Alastair Craw, Diane Maclagan
The GIT chamber decomposition arising from a subtorus action on a quasiprojective toric variety is a polyhedral complex. Denote by Sigma the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety defined by the fan Sigma is the normalization of the toric Chow quotient of a closely related affine toric variety by a complementary torus.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In the article math.AG/0102085, Lemma 5.5 was false. Hence the decomposition theorem (Theorem 5.8) and all statements in Section 7 are not proved.
We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $Γ$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $Γ$-admissible fans admit $Γ$-admissible completions. In order to show this we prove a combinatorial analog of noeth…
We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bun…
This paper studies two related subjects. One is some combinatorics arising from linear projections of polytopes and fans of cones. The other is quotient varieties of toric varieties. The relation is that projections of polytopes are related to quotients of projective toric varieties and projection of fans are related to quotients of general toric varieties. Despite its relation to geometry the first part is purely combinatorial and should be of interest in its own right.
Nonrational polytopes and fans in toric geometry
本研究在环面几何框架下考察非有理凸多面体与非有理扇的概念，梳理该领域最新进展及相互关联，为相关研究提供参考。
We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric $g$ and $h$ polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two com…
In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. For this purpose, we obtain two main results. The first is to investigate a combinatorial aspect of the 2-category of toric algebraic stacks defined in \cite{I2}. We establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second is to give a geometric characterization theorem for toric algebraic stacks.
Gröbner Fans and Minimal Embedded Toric Resolutions of Rational Double Point Singularities
本文通过显式构造ADE奇点的Gröbner扇并利用profile概念进行加细，重构了有理二重点奇点的嵌入环面消解。
We show that the mirror constructions of Greene-Plesser, Berglund-Hubsch, Batryev-Borsov, Givental and Hori-Vafa can be expressed in terms of what we call dual fans. To do this, we associate to a pair of dual fans a pair of toric Landau-Ginzburg models, and we describe a process by which each of the mirror constructions listed also produces a pair of toric Landau-Ginzburg models. Replacing mirror pairs by toric Landau-Ginzburg models is reversible, and our main result is the dual fan models and the mirror pairs models coincide.
The purpose of this paper and its sequel (Toric Stacks II) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. In this paper, we define a \emph{toric stack} as a quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a \emph{stacky fan}. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stress…
We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These…
We study the category of KM fans - a "stacky" generalization of the category of fans considered in toric geometry - and its various realization functors to "geometric" categories. The "purest" such realization takes the form of a functor from KM fans to the 2-category of stacks over the category of fine fans, in the "characteristic-zero-étale" topology. In the algebraic setting, over a field of characteristic zero, we have a realization functor from KM fans to (log) Deligne-Mumford stacks. We prove that this realization functor gives rise to an equivalence of categories between (lattice) KM fa…
Higher rank Gelfand-Kapranov-Zelevinsky fans
本文研究高阶GKZ扇形，通过点配置构造高阶拟估值，将相关紧致簇平坦退化为减少的紧致簇联合体，编码点配置的多面体细分。
We construct one-parameter complex analytic families whose special fibers are complete toric varieties. Under some assumptions, the general fibers of these families also become toric varieties and we can explicitly describe the corresponding fans from the data of the fans corresponding to the special fibers. Using these families, we give a deformation family for a certain toric weakened Fano 3-fold. Moreover, we get some examples of toric weakened Fano 4-folds.
In this paper we study the geometry and combinatorics of the possible rational polyhedral fans with a given set of rays. The main questions we consider are when such fans are projective, complete, or simplicial. To answer these questions we use techniques of invariant theory to characterize the quotients of maximal saturated open sets in torus representations.
We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field $k$. Then we provide some situations where toric varieties over $k$ are classified by Galois-stable fans, and spherical embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonspli…
In this article, we first give some elementary proprieties of monoids and fans, then construct a toric scheme over an arbitrary ring, from a given fan. Using Valuative Criterion, we prove that this scheme is separated and give the sufficient and necessary condition when it is proper. We also study the regularity and logarithmic regularity of it. Finally we study the morphisms of toric schemes induced by the homomorphisms of fans.
