Advances in Geometry最新文献

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Inequalities for f *-vectors of lattice polytopes 格状多面体 f * 向量的不等式

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0002

Matthias Beck, Danai Deligeorgaki, Max Hlavacek, Jerónimo Valencia-Porras

The Ehrhart polynomial ehr P (n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f *-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr P (n) with respect to the binomial coefficient basis

$begin{array}{} bigl{binom{n-1}{0},binom{n-1}{1},dots,binom{n-1}{d}bigr}, end{array}$

where d = dim P. Similarly to h/h *-vectors, the f *-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f *-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f *-coefficients increases and the last quarter decreases. Even though f *-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h *-vector, there is a polytope with the same h *-vector whose f *-vector is unimodal.

由 Felix Breuer 于 2012 年提出的 P 的 f * 向量是 ehr P (n) 关于二项式系数基础 $begin{array}{} 的系数向量。与 h/h *-向量类似，P 的 f *-向量与其单模态三角剖分（如果存在的话）的 f -向量重合。我们提出了格状多面体的 f *-向量系数之间的几个不等式。这些不等式与简单多面体 f *-向量系数的现有不等式有惊人的相似之处；例如，f *-系数的前半部分会增加，而后四分之一会减少。尽管多面体的 f *-vectors 并不总是单峰的，但有几个多面体族具有单峰特性。我们还证明，对于任何具有给定艾尔哈特 h *向量的多面体，都有一个具有相同 h *向量的多面体，其 f *向量是单峰的。

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引用次数: 0

Variations on the Weak Bounded Negativity Conjecture 弱边界否定性猜想的变体

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2023-0027

Ciro Ciliberto, Claudio Fontanari

We present two applications of Hao’s proof of the Weak Bounded Negativity Conjecture. First, we address the so-called Weighted Bounded Negativity Conjecture and we prove that all but finitely many reduced and irreducible curves C on the blow-up of ℙ2 at n points satisfy the inequality

$begin{array}{} displaystyle C^2 ge min bigl{-frac{1}{12} n (C.L +27), -2 bigr}, end{array}$

where L is the pull-back of a line. Next, we turn to the widely open conjecture that the canonical degree C.KX of an integral curve on a smooth projective surface X is bounded from above by an expression of the form A(g − 1) + B, where g is the geometric genus of C and A, B are constants depending only on X. We prove that this conjecture holds with A = − 1 under the assumptions h 0(X, −KX ) = 0 and h 0(X, 2KX + C) = 0.

我们介绍了郝氏对弱有界否定猜想证明的两个应用。首先，我们讨论了所谓的加权有界否定猜想，并证明了在ℙ2的炸开上，除了有限多的还原曲线和不可还原曲线C之外，所有在n个点上的还原曲线和不可还原曲线C都满足不等式 $begin{array}{}displaystyle C^2 ge min bigl{-frac{1}{12} n (C.L +27), -2 bigr}, end{array}$ 其中 L 是直线的回拉。接下来，我们将讨论一个广为流传的猜想，即光滑投影面 X 上积分曲线的规范度 C.KX 是由 A(g - 1) + B 形式的表达式从上而下限定的，其中 g 是 C 的几何属，A、B 是仅取决于 X 的常数。

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引用次数: 0

Poisson Structures on moduli spaces of Higgs bundles over stacky curves 堆叠曲线上希格斯束模态空间的泊松结构

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0004

Georgios Kydonakis, Hao Sun, Lutian Zhao

We demonstrate the construction of Poisson structures via Lie algebroids on moduli spaces of twisted stable Higgs bundles over stacky curves. The construction provides new examples of Poisson structures on such moduli spaces. Special attention is paid to moduli spaces of parabolic Higgs bundles over a root stack.

