Takayuki Hibi, Akihiro Higashitani
Gorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem which partially ordered sets yield smooth Fano polytopes.
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The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any three-dimensi…
We introduce the notion of barycentric transformation of Fano polytopes, from which we can assign a certain type to each Fano polytope. The type can be viewed as a measure of the extent to which the given Fano polytope is close to be Kähler-Einstein. In particular, we expect that every Kähler-Einstein or symmetric Fano polytope is of type $B_\infty$. We verify this expectation for some low dimensional cases. We emphasize that for a Fano polytope $X$ of dimension $1$, $3$ or $5$, $X$ is Kähler-Einstein if and only if it is of type $B_\infty$.
In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit Kähler--Einstein metrics.
Let $\mathcal{F}(n)$ be the set of smooth Fano $n$-polytopes up to unimocular equivalence. In this paper, we consider the F-equivalence or I-equivalence classes for $\mathcal{F}(n)$ and introduce F-isolated or I-isolated smooth Fano $n$-polytopes. First, we describe all of F-equivalence classes and I-equivalence classes for $\mathcal{F}(5)$. We also give a complete characterization of F-equivalence classes (I-equivalence classes) for smooth Fano $n$-polytopes with $n+3$ vertices and construct a family of I-isolated smooth Fano polytopes.
We classify the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices. In particular, it turns out that these are all smooth Fano polytopes. This improves on previous results of Casagrande in 2006 and Oebro in 2008. Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.
The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth $d$-dimensional Fano polytope has at most $3d$ vertices. Smooth Fano polytopes in dimension $d$ with at least $3d-2$ vertices are completely known. The main result of this paper deals with the case of $3d-k$ vertices for $k$ fixed and $d$ large. It implies that there …
The Fine interior $Δ^{\text{FI}}$ of a $d$-dimensional lattice polytope $Δ$ is a rational subpolytope of $Δ$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope $Δ$. This paper presents some computational results on the Fine interior of all $674,\!688$ three-dimensional canonical Fano polytopes.
Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
We introduce the notion of cracked polytope, and - making use of joint work with Coates and Kasprzyk - construct the associated toric variety $X$ as a subvariety of a non-singular toric variety $Y$ under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of $X$ to exist inside $Y$. We exhibit a relative anti-canonical divisor for this smoothing of $X$, and show that the general member is simple normal crossings.
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。
Bounds for smooth Fano polytopes
研究光滑Fano多面体的delta向量和f向量特性，证明其delta向量单峰性并给出体积上下界。
Let $Δ$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_Δ$ associated with the polytope $Δ$. Using the stringy Libgober-Wood identity and our formula, we generalize the well-known combinatorial identity $\sum_{θ\preceq Δ\atop \dim (θ) =1} v(θ) \cdot v(θ^*) = 24$ holding in the case of $3$-dimensional reflexive polytopes $Δ$.
A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications. This generalizes a result of Ewald on pseudo-symmetric nonsingular toric Fano varieties.
It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension $d$ is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart …
We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional toric Fano variety X with at worst canonical singularities.
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.
The symmetric edge polytopes of odd cycles (del Pezzo polytopes) are known as smooth Fano polytopes. In this paper, we show that if the length of the cycle is 127, then the Ehrhart polynomial has a root whose real part is greater than the dimension. As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the roots of Ehrhart polynomials.
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…
Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there e…
We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has "many" pairs of centrally symmetric vertices.
Unimodular Smooth Fano Polytopes and their Relation with Ewald Conditions
本文引入幺模光滑 Fano 多面体（USFP），证明其满足三种 Ewald 条件，并利用 Seymour 分解定理给出刻画，同时将深度单调多面体的对偶性质推广到 USFP。
V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.
Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.
Symmetric and Kähler--Einstein Fano polygons
本文研究奇异对称与 Kähler--Einstein Fano 多胞体，证明所有对称 Fano 多胞体均为 Kähler--Einstein，并详细刻画了对称与 Kähler--Einstein Fano 多边形的自同构群。
We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.
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 characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.
Richard Stanley introduced the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\mathcal{O}(P)$ is equal to that of $\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\mathcal{O}(P)$ and $\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $Γ(\mathcal{O}(P), -\mathcal{O}(Q))$, $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ and $Γ(\mathcal{C}(P), -\mathcal{C}(Q))$, where $P$ and $Q$ are partially ordered sets wi…
Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list …
Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes
本文证明了伪对称单纯反射多胞体的极对偶的Ehrhart多项式具有魔幻正性，并进一步证明伪对称光滑Fano多胞体的魔幻系数严格为正，从而得出其极对偶Ehrhart正且$h^*$-多项式实根且$\u03b3$-正，同时解决了Konoike关于循环对称边多胞体的问题。
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。
The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ but is not always equal to it: this situation is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. Using techniques from algebraic geometry - specifically the $\mathbb{Q}$-Gorenstein deformation theory of orbifold del Pezzo surfaces - we explain quasi-period collapse for ra…
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts ab…
Fano and Reflexive Polytopes from Feynman Integrals
本文研究了从准有限费曼积分中出现的Fano和反射多面体，通过计算相关双变量Ehrhart多项式确定内部点数，发现这些多面体在高维情况下稀少，并揭示了一种与Calabi-Yau流形相关的几何结构。
Let $P$ be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope ${\mathcal O}(P)$ is equal to that of the chain polytope ${\mathcal C}(P)$. Furthermore, it will be shown that the degree sequence of the finite simple graph which is the $1$-skeleton of ${\mathcal O}(P)$ is equal to that of ${\mathcal C}(P)$ if and only if ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent.
Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furt…
This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.
Residue polytopes
本文为层级图定义留数多胞体，证明其在有序划分加粗下具有面包含相容性，并给出平凡划分下留数多胞体的全部面结构，用于描述黎曼面族退化时阿贝尔微分空间的极限。
Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $Γ(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give t…
Though much is known about ${\bf s}$-lecture hall polytopes, there are still many unanswered questions. In this paper, we show that ${\bf s}$-lecture hall polytopes satisfy the integer decomposition property (IDP) in the case of monotonic ${\bf s}$-sequences. Given restrictions on a monotonic ${\bf s}$-sequence, we discuss necessary and sufficient conditions for the Fano, reflexive and Gorenstein properties. Additionally, we give a construction for producing Gorenstein/IDP lecture hall polytopes.
We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.
The problem when the order polytope and the chain polytope of a finite partially ordered set are unimodularly equivalent will be solved.
The order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ which is the convex hull of $\mathcal{O}(P) \cup (-\mathcal{C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.
Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an object which corresponds to the equivariant intersection cohomology complex. Our first main result is the ``elementary'' decomposition theorem for the direct image under subdivision of fans We also develop the Borel-Moore- Verdier duality in the derived category of sheaves on the fan.
The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. \cite[Conjecture 4.10]{MHNOH}, says that all roots $α$ of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension $d$ satisfy $-\frac{d}{2} \leq \Re(α) \leq \frac{d}{2} -1$. In this paper, we observe the behaviors of roots of SSNN polynomials which are a wider class of the polynomials containing all the Ehrhart polynomials of Gorenstein Fano polytopes. As a result, we verify that this conjecture is true when the roots are real numbers or when $d \leq 5$.
A Gorenstein polytope of index r is a lattice polytope whose r-th dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper, we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify d-dimensional smooth Gorenstein polytopes with index larger than (d+3)/3. Moreover, we use a modification of Oebr…
The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensio…
In 2010, M. Studený, R. Hemmecke, and S. Linder explored a new algebraic description of graphical models, called characteristic imsets. Compare with standard imsets, characteristic imsets have several advantages: they are still unique vector representative of conditional independence structures, they are 0-1 vectors, and they are more intuitive in terms of graphs than standard imsets. After defining a characteristic imset polytope (cim-polytope) as the convex hull of all characteristic imsets with a given set of nodes, they also showed that a model selection in graphical models, which maximize…
A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on …
It will be shown that the toric ring of the chain polytope of a finite partially ordered set is an algebra with straightening laws on a finite distributive lattice. Thus in particular every chain polytope possesses a regular unimodular triangulation arising from a flag complex.
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $α$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(α) \le D - 1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle $|z+\tfrac{d}{4}| \le \tfrac{d}{4}$ or are negative int…
Stanley (1986) introduced the order polytope and chain polytope of a partially ordered set and showed that they are related by a piecewise-linear homeomorphism. In this paper we view order and chain polytopes as instances of distributive and anti-blocking polytopes, respectively. Both these classes of polytopes are defined in terms of the componentwise partial order on $\mathbb R^n$. We generalize Stanley's PL-homeomorphism to a large class of distributive polyhedra using infinite walks in marked networks.
The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen--Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen--Macaulay poset of the same rank. A corresponding notion of edgewise Cohen--Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.
Equatorial Flow Triangulations of Gorenstein Flow Polytopes
本文应用Bruns和Römer的赤道三角剖分技术，给出Gorenstein流多面体未被研究的三角剖分的纯组合描述，证明其与框架法所得三角剖分不同，还找到投影自反多面体的面描述，并在强平面偏序集下生成序多面体新幺模三角剖分。
Let $K[\mathcal{O}(P)]$ denote the toric ring of the order polytope $\mathcal{O}(P)$ of a finite partially ordered set $P$ and $K[\mathcal{C}(P)]$ that of the chain polytope $\mathcal{C}(P)$. It will be shown that $β_{p, p+j}(K[\mathcal{O}(P)]) = β_{p, p+j}(K[\mathcal{C}(P)])$ for all $j \geq 0$, where $p$ is the projective dimension of $K[\mathcal{O}(P)]$ (and that of $K[\mathcal{C}(P)]$). In particular, $K[\mathcal{O}(P)]$ is level if and only if $K[\mathcal{C}(P)]$ is level.
Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties
本文证明了 Gorenstein 环面 Fano 簇的渐近 Chow 半稳定性蕴含关于环面测试构型的 Ding 多稳定性，推广了已有结果，并给出了 Mabuchi 常数的可加性公式及相对 K-稳定性与 Ding 稳定性差异的系统实例。
Stellahedral geometry of partially ordered sets
本文提出偏序集的星状变换，保持欧拉性等性质，并证明其与环面h-多项式联系，从而得出增广Chow多项式的正性结果，并否定两个开放问题。
The extremals of Stanley's inequalities for partially ordered sets
本文完整刻画了偏序集上 Stanley 不等式的等号成立情形，通过建立偏序集组合学与凸多面体几何之间的新“字典”来刻画极值结构。
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