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric …
A Functorial Classification of Toric Stacks
本文通过引入栈扇2-范畴$\textsf{StkFan}$，构造其与环面栈2-范畴$\textsf{TStk}$的等价，解决了环面栈范畴化构造的错误或局限问题，还发展了环面范畴理论并为Gale对偶提供公理框架。
Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials
本文推导了有限群作用于完备单纯形扇的Stanley-Reisner环的Artinian约简的分层特征公式，基于等变h多项式，并研究了该群在关联的典范流形上同调的表示。
The purpose of the present paper is threefold. First: giving a treatise on weighted projective spaces by the toric point of view. Second: providing characterizations of fans and polytopes giving weighted projective spaces, with particular focus on a kind of \emph{recognition process} of toric data like fans and polytopes. Third: building a mathematical framework for the algorithmic and computational approach to wps's realized in [23,24].
We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.
We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme of finite type when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of a matroid fan is a torus torsor over the realization space of the matroid. As a consequence, we deduce that these tropical realization spaces satisfy Murphy's Law.
The goal of this article is to give a Lefschetz type decomposition for the cd-index of a complete fan. To a complete simplicial fan one can associate a toric variety X, the even Betti numbers h_i of X and the numbers g_i = h_i-h_{i-1}. If the fan is projective, then non-negativity of g_i follows from the Lefschetz decomposition of the cohomology. In the case of a nonsiplicial complete fan one can analogously compute the flag h-numbers h_S and, by a change of variable formula, the cd-index. We give an analogue of the Lefschetz operation for the cd-index. This gives another proof of the non-nega…
Foldable fans, cscK surfaces and local K-moduli
本文引入并分类格自同构群含非平凡循环子群的光滑环面可折叠曲面，证明其均存在cscK度量，并刻画极化cscK模空间在相应点附近的局部有限商模型。
We study certain foliated complex manifolds that behave similarly to complete nonsingular toric varieties. We classify them by combinatorial objects that we call marked fans. We describe the basic cohomology algebras of them in terms of corresponding marked fans. We also study the basic Dolbeault cohomology algebras of them.
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over tori…
Reduced superschemes and the combinatorics of toric supervarieties
本文通过重新定义约化超环与超概形，建立了环面超簇与装饰多面体扇之间的范畴等价，并利用装饰多胞体刻画了异构超格拉斯曼流形中的自然环面超簇。
Conic modules, secondary fans and non-commutative resolutions
本文研究锥模与二次扇的关系，提出非交换解析的构造方法，并探讨了锥模在非交换解析中的应用条件。
As an alternative to the description of a toric variety by a fan in the lattice of one parameter subgroups, we present a new language in terms of what we call bunches -- these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the usual method by fans does. For example, the numerically effective cone and the ample cone of a toric variety can be rea…
On the classification of smooth toric surfaces with exactly one exceptional curve
本文分类了恰含一条例外曲线的光滑射影环曲面，证明其同构于 $\mathbb{F}_1$ 或由有理数 $r$ 定义的 $S_r$，并建立了 $S_r$ 的扇与相关三维簇扇的组合联系。
We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construct…
Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the quotient group N/H, where N is the normalizer of H in G. The resulting Chow quotient is equal to (a blowup of) the simple toroidal compactification of G/(H N^0). In the horospherical case, for example, it is equal to a flag variety, and the slices (coefficients) of the divisorial fan are merely shifts of the colored fan along the colors.
Toric supervarieties with one odd dimension
本文推广经典环面簇概念提出环面超簇，用带装饰的多面体扇给出单奇维度拟正规环面超簇范畴的组合解释，以此计算其不变量并提取几何信息。
We use Altmann's toric fan description of P-resolutions to formulate a new description of deformation theory invariants for two-dimensional cyclic quotient singularities. In particular, we show how to calculate the dimensions of the (reduced) versal base space components as well as Milnor numbers of smoothings over them.
