双有理变换对应二聚体可积系统间的双有理等价，并通过两个阿贝尔轨道簇实例验证了谱曲线与哈密顿量的关联。

反射多边形对应的二元可积系统的分类与birational等价

Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons

本文对反射多边形对应的二元可积系统进行分类与birational等价研究，揭示了其与膜铺砌之间的关系及Seiberg对偶的等价类。

来自Vex三角剖分的Calabi-Yau三维流形

Calabi-Yau Threefolds from Vex Triangulations

本文通过研究四维自反多胞体的精细正则三角剖分（包括FRST和vex三角剖分），证明了vex三角剖分对应的非弱Fano环面簇及其Calabi-Yau超曲面是光滑的，并枚举了Kreuzer-Skarke数据库中h^{1,1}≤7的所有精细正则三角剖分，表明vex三角剖分产生了大量光滑Calabi-Yau三维流形并提供了Kähler模空间新区域的环面描述。

Factorization of normalization of Nash blow-up of order n of An by the minimal resolution

We show that the normalization of the Nash blow-up of order n of the toric surface singularity An can be factorized by the minimal resolution of An. The result is obtained using the combinatorial description of these objects.

双有理映射、PBW退化旗与偏序集多面体

Birational maps, PBW degenerate flags and poset polytopes

本研究将射影空间与格拉斯曼流形间双有理映射的图闭包结果推广至PBW退化旗簇，分析其组合、代数与几何性质，验证了环面退化的普适性，并通过箭图表示刻画旗簇图闭包。

Factorization Theorem for Projective Varieties with Finite Quotient Singularities

In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.

Minimal model theorem for toric divisors

Minimal model conjecture for a proper variety $X$ is that if $κ(X)\geq 0$, then $X$ has a minimal model with the abundance and if $κ=-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture for a $\D$-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of ambient toric variety. Furthermore, for such a divisor $X$ with $κ(X)\geq 0$ we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of…

到 Grassmann 簇的双有理映射、表示与偏序集多胞体

Birational maps to Grassmannians, representations and poset polytopes

本文研究射影空间到 Grassmann 簇的双有理映射的图闭包，给出其显式描述、纤维计算及到退化李代数循环表示射影化的嵌入，并利用新偏序集多胞体推广 FFLV 基，得到平坦环状退化。

可积系统与广义环形多边形及被Higgs化的5维N=1理论

Integrable Systems for Generalized Toric Polygons and Higgsed 5d N=1 Theories

研究广义环形多边形的可积系统及其与5维量子场论的关系，通过精细的birational变换构建Higgs化的N=1理论。

The Pfaffian-Grassmannian equivalence revisited

We give a new proof of the 'Pfaffian-Grassmannian' derived equivalence between certain pairs of non-birational Calabi-Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three steps by passing through some intermediate categories of (global) matrix factorizations. The first step is global Knoerrer periodicity, the second comes from a birational map between Landau-Ginzburg B-models, and for the third we develop some new techniques.

Quivers from Matrix Factorizations

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factor…

环面曲面映射的无稳定性等分布

Equidistribution without stability for toric surface maps

本文针对无法双有理共轭到代数稳定映射的一类环面曲面有理映射，证明了曲线迭代原像的等分布结果，并在小拓扑度条件下证明了迭代前向像的等分布，为此系统发展了环面曲面逆向极限上的正闭(1,1)流理论框架。

Matrix factorizations from SYZ transformations

It is known that Lagrangian torus fibers of the moment map of a toric Fano manifold $X$, equipped with flat $U(1)$-connections, are mirror to matrix factorizations of the mirror superpotential $W:\check{X}\rightarrow\bC$. Via SYZ mirror transformations, we describe how this correspondence, when $X$ is $\bP^1$ or $\bP^2$, can be explained in a geometric way.

基于关于吹胀的多项式指标的双有理映射的动力次数 II. 三维示例

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

本文提出一种利用多项式指标与拉回因式分解计算双有理映射迭代次数的新方法，通过三维示例验证其灵活性，无需构造代数稳定提升，优于传统皮卡群方法。

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

In this paper we address the problem of computing $\text{deg}(f^n)$, the degrees of iterates of a birational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$. For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of $f$, based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by $f$, and the propagation of these indices along orbits of $f$. For maps admitting algebraically stable modifications $f_X:X\rightarrow X$, where $X$ is a variety obtained from $\mathbb P^N$ by a finite number…

高维空间中一种有效的van der Corput型方法

An effective van der Corput-type method in higher dimensions

本文提出了一种在高维空间中有效进行van der Corput型分析的方法，通过有理映射推导出Varchenko渐近展开式的主项，并展望其在振荡积分相关问题中的潜在应用。