代数几何

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[1] arXiv:2508.02848 [中文pdf, pdf, html, 其他]

标题： 相对杜波伊斯复形的一般基变换 显示英文标题

标题： General base change for relative Du Bois complexes

Caleb Ji, Sándor Kovács

评论： 7页

主题： 代数几何 (math.AG)

对Kovács和Taji在arxiv:2307.07192中提出的问题给出了部分回答，即由非奇异曲线参数化的族的相对Du Bois复形在基底的一般点上的基变换下保持不变。 还表明，这一性质在特殊点上通常不成立。 显示英文摘要

A partial answer is given to a question raised by Kov\'acs and Taji in arxiv:2307.07192, namely that the relative Du Bois complex of a family parametrized by a non-singular curve commutes with base change to a general point on the base. It is also shown that this property usually fails for special points.

[2] arXiv:2508.02893 [中文pdf, pdf, html, 其他]

标题： 关于Enriques流形的Nielsen实现问题的一篇注记 显示英文标题

标题： A note on the Nielsen realization problem for Enriques manifolds

Simone Billi

评论： 11页，欢迎提出评论！

主题： 代数几何 (math.AG) ; 几何拓扑 (math.GT)

我们给出一个数值准则，用于解决Enriques流形的Nielsen实现问题，该准则基于超凯勒流形的Birman-Hilden理论的最新发展以及超凯勒流形的Nielsen实现。我们将该准则应用于已知的Enriques流形例子，以获得可以实现或无法实现的显式群，并对与Nielsen实现问题相关的问题进行评论。 显示英文摘要

We give a numerical criterion for the Nielsen realization problem for Enriques manifolds, based on the recent developments on the Birman-Hilden theory for hyper-K\"ahler manifolds and on Nielsen realization for hyper-K\"ahler manifolds. We apply the criterion to known examples of Enriques manifolds to get explicit groups that can be realized or not realized, and comment on questions related to the Nielsen realization problem.

[3] arXiv:2508.02941 [中文pdf, pdf, html, 其他]

标题： 类型C的热带簇簇 varieties 显示英文标题

标题： Tropical cluster varieties of type C

Igor Makhlin

主题： 代数几何 (math.AG) ; 组合数学 (math.CO)

我们明确描述了有限类型C的簇流形的热带化，将其视为轴对称系统发育树的空间。 我们还找到了簇流形和簇配置空间中坐标出现的所有符号模式。 我们将相应的带符号的热带化实现为热带化中的子扇，每个这样的子扇与一个循环多面体或结合多面体对偶。 作为附加结果，我们构建了两个流形定义理想的格里纳尔和热带基，并分类了出现的环面退化。 显示英文摘要

We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and the cluster configuration space. We realize the corresponding signed tropicalizations as subfans of the tropicalization, with each such subfan dual to either a cyclohedron or an associahedron. As additional results, we construct Gr\"obner and tropical bases for the defining ideals of both varieties, and classify the arising toric degenerations.

[4] arXiv:2508.03103 [中文pdf, pdf, html, 其他]

标题： 关于紧化雅可比簇的陈类的顽固性 显示英文标题

标题： On the Perversity of Chern Classes for Compactified Jacobians

Soumik Ghosh

评论： 25页。欢迎提出评论！

主题： 代数几何 (math.AG)

我们证明了紧化雅可比纤维丛的陈类的一些 perverse 界，即紧化雅可比的$k$-阶陈类具有 perverse$\leq k$。我们的结果具有动机性质，并且我们还证明了在 arxiv:2402:08861 中提出的猜想的一个分层版本。 显示英文摘要

We prove some perversity bounds for the Chern classes of a compactified Jacobian fibration, namely the $k$-th Chern class of the compactified Jacobian has perversity $\leq k$. Our results are motivic in nature, and we also prove a filtration version of a conjecture raised in arxiv:2402:08861.

[5] arXiv:2508.03145 [中文pdf, pdf, html, 其他]

标题： Theta-范畴和Tannakian对偶性 显示英文标题

标题： Theta-Categories and Tannakian duality

Joost Nuiten, Bertrand Toen

评论： 33页

主题： 代数几何 (math.AG)

我们引入了$\Theta$-范畴的概念，这是对对称单子$\infty$-范畴概念的改进。 我们使用这个概念来证明一个Tannakian对偶陈述，在$\Theta$-范畴的背景下，通过某种纤维函子堆栈将$\Theta$-范畴与fpqc-堆叠联系起来。 这提供了一个强大的联系，在任意特征的基环上，Tannakian$\Theta$-范畴与形式基本群类型之间。 显示英文摘要

We introduce a notion of $\Theta$-categories, which is a refinement of the notion of symmetric monoidal $\infty$-categories. We use this notion to prove a Tannakian duality statement, relating $\Theta$-categories with fpqc-stacks by means of a certain stack of fiber functors in the context of $\Theta$-categories. This provides, over a base ring of arbitrary characteristic, a strong link between Tannakian $\Theta$-categories and the schematic homotopy types.

[6] arXiv:2508.03299 [中文pdf, pdf, html, 其他]

标题： 关于卡拉比-丘算子的构造 显示英文标题

标题： On the construction of Calabi-Yau operators

Stefan Reiter

主题： 代数几何 (math.AG)

给定一个几何来源的微分算子，存在一系列保持该性质的操作，例如，张量积、拉回、前推和中间卷积。 我们应用这些操作的某些序列来构造已知的和新的卡拉比-丘算子。 显示英文摘要

Given a differential operator of geometric origin there exists a list of operations that preserve this property, e.g., tensor products, pull-backs, push-forwards and the middle convolution. We apply certain sequences of these operations to construct known and new examples of Calabi-Yau operators.

[7] arXiv:2508.03623 [中文pdf, pdf, html, 其他]

标题： 非有理变体与互质次数的有理参数化 显示英文标题

标题： Nonrational varieties with unirational parametrizations of coprime degrees

Song Yang, Xun Yu, Zigang Zhu

评论： 10页

主题： 代数几何 (math.AG)

我们证明存在一个$2$维的光滑三次超曲面族，它们接受互质次数的有理参数化。 这与 Clemens--Griffiths 的工作一起解决了长期存在的开放问题，即是否存在一个非有理簇具有互质次数的有理参数化。 我们的证明使用了一种新的方法，称为 Noether--Cremona 方法，用于确定超曲面商的有理性。 显示英文摘要

We show that there exists a $2$-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This together with Clemens--Griffiths' work solves the long standing open problem whether there exists a nonrational variety with unirational parametrizations of coprime degrees. Our proof uses a new approach, called the Noether--Cremona method, for determining the rationality of quotients of hypersurfaces.

交叉提交 (展示 6 之 6 条目 )

[8] arXiv:2508.02800 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]

标题： 从Zeta生成元出发的单值共作用研究 显示英文标题

标题： Towards Motivic Coactions at Genus One from Zeta Generators

Axel Kleinschmidt, Franziska Porkert, Oliver Schlotterer

评论： 46 + 17页

主题： 高能物理 - 理论 (hep-th) ; 代数几何 (math.AG) ; 数论 (math.NT)

多重zeta值和多重多对数的动机共作用在各种量子场论和弦理论中的散射幅度的结构洞察和计算方法中编码了重要的信息。 在本工作中，我们提出了来自 genus 一配置空间积分的全纯Eisenstein级数的迭代积分的共作用公式。 我们的提议是基于最近通过zeta生成元重新表述的零genus下多重多对数的动机共作用和单值映射之间的形式相似性。 然后，通过类比于利用zeta生成元在genus 一构造单值迭代Eisenstein积分的方法，提出了本工作中的genus一共作用。 我们证明了我们的提议表现出共作用预期的性质，并推导了从正则化极限得到的多重模值的$f$-字母分解。 显示英文摘要

The motivic coaction of multiple zeta values and multiple polylogarithms encodes both structural insights on and computational methods for scattering amplitudes in a variety of quantum field theories and in string theory. In this work, we propose coaction formulae for iterated integrals over holomorphic Eisenstein series that arise from configuration-space integrals at genus one. Our proposal is motivated by formal similarities between the motivic coaction and the single-valued map of multiple polylogarithms at genus zero that are exposed in their recent reformulations via zeta generators. The genus-one coaction of this work is then proposed by analogies with the construction of single-valued iterated Eisenstein integrals via zeta generators at genus one. We show that our proposal exhibits the expected properties of a coaction and deduce $f$-alphabet decompositions of the multiple modular values obtained from regularized limits.

[9] arXiv:2508.02859 (交叉列表自 math.GR) [中文pdf, pdf, html, 其他]

标题： 线性群的$2$维度 显示英文标题

标题： The essential $2$-dimension of the linear groups

Hannah Knight

评论： arXiv管理员注释：与arXiv:2109.02698和arXiv:2204.13227文本重叠

主题： 群论 (math.GR) ; 代数几何 (math.AG) ; 表示理论 (math.RT)

在本文中，我们计算了当定义素数为奇数时一般线性群、射影一般线性群、当$n$为奇数或$n = 2$时的特殊线性群，以及在$q \equiv 1 \mod 4$、$s = v_2(q-1)$和$\Gamma = \text{Gal}(k(\zeta_{2^s})/k)$为平凡的情况下特殊线性群及其商群（如射影特殊线性群）的本质$2$维度。 显示英文摘要

In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear groups and quotients of it (such as the projective special linear groups) in the case case $q \equiv 1 \mod 4$, $s = v_2(q-1)$, and $\Gamma = \text{Gal}(k(\zeta_{2^s})/k)$ is trivial.

[10] arXiv:2508.02891 (交叉列表自 math.CO) [中文pdf, pdf, 其他]

标题： Plabic 穿插和簇促进映射 显示英文标题

标题： Plabic Tangles and Cluster Promotion Maps

Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, Lauren Williams

评论： 80页，13图

主题： 组合数学 (math.CO) ; 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 代数几何 (math.AG)

受对amplituhedron的tilings的BCFW递归的启发，我们引入了“plabic tangles”的一般框架，该框架利用plabic图在称为“promotions”的Grassmannian乘积之间定义有理映射。 论文的核心猜想是promotion映射是准簇同态，我们已对该类promotion的几个类别进行了证明。 为了定义promotion映射，我们利用plabic图上的$m$-向量关系配置（$m$-VRCs）。 我们将$m$-VRCs与amplituhedron映射在positroid variety上的次数（也称为“交点数”）相关联，并表征所有交点数为一的plabic树及其VRCs。 最后，我们表明promotion映射具有操作符结构，并且通过“$4$-质量框”promotion的类别，我们指出了超出簇代数的非有理映射的新正性性质。 Promotion映射与amplituhedron的几何和簇结构以及平面$\mathcal{N}=4$超Yang-Mills理论中散射幅度的奇点有重要联系。 显示英文摘要

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotion, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory.

[11] arXiv:2508.02907 (交叉列表自 math.CO) [中文pdf, pdf, html, 其他]

标题： 洛伦兹多项式和三角超域上的拟阵 1：拓扑方面 显示英文标题

标题： Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects

Matthew Baker, June Huh, Mario Kummer, Oliver Lorscheid

主题： 组合数学 (math.CO) ; 代数几何 (math.AG)

洛伦兹多项式在连续和离散凸性之间架起了一座桥梁，连接了分析学和组合学。 在本文中，我们研究了空间$\mathbb{P}\textrm{L}_J$在$J$上模$\mathbb{R}_{>0}$的拓扑结构，当且仅当$J$是一个拟多面体的基集合时该空间非空。 我们证明了$\mathbb{P}\textrm{L}_J$是一个边界为维数等于$J$的 Tutte 秩的流形，并且更准确地说，它与移除了$J$的 Dressian 边界后的闭欧几里得球面同胚。 Furthermore, we show that $\mathbb{P}\textrm{L}_J$ is homeomorphic to the thin Schubert cell $\textrm{Gr}_J(\mathbb{T}_q)$ of $J$ over the triangular hyperfield $\mathbb{T}_q$, introduced by Viro in the context of tropical geometry and Maslov dequantization, for any $q>0$. This identification enables us to apply the representation theory of polymatroids developed in a companion paper, as well as earlier work by the first and fourth authors on foundations of matroids, to give a simple explicit description of $\mathbb{P}\textrm{L}_J$ up to homeomorphism in several key cases. Our results show that $\mathbb{P}\textrm{L}_J$ always admits a compactification homeomorphic to a closed Euclidean ball. 它们还可以用来以否定方式回答Brändén提出的问题，通过证明在所有多项式模$\mathbb{R}_{>0}$的空间中，$\mathbb{P}\textrm{L}_J$的闭包通常不是与闭欧几里得球同胚的。 此外，我们引入了Lorentzian多项式的缩放类空间的Hausdorff紧化，并证明复数Grassmannian的Chow商自然地映射到这个紧化空间。 这提供了一个几何框架，将Lorentzian多项式空间的渐近结构与代数几何中的经典构造联系起来。 显示英文摘要

Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mathbb{P}\textrm{L}_J$ of Lorentzian polynomials on $J$ modulo $\mathbb{R}_{>0}$, which is nonempty if and only if $J$ is the set of bases of a polymatroid. We prove that $\mathbb{P}\textrm{L}_J$ is a manifold with boundary of dimension equal to the Tutte rank of $J$, and more precisely, that it is homeomorphic to a closed Euclidean ball with the Dressian of $J$ removed from its boundary. Furthermore, we show that $\mathbb{P}\textrm{L}_J$ is homeomorphic to the thin Schubert cell $\textrm{Gr}_J(\mathbb{T}_q)$ of $J$ over the triangular hyperfield $\mathbb{T}_q$, introduced by Viro in the context of tropical geometry and Maslov dequantization, for any $q>0$. This identification enables us to apply the representation theory of polymatroids developed in a companion paper, as well as earlier work by the first and fourth authors on foundations of matroids, to give a simple explicit description of $\mathbb{P}\textrm{L}_J$ up to homeomorphism in several key cases. Our results show that $\mathbb{P}\textrm{L}_J$ always admits a compactification homeomorphic to a closed Euclidean ball. They can also be used to answer a question of Br\"and\'en in the negative by showing that the closure of $\mathbb{P}\textrm{L}_J$ within the space of all polynomials modulo $\mathbb{R}_{>0}$ is not homeomorphic to a closed Euclidean ball in general. In addition, we introduce the Hausdorff compactification of the space of rescaling classes of Lorentzian polynomials and show that the Chow quotient of a complex Grassmannian maps naturally to this compactification. This provides a geometric framework that connects the asymptotic structure of the space of Lorentzian polynomials with classical constructions in algebraic geometry.

[12] arXiv:2508.03416 (交叉列表自 math.CV) [中文pdf, pdf, html, 其他]

标题： 关于Christoffel-Darboux核的局部化 显示英文标题

标题： On the Localization of the Christoffel-Darboux Kernel

Siarhei Finski

评论： 14页；在其他结果中，我们表明，某些结论来自arXiv:2506.01610，在不假设Bernstein-Markov性质的情况下仍然有效，只要对Christoffel-Darboux核进行更细致的分析。

主题： 复变量 (math.CV) ; 代数几何 (math.AG) ; 经典分析与常微分方程 (math.CA) ; 泛函分析 (math.FA)

我们证明了对于非全纯测度，Christoffel-Darboux（或Bergman）核在对角线上局部化。 这导致了在Toeplitz算子理论中的几个应用，特别是关于其渐近代数性质和谱等分布。 显示英文摘要

We establish that the Christoffel-Darboux (or Bergman) kernel localizes along the diagonal for non-pluripolar measures. This leads to several applications in the theory of Toeplitz operators, particularly concerning their asymptotic algebra property and spectral equidistribution.

[13] arXiv:2508.03568 (交叉列表自 math.CO) [中文pdf, pdf, html, 其他]

标题： 通过导数计算幂级数分解 显示英文标题

标题： Computing plethysms via derivations

Alessandro D'Andrea, Enrico Fatighenti, Claudio Onorati

评论： 18页。欢迎提出意见

主题： 组合数学 (math.CO) ; 代数几何 (math.AG)

We consider a derivation D on the ring ${\Lambda}$ of symmetric functions and investigate its combinatorial, algebraic and geometric properties. More precisely, we show that D restricts to a quasi-isometry, with respect to the Hall product, on the graded component of ${\Lambda}$ of each positive degree and provide a chain-rule formula with respect to the plethysm operation. Furthermore, we relate the geometric shape of D(f), where f $\in {\Lambda}$ is an homogeneous symmetric function, to that of f. An application to the shape of the partitions appearing in a given plethysms is proved. 显示英文摘要

We consider a derivation D on the ring ${\Lambda}$ of symmetric functions and investigate its combinatorial, algebraic and geometric properties. More precisely, we show that D restricts to a quasi-isometry, with respect to the Hall product, on the graded component of ${\Lambda}$ of each positive degree and provide a chain-rule formula with respect to the plethysm operation. Furthermore, we relate the geometric shape of D(f), where f $\in {\Lambda}$ is an homogeneous symmetric function, to that of f. An application to the shape of the partitions appearing in a given plethysms is proved.

替换提交 (展示 15 之 15 条目 )

[14] arXiv:1802.10202 (替换) [中文pdf, pdf, html, 其他]

标题： 伪有效4-fold翻转的终止 显示英文标题

标题： Termination of pseudo-effective 4-fold flips

Joaquín Moraga

评论： 最终版本

主题： 代数几何 (math.AG)

设$(X,\Delta)$是在特征零的代数闭域上的对数规范$4$-流形。 我们证明任何序列的$(K_X+\Delta)$-翻转变换都会终止。 显示英文摘要

Let $(X,\Delta)$ be a log canonical $4$-fold over an algebraically closed field of characteristic zero. We prove that any sequence of $(K_X+\Delta)$-flips terminates.

[15] arXiv:2309.16187 (替换) [中文pdf, pdf, html, 其他]

标题： 有理性问题对于范数一环面在$A_5$和${\rm PSL}_2(\mathbb{F}_8)$扩展中 显示英文标题

标题： Rationality problem for norm one tori for $A_5$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions

Akinari Hoshi, Aiichi Yamasaki

评论： 26页。猜想1.4已修改。摘要也已修改。arXiv管理员注：与arXiv:2302.06231、arXiv:1811.01676、arXiv:1811.02145、arXiv:1210.4525存在文本重叠。

主题： 代数几何 (math.AG) ; 数论 (math.NT) ; 环与代数 (math.RA)

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions. 特别是，我们通过使用GAP计算并在PARI/GP的帮助下证明了当$G={\rm Gal}(L/k)\simeq {\rm PSL}_2(\mathbb{F}_{8})$、$H={\rm Gal}(L/K)\simeq (C_2)^3$和$H\simeq (C_2)^3\rtimes C_7$时，$T$是稳定的$k$-有理的，其中$C_n$是阶为$n$的循环群。 基于结果，我们猜想 $T$ 对于 $G\simeq {\rm PSL}_2(\mathbb{F}_{2^d})$, $(C_2)^d\leq H\leq (C_2)^d\rtimes C_{2^d-1}$是稳定 $k$-有理的。 其他一些情况$G\simeq A_n$, $S_n$, ${\rm GL}_n(\mathbb{F}_{p^d})$, ${\rm SL}_n(\mathbb{F}_{p^d})$, ${\rm PGL}_n(\mathbb{F}_{p^d})$, ${\rm PSL}_n(\mathbb{F}_{p^d})$和 $H\lneq G$也针对较小的 $n$ 和 $p^d$ 进行了研究。 显示英文摘要

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions. In particular, we prove that $T$ is stably $k$-rational for $G={\rm Gal}(L/k)\simeq {\rm PSL}_2(\mathbb{F}_{8})$, $H={\rm Gal}(L/K)\simeq (C_2)^3$ and $H\simeq (C_2)^3\rtimes C_7$ where $C_n$ is the cyclic group of order $n$ by using GAP computations with the aid of PARI/GP. Based on the result, we conjecture that $T$ is stably $k$-rational for $G\simeq {\rm PSL}_2(\mathbb{F}_{2^d})$, $(C_2)^d\leq H\leq (C_2)^d\rtimes C_{2^d-1}$. Some other cases $G\simeq A_n$, $S_n$, ${\rm GL}_n(\mathbb{F}_{p^d})$, ${\rm SL}_n(\mathbb{F}_{p^d})$, ${\rm PGL}_n(\mathbb{F}_{p^d})$, ${\rm PSL}_n(\mathbb{F}_{p^d})$ and $H\lneq G$ are also investigated for small $n$ and $p^d$.

[16] arXiv:2404.03935 (替换) [中文pdf, pdf, html, 其他]

标题： 与Kodaira循环和正交簇相关的Feigin-Odesskii括号 显示英文标题

标题： Feigin-Odesskii brackets associated with Kodaira cycles and positroid varieties

Zheng Hua, Alexander Polishchuk

评论： 在此版本中，我们更正了旧定理3.2.1中的一个错误。它现在被新的定理3.2.2所替代。Feigin-Odesskii括号和标准括号应该通过在极大环面上的双向量场的扭转变换而有所不同。

主题： 代数几何 (math.AG) ; 组合数学 (math.CO) ; 量子代数 (math.QA) ; 表示理论 (math.RT)

我们将格拉斯曼流形中的开放正则素簇$G(k,n)$与某些向量丛复形的模空间在科达拉环$C^n$上建立联系，利用后者模空间上的偏移泊松结构，并将其与$G(k,n)$上标准泊松结构的某种扭变相关联。 %通过其极大环上的双矢量场。 这种联系使我们能够解决向量丛在$C^n$上的扩张分类问题。 基于此解决方案，我们进一步对$G(k,n)$中所有正则素簇在扭曲的标准泊松结构下的辛叶进行了分类。 此外，我们得到了在扭曲的标准泊松结构下$G(k,n)$的辛叶模堆栈的显式描述，作为向量丛堆栈的一个开子堆栈$C^n$。 显示英文摘要

We establish a link between open positroid varieties in the Grassmannians $G(k,n)$ and certain moduli spaces of complexes of vector bundles over Kodaira cycle $C^n$, using the shifted Poisson structure on the latter moduli spaces and relating them to a certain twist of the standard Poisson structure on $G(k,n)$. %by a bivector field on its maximal torus. This link allows us to solve a classification problem for extensions of vector bundles over $C^n$. Based on this solution we further classify the symplectic leaves of all positroid varieties in $G(k,n)$ with respect to the twisted standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of $G(k,n)$ with the twisted standard Poisson structure as an open substack of the stack of vector bundles on $C^n$.

[17] arXiv:2405.17113 (替换) [中文pdf, pdf, html, 其他]

标题： 有理同伦与主$G$-丛模堆的霍奇理论 显示英文标题

标题： Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles

Pedro L. del Angel R., Frank Neumann

评论： 14页

期刊参考： 新的数学分析工具及其应用——第14届ISAAC大会论文集，巴西里贝朗普雷托，2023年，{\em 数学趋势}，{\em 研究视角}，Springer--Birkh\"auser 2025年，35--49

主题： 代数几何 (math.AG) ; 代数拓扑 (math.AT)

对于半单复代数群$G$，我们使用底层拓扑堆栈的同伦理论，确定了在连通光滑复射影簇$X$上主$G$-丛的模堆栈${\mathscr B}un_{G,X}$的有理上同调和 Hodge-Tate 结构。 显示英文摘要

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack ${\mathscr B}un_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of special type using the homotopy theory of the underlying topological stack.

[18] arXiv:2407.17701 (替换) [中文pdf, pdf, html, 其他]

标题： 几何、算术和混沌 显示英文标题

标题： On Geometry, Arithmetics and Chaos

Lars Andersen

主题： 代数几何 (math.AG) ; 微分几何 (math.DG) ; 动力系统 (math.DS)

我们的主要结果是，维度$n+1$中的混沌是一个一维的几何对象，嵌入在一个维度为$n$的几何对象中，这对应于一个维度为$n$的对象，该对象可能是奇异的或非奇异的。 我们的主要结果是，在第一种情况下，这种混沌发生在孤立奇点或非孤立奇点上。 在第一种情况下，这种混沌要么是边界混沌，要么是球面混沌，这在非奇异情况下也发生。 在孤立奇点几何的情况下，会出现混沌，这种混沌可以是边界混沌、球面混沌或管状混沌。 我们进一步证明，质数表现出量子行为。 显示英文摘要

Our main result is that chaos in dimension $n+1$ is a one-dimensional geometrical object embedded in a geometrical object of dimension $n$ which corresponds to a $n$ dimensional object which is either singular or non-singular. Our main result is then that this chaos occurs in the first case as either on an isolated or non-isolated singularity. In the first case this chaos is either boundary chaos or spherical chaos which is what happens also in the non-singular case. In the case of an isolated singular geometry one has chaos which can either be boundary, spherical or tubular chaos. We furthermore prove that the prime numbers display quantum behaviour.

[19] arXiv:2409.08370 (替换) [中文pdf, pdf, html, 其他]

标题： Hrushovski定理的代数证明 显示英文标题

标题： An Algebraic Proof of Hrushovski's Theorem

Thomas Wisson

评论： 备注1.1、1.5、1.6和1.8被添加以帮助解释定理1.3的背景，该定理本身也进行了略微改写以增强清晰度。

主题： 代数几何 (math.AG) ; 数论 (math.NT)

在他的关于Mordell-Lang猜想的论文中，Hrushovski运用了模型论的技术来证明该猜想的函数域版本。 在这样做时，他能够回答Voloch的一个相关问题，我们此后将其称为Hrushovski定理。 在本文中，我们将给出该定理在特征$p$情形下的另一种证明，但仅使用纯粹的代数几何思想。 显示英文摘要

In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques from model theory to prove the function field version of the conjecture. In doing so he was able to answer a related question of Voloch, which we refer to henceforth as Hrushovski's theorem. In this paper we shall give an alternative proof of said theorem in the characteristic $p$ setting, but using purely algebro-geometric ideas.

[20] arXiv:2507.08522 (替换) [中文pdf, pdf, html, 其他]

标题： 奇异代数簇的Miyaoka-Yau不等式，其典范或反典范除子为大除子 显示英文标题

标题： The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

Masataka Iwai, Satoshi Jinnouchi, Shiyu Zhang

评论： v2：62页；小修。（修订了引言并增加了命题4.25。）v1：61页；欢迎提出意见

主题： 代数几何 (math.AG) ; 复变量 (math.CV) ; 微分几何 (math.DG)

我们建立了对于具有大典范除子 $K_X$的 $n$维射影 klt 变体的 Miyaoka-Yau 不等式： \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \]。我们也证明了对于具有大反典范除子 $-K_X$的 K-半稳定射影 klt 变体的 Miyaoka-Yau 不等式。作为我们方法的一部分，我们定义了奇异变体上的非多极化积 $\langle \alpha_1 \cdots \alpha_p \rangle$，并针对一个大类 $\alpha$建立了 $\langle \alpha^{n-1} \rangle$-半稳定 Higgs 纹层的 Bogomolov-Gieseker 类型不等式。 此外，我们研究在$K_X$或$-K_X$为 nef 的情况下第二陈类不等式。 显示英文摘要

We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle \alpha_1 \cdots \alpha_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle \alpha^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $\alpha$. In addition, we investigate second Chern class inequalities in the cases where $K_X$ or $-K_X$ is nef.

[21] arXiv:2507.19163 (替换) [中文pdf, pdf, html, 其他]

标题： Fano 算法的次最大初等对称函数 显示英文标题

标题： Fano schemes of sub-maximal elementary symmetric functions

Alexandru Chirvasitu

评论： 8页+参考文献；v2更新了参考文献并添加了推论0.3

主题： 代数几何 (math.AG) ; 组合数学 (math.CO) ; 环与代数 (math.RA) ; 表示理论 (math.RT)

Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. 我们描述了射影空间$(d-1)$在$E_{\dim V-1}$零点中的 Fano 模式$F_{d-1}(Z(E_{\dim V-1}))$上的有理点。孤立点仅在$\dim V=2d$时存在，在这种情况下，它们与$1\cdot 3\cdots (2d-1)$在$2d$元素集上的配对一一对应。这尤其证实了 Ambartsoumian、Auel 和 Jebelli 的一个猜想，即（在$\mathbb{R}$上）所有孤立点都可以通过适当的符号的积分星变换恢复。 显示英文摘要

Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over $\mathbb{R}$) all isolated points are recoverable via integral star transforms with appropriate symbols.

[22] arXiv:2507.22408 (替换) [中文pdf, pdf, html, 其他]

标题： 坐标代数的生成元的仿射ind-概形 显示英文标题

标题： On the generators of coordinate algebras of affine ind-varieties

Alexander Chernov

评论： 6页，0图

主题： 代数几何 (math.AG) ; 交换代数 (math.AC)

在本文中，我们研究仿射 ind-流形的坐标环的结构。 我们证明，任何不是同构于仿射代数簇的仿射 ind-流形的坐标环都不具有可数生成元集。 此外，我们证明仿射 ind-流形的坐标环具有一个处处稠密的可数维子空间。 显示英文摘要

In this paper we study the structure of the coordinate ring of an affine ind-variety. We prove that any coordinate ring of an affine ind-variety which is not isomorphic to an affine algebraic variety does not have a countable set of generators. Also we prove that coordinate rings of affine ind-varieties have an everywhere dense subspace of countable dimension.

[23] arXiv:2508.00729 (替换) [中文pdf, pdf, html, 其他]

标题： 树分解的小图以及将它们作为实代数函数的Reeb图实现的图 显示英文标题

标题： Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions

Naoki Kitazawa

评论： 10页，2图，“图2”已更改，我们证明中的一种错误已更正，我们的主要结果未变

主题： 代数几何 (math.AG) ; 组合数学 (math.CO) ; 度量几何 (math.MG)

我们一直对图以及将它们作为显式实代数函数的Reeb图感兴趣。 可微函数的Reeb图是定义域流形的商空间，被视为所有单点的原像的所有连通分支组成的空间。 自20世纪上半叶Morse函数理论诞生以来，Reeb图一直是流形几何中的基本而强大的工具。 我们可以很容易地看出，维度至少为$2$的单位球面的自然高度的Reeb图是一个恰好有一条边和两个顶点的图。 我们关注的是可以被良好分解为树的图的实现，每个顶点对应一个恰好有一条边和两个顶点的图或者一个恰好有两条边且与圆同胚的图。 显示英文摘要

We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the natural height of the unit sphere whose dimension is at least $2$ is a graph with exactly one edge and two edges. We are concerned with realizations of graphs decomposed into trees nicely, each vertex of which corresponds to a graph with exactly one edge and two edges or a graph with exactly two edges homeomorphic to a circle.

[24] arXiv:2406.00964 (替换) [中文pdf, pdf, html, 其他]

标题： 椭圆量子环面代数 U_{t_1,t_2,p}(gl_{1，tor}), 顶点算子和 L-算子 显示英文标题

标题： Elliptic Quantum Toroidal Algebra U_{t_1,t_2,p}(gl_{1,tor}), Vertex Operators and L-operators

Hitoshi Konno, Andrey Smirnov

评论： 55页，即将发表于《纯粹数学高级研究》

主题： 表示理论 (math.RT) ; 高能物理 - 理论 (hep-th) ; 代数几何 (math.AG) ; 量子代数 (math.QA)

我们通过结合U_{t_1,t_2,p}(gl_{1，tor})的表示和瞬子模空间M(n,r)的椭圆稳定包的概念，提出了椭圆量子环面代数U_{t_1,t_2,p}(gl_{1，tor})的新顶点算子，包括I型和II型对偶。 这些顶点算子通过它们的复合再现了椭圆稳定包的混排乘积公式。 我们还证明了顶点算子复合的真空期望值给出了M(n,r)的K理论顶点函数的正确公式。 随后，我们推导了顶点算子之间的交换关系，并构造了一个L算子，该算子满足RLL=LLR^*关系，其中R和R^*是作为椭圆稳定包的过渡矩阵定义的椭圆动态瞬子R矩阵。 假设L的通用形式，我们以它为基础定义了一个余乘法\Delta 。 结果表明，新的顶点算子是关于\Delta 的 U_{t_1,t_2,p}(gl_{1，tor})-模的交织算子。 显示英文摘要

We propose new vertex operators, both the type I and the type II dual, of the elliptic quantum toroidal algebra U_{t_1,t_2,p}(gl_{1,tor}) by combining representations of U_{t_1,t_2,p}(gl_{1,tor}) and the notions of the elliptic stable envelopes for the instanton moduli space M(n,r). The vertex operators reproduce the shuffle product formula of the elliptic stable envelopes by their composition. We also show that the vacuum expectation value of a composition of the vertex operators gives a correct formula of the K-theoretic vertex function for M(n,r). We then derive exchange relations among the vertex operators and construct a L-operator satisfying the RLL=LLR^* relation with R and R^* being elliptic dynamical instanton R-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of L, we define a comultiplication \Delta in terms of it. It turns out that the new vertex operators are intertwining operators of the U_{t_1,t_2,p}(gl_{1,tor})-modules w.r.t \Delta.

[25] arXiv:2504.21081 (替换) [中文pdf, pdf, html, 其他]

标题： 双有理变换在Dimer可积系统上的应用 显示英文标题

标题： Birational Transformations on Dimer Integrable Systems

Minsung Kho, Norton Lee, Rak-Kyeong Seong

评论： 6页，2图。v2：发表版本

期刊参考： 物理评论D 112, L041901 (2025)

主题： 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 代数几何 (math.AG)

我们证明，当两个toric Calabi-Yau 3-folds及其对应的toric varieties通过有理变换相关联时，它们与一对定义dimer可积系统的2-torus上的dimer模型相关联，这些系统本身也变为有理等价的。 由Goncharov和Kenyon首次引入了由dimer模型定义的这些可积系统。 我们使用一对对应于形式为C^3/Z_4 x Z_2的阿贝尔轨道的dimer可积系统来明确说明这种等价性，其轨道作用为(1,0,3)(0,1,1)，以及对应于C/Z_2 x Z_2的轨道作用为(1,0,0,1)(0,1,1,0)的dimer可积系统，其谱曲线和哈密顿量被证明通过有理变换相关联。 显示英文摘要

We show that when two toric Calabi-Yau 3-folds and their corresponding toric varieties are related by a birational transformation, they are associated with a pair of dimer models on the 2-torus that define dimer integrable systems, which themselves become birationally equivalent. These integrable systems defined by dimer models were first introduced by Goncharov and Kenyon. We illustrate this equivalence explicitly using a pair of dimer integrable systems corresponding to the abelian orbifolds of the form C^3/Z_4 x Z_2 with orbifold action (1,0,3)(0,1,1) and C/Z_2 x Z_2 with action (1,0,0,1)(0,1,1,0), whose spectral curves and Hamiltonians are shown to be related by a birational transformation.

[26] arXiv:2505.02950 (替换) [中文pdf, pdf, html, 其他]

标题： RNS 超弦测度 3 阶曲面 显示英文标题

标题： RNS superstring measure for genus 3

Petr Dunin-