We consider subtorus actions on complex toric varieties. A natural candidate for a categorical quotient of such an action is the so-called toric quotient, a universal object constructed in the toric category. We prove that if the toric quotient is weakly proper and if in addition the quotient variety is of expected dimension then the toric quotient is in fact a categorical quotient in the category of algebraic varieties. For example, weak properness always holds for the toric quotient of a subtorus action on a toric variety whose fan has a convex support.
We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semigroup rings are quadratic.
We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn, follows from a theorem that we prove for toric degenerations of more general Chow quotients. Along the way, we also argued that GIT quotient of a flat family is again flat. In particular, all GIT quotients of flag varieties by maximal tori can be flatly degenerated to projective toric varieties ([9] and [12]).
We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second o…
In this paper we show that quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$ are toric varieties.
Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors $V$ as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by $V$ and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels' …
An infinite type toric variety is a normal toric variety given by a fan with infinitely many cones. We construct examples in this paper coming from representation theory of loop groups. The fans that appear are cones on Voronoi tilings on a vector space equipped with an inner product. We also explain the affine analogue of the connection between a generic torus orbit closure in a flag variety and the closure a maximal torus in the wonderful compactification.
We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quotients. We construct a good model, i.e. a dominant toric morphism from the given toric variety to some ``maximal'' toric variety having a good quotient by the induced action of the given subtorus.
The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continui…
P-resolutions of two-dimensional, cyclic quotient singularities have been introduced to study deformation theory. Those P-resolutions (as well as the singularities themselves) are toric varieties. In the present paper we give a straight, elementary description of them just by their defining fans.
On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coinci…
Sharp Bounds and Canonical Blow-ups for Toric Projectivization
本文提出一种显式的投影化算法，通过基典范细化将任意光滑完全扇转化为光滑完全投影扇，且环态射分解为有限次余维二中心的正则爆裂。
Let $X$ be a projective normal toric variety and $T_0$ a rank one subtorus of the defining torus of $X$. We show that the normalization of the Chow quotient $X//T_0$, in the sense of Kapranov-Sturmfels-Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$.
We apply a Mayer-Vietoris sequence argument to identify the Atiyah-Segal equivariant complex K-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension 2, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with "distant singular cones," and prove that the identification h…
In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne using different techniques.
The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration A of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of A. For this, we develop a theory of irrational toric varieties associated to arb…
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.
We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. An important tool for the proof is a universal reduction of an arbitrary toric variety to a divisorial one. This is done in terms of support maps, a notion generalizing support functions on a polytopal fan. A further essential …
Multi-fan is an analogous notion of fan. As a fan is associated to a toric variety a multi-fan is associated to a torus orbifold. Orbifold elliptic class and orbifold elliptic genus are defined for a triple of a multi-fan, a set of generating integral vectors of one dimensional cones and a $\mathbb{Q}$ divisor. They are shown to behave functorially with respect to birational morphisms between these triples. The result may be considered as a combinatorial or topological analogue of the main result of Borisov and Libgober, McKay correspondence for elliptic genera, Ann. of Math., 161 (2005).
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。
In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of Demushkin on affine toric varieties.
This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks $[M/G]$, where $M$ is a smooth manifold equipped with a smooth proper action by a Lie group $G$. The characterization is described in terms of the action of the connected component $G_0$ on $M$ and is related to (stacky) fundamental group and covering theory. This characterization is then applied to smooth toric Deligne-Mumford stacks, and global quotients among toric DM stacks are then characterized in terms of their associated combinatorial data of stac…
When a reductive group acts on an algebraic variety, a linearized ample line bundle induces a stratification on the variety where the strata are ordered by the degrees of instability. In this paper, we study variation of stratifications coming from the group actions in the GIT quotient construction for projective toric varieties. Cox showed that each projective toric variety is a GIT quotient of an affine space by a diagonalizable group with respect to linearizations that come from ample divisors on the toric variety. We provide a sufficient conditions for two ample divisors to induce the same…
This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a charact…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。