Polytope duality for families of $K3$ surfaces and coupling

Period differential equations for the families of $K3$ surfaces with $2$ parameters derived from the reflexive polytopes

In this paper, we study the period mappings for the families of $K3$ surfaces derived from the $3$-dimensional $5$-verticed reflexive polytopes. We determine the lattice structures, the period differential equations and the projective monodromy groups. Moreover, we show that one of our period differential equations coincides with the unifomizing differential equation of the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$.

Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results ge…

来自多面体突变的退化几何族

Geometric families of degenerations from mutations of polytopes

本文提出多联晶格概念，用分片线性双射关联晶格族，并研究其对应多面体所编码的仿射簇紧化，证明在适当条件下这些紧化具有算术Cohen-Macaulay性及有限生成类群和Cox环。

Infinite families of hypertopes from centrally symmetric polytopes

We construct infinite families of abstract regular polytopes of type $\{4,p_1,\ldots,p_{n-1}\}$ from extensions of centrally symmetric spherical abstract regular $n$-polytopes. In addition, by applying the halving operation, we obtain infinite families of both locally spherical and locally toroidal regular hypertopes of type $\left\{{p_1 \atop p_1},\ldots,p_{n-1}\right\}$.

Completions, Reversals, and Duality for Tropical Varieties

The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of tropical hypersurfaces the corresponding point symmetry is obtained for their Newton polytopes and lattice polytopes.

Lattice duality for families of $K3$ surfaces associated to transpose duality

The aim of this article is to show that the transpose-dual pairs in the sense of Ebeling-Ploog of singularities $(Z_{1,0},\, Z_{1,0})$, $(U_{1,0},\, U_{1,0})$, $(Q_{17},\, Z_{2,0})$, $(W_{1,0},\, W_{1,0})$ that are concluded to be polytope-dual by the author are actually lattice-dual.

A mirror duality for families of $K3$ surfaces associated to bimodular singularities

Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodular singularities which is part of the Berglund--H$\ddot{\textnormal{u}}$bsch mirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to a polytope mirror symmetry of families of $K3$ surfaces. We discuss in this article how this symmetry extends to a symmetry between lattices.

Difference between families of weakly and strongly maximal integral lattice-free polytopes

A $d$-dimensional closed convex set $K$ in $\mathbb{R}^d$ is said to be lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^d$. We consider the following two families of lattice-free polytopes: the family $\mathcal{L}^d$ of integral lattice-free polytopes in $\mathbb{R}^d$ that are not properly contained in another integral lattice-free polytope and its subfamily $\mathcal{M}^d$ consisting of integral lattice-free polytopes in $\mathbb{R}^d$ which are not properly contained in another lattice-free set. It is known that $\mathcal{M}^d = \mathcal{L}^d$ holds for $d \le 3$ and, for e…

Combinatorial aspects of mirror symmetry

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…

Strange duality and polar duality

We describe a relation between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M.~Kobayashi. We show that this relation continues to hold for the extension of Arnold's strange duality found by C.~T.~C.~Wall and the author. By a method of Ehlers-Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. We generalize this method to the isolated complete intersection singularities embraced in the extended duality. We use this to explain the duality of characteristic polynomials …

Reflexive Polytopes and Lattice-Polarized K3 Surfaces

In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $Δ$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $Δ$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $Δ$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(Δ, Δ^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of M…

偏序集多胞体与管道梦：C型和B型

Poset polytopes and pipe dreams: types C and B

该研究利用偏序集多胞体与管道梦的组合联系，在C型和B型中构造并统一了表示基、Newton–Okounkov体及旗簇环面退化。

环面完全交的分次对偶与镜像对称

Framed duality and mirror symmetry for toric complete intersections

本文将完全环面簇中的超曲面族$f$-镜像对称系统扩展至完全交子簇族，基于分次对偶发展了相关理论，并探讨了广义Landau-Ginzburg/完全交对应。

Duality of planar and spacial curves: new insight

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for a regular behavior of the equiclassical stratification. We also discuss duality of curves in higher-dimensional projective spaces and in Grassmannians with focus on similar questions of the regularity of equiclassical families of spacial curves.

Mirror constructions for K3 surfaces from bimodal singularities

We study lattice polarizations of five exceptional pairs of families of K3 surfaces obtained via compactifications of strange dual pairs of bimodal singularities. We show that the polarizations induced by embedding these paired families in paired toric varieties obtained from polar dual reflexive polytopes cannot be mirror lattices and identify mirror sublattices.

Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space $V$, namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in $V$. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in $V$ gives rise to an effectively computable \SI-interpolator (and a local Euler-MacLaurin formula), an \IS-inte…

自反多胞形的似然几何

Likelihood Geometry of Reflexive Polytopes

本文研究自反多胞形定义的统计模型的ML度，计算了所有4319类三维自反多胞形的ML度，发现与环簇度的间隙，并给出超立方体等族的ML度公式及ML度为1的嵌入。

Cluster duality and mirror symmetry for Grassmannians

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $\mathbb X=Gr_{n-k}(\mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(\check{\mathbb X}^\circ, W_q:\check{\mathbb X}^\circ \to \mathbb{C})$, where $\check{\mathbb X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{\mathbb X} = Gr_k((\mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expr…

Families of distributions and Pfaff systems under duality

A singular distribution on a non-singular variety $X$ can be defined either by a subsheaf $D$ of the tangent sheaf, or by the zeros of a subsheaf $D^0$ of the sheaf of 1-forms. Although both definitions are equivalent under mild conditions on $D$, they give rise, in general, to non-equivalent notions of flat families. In this work we investigate conditions under which both notions of flat families are equivalent. In the last sections we focus on the case where the distribution is integrable, and we use our results to generalize a theorem of Cukierman and Pereira.

渐近不变量的渐进族对偶性

Duality for asymptotic invariants of graded families

本文研究了渐进族的对偶性，通过交换子加性和超加性序列并反转其渐近增长常数，揭示了其在代数几何中的应用，包括Macaulay-Matlis对偶性和jet分离序列的对偶关系。

论好的无穷Toric码族或其缺失

On Good Infinite Families of Toric Codes or the Lack Thereof

本文研究Toric码的无穷族，通过多胞体运算构造，给出不存在信息率和相对最小距离均不趋于0的好码族的条件与证据。