我们展示了通过在堆叠曲线上的扭曲稳定希格斯束的模空间上的Lie algebroids构建泊松结构。该构造为此类模空间上的泊松结构提供了新的范例。我们特别关注根堆栈上抛物线希格斯束的模空间。

引用次数: 0

Deformation cones of Tesler polytopes 特斯勒多边形的变形锥

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0003

Yonggyu Lee, Fu Liu

For a ∈

$begin{array}{} displaystyle mathbb{R}_{geq 0}^{n} end{array}$

, the Tesler polytope Tes n ( a ) is the set of upper triangular matrices with non-negative entries whose hook sum vector is a . We first give a different proof of the known fact that for every fixed a 0 ∈

$begin{array}{} displaystyle mathbb{R}_{ gt 0}^{n} end{array}$

, all the Tesler polytopes Tes n ( a ) are deformations of Tes n ( a 0). We then calculate the deformation cone of Tes n ( a 0). In the process, we also show that any deformation of Tes n ( a 0) is a translation of a Tesler polytope. Lastly, we consider a larger family of polytopes called flow polytopes which contains the family of Tesler polytopes and chracterize the flow polytopes which are deformations of Tes n ( a 0).

For a ∈ $begin{array}{}displaystyle mathbb{R}_{geq 0}^{n}end{array}$ ，Tesler 多面体 Tes n ( a ) 是具有非负条目的上三角矩阵的集合，其钩和向量是 a 。我们首先给出一个不同的证明，即对于每一个固定的 a 0 ∈ $begin{array}{} 的已知事实。displaystyle mathbb{R}_{ gt 0}^{n}end{array}$ ，所有的 Tesler 多面体 Tes n ( a ) 都是 Tes n ( a 0) 的变形。然后我们计算 Tes n ( a 0) 的变形锥。在此过程中，我们还证明了 Tes n ( a 0) 的任何变形都是 Tesler 多面体的平移。最后，我们考虑了一个更大的多面体族，称为流多面体，它包含了 Tesler 多面体族，并对作为 Tes n ( a 0) 变形的流多面体进行了分析。

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引用次数: 0

Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs 曲面上大地流的分数线性积分和中井大地四网

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0008

Sergey I. Agafonov, Thaís G. P. Alves

We prove that if the geodesic flow on a surface has an integral which is fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for the existence of fractional-linear integrals: such an integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.

我们证明，如果曲面上的大地流具有矩分数线性积分，那么这种积分的空间维度要么是 3，要么是 5，后一种情况对应于恒定高斯曲率。我们还给出了分数线性积分存在的几何标准：当且仅当表面上有一个大地4网，且与网叶相切的四个方向的交叉比恒定时，这样的积分才存在。

引用次数: 0

Lower bound on the translative covering density of octahedra 八面体平移覆盖密度下限

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0006

Yiming Li, Yanlu Lian, Miao Fu, Yuqin Zhang

Based on Zong’s work [26] on translative packing densities of 3-dimensional convex bodies, we present a local method to estimate the density θt (C 3) of the densest translative covering of an octahedron. As a consequence we prove that θt (C 3) ≥ 1 + 6.6 × 10–8, which is the first non-trivial lower bound for this density.

基于宗庆后关于三维凸体平移堆积密度的研究[26]，我们提出了一种局部方法来估计八面体最密平移覆盖的密度θt (C 3)。因此，我们证明了 θt (C 3) ≥ 1 + 6.6 × 10-8，这是该密度的第一个非难下限。

引用次数: 0

Characterization of the sphere and of bodies of revolution by means of Larman points 通过拉曼点确定球体和旋转体的特征

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0007

M. Angeles Alfonseca, M. Cordier, J. Jerónimo-Castro, E. Morales-Amaya

Let n ≥ 3 and let K ⊂ ℝ n be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p, the section Π ∩ K has an (n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K, p is in the corresponding (n – 2)-plane of symmetry, then we call p a revolution point of K. We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝ n is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in [7]. We also show that if p is a Larman point of K ⊂ ℝ3 and there exists a line L such that p ∉ L and, for every plane Π passing through p, the line of symmetry of the section Π ∩ K intersects L, then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝ n with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D, then K is a body of revolution with axis D.

设 n ≥ 3，且 K ⊂ ℝ n 是一个凸体。如果经过 p 的每一个超平面 Π 的截面 Π ∩ K 都有一个 (n - 2) 对称面，则称点 p∈ int K 为 K 的拉曼点。如果 p 是 K 的一个拉曼点，并且对于每一段 Π ∩ K，p 都在相应的（n - 2）对称面上，那么我们称 p 为 K 的一个旋转点。我们猜想，如果 K 包含一个不是旋转点的拉曼点，那么 K 要么是一个椭圆体，要么是一个旋转体。这概括了贝兹德克对 n = 3 的猜想。我们证明了与严格凸原点对称体猜想相关的几个结果。也就是说，如果 K ⊂ ℝ n 是一个严格凸原点对称体，其中包含一个非原点的旋转点 p，那么 K 是一个旋转体。这概括了 [7] 中的假旋转轴定理。我们还证明，如果 p 是 K ⊂ ℝ3 的一个拉曼点，并且存在一条直线 L，使得 p ∉ L，并且对于经过 p 的每一个平面 Π，截面 Π ∩ K 的对称线都与 L 相交，那么 K 是一个旋转体（在某些情况下，K 是一个球体）。此外，对于 K ⊂ ℝ n（n≥4），我们证明了如果 K 的每个超平面截面或投影都是一个旋转体，并且 K 有唯一的直径 D，那么 K 是一个以 D 为轴的旋转体。

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引用次数: 0

Generalized Shioda–Inose structures of order 3 广义的 3 阶汐达-伊诺斯结构

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0005

Alice Garbagnati, Yulieth Prieto-Montañez

A Shioda–Inose structure is a geometric construction which associates to an Abelian surface a projective K3 surface in such a way that their transcendental lattices are isometric. This geometric construction was described by Morrison by considering special symplectic involutions on the K3 surfaces. After Morrison several authors provided explicit examples. The aim of this paper is to generalize Morrison’s results and some of the known examples to an analogous geometric construction involving not involutions, but order 3 automorphisms. Therefore, we define generalized Shioda–Inose structures of order 3, we identify the K3 surfaces and the Abelian surfaces which appear in these structures and we provide explicit examples.

Shioda-Inose 结构是一种几何构造，它将投影 K3 曲面与阿贝尔曲面关联起来，使它们的超越网格等距。莫里森通过考虑 K3 曲面上的特殊交映渐开线描述了这种几何构造。在莫里森之后，又有几位学者提供了明确的例子。本文的目的是将莫里森的结果和一些已知的例子推广到一个类似的几何构造中，其中涉及的不是渐开线，而是阶 3 自变量。因此，我们定义了广义的 3 阶汐达-伊诺斯结构，确定了出现在这些结构中的 K3 曲面和阿贝尔曲面，并提供了明确的例子。

引用次数: 0

The feet of orthogonal Buekenhout–Metz unitals Buekenhout-Metz 正交单元脚

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0001

S.G. Barwick, W.-A. Jackson, P. Wild

In this article we look at the geometric structure of the feet of an orthogonal Buekenhout–Metz unital 𝓤 in PG(2, q 2). We show that the feet of each point form a set of type (0, 1, 2, 4). Further, we discuss the structure of any 4-secants, and determine exactly when the feet form an arc.

本文研究了 PG(2, q 2) 中正交布肯豪特-梅兹单元𝓤 脚的几何结构。我们证明，每个点的脚都构成一个类型为 (0, 1, 2, 4) 的集合。此外，我们还讨论了任何 4-secants 的结构，并准确地确定了脚何时形成弧。

引用次数: 0

Some observations on conformal symmetries of G 2-structures 关于 G 2 结构共形对称性的几点观察

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry

Pub Date : 2024-08-13 DOI: 10.1515/advgeom-2024-0009

Christopher Lin

On a 7-manifold with a G 2-structure, we study conformal symmetries — which are vector fields whose flow generate conformal transformations of the G 2-structure. In particular, we focus on compact 7-manifolds and the condition that the Lee form of the G 2-structure is closed. Among other observations, we show that conformal symmetries are determined within a conformal class of the G 2-structure by the symmetries of a unique (up to homothety) G 2-structure whose Lee form is harmonic. On a related note, we also demonstrate that symmetries are split along fibrations when the Lee vector field is itself a symmetry.

在具有 G 2 结构的 7-manifold 上，我们研究共形对称性--即其流动产生 G 2 结构共形变换的向量场。我们尤其关注紧凑的 7-manifolds，以及 G 2-结构的李形式是封闭的这一条件。除其他观察结果外，我们还证明了共形对称性是在 G 2-结构的共形类中由唯一（同性）G 2-结构的对称性决定的，而该结构的李形式是谐波的。与此相关，我们还证明了当李向量场本身是一个对称性时，对称性会沿着纤维分裂。

引用次数: 0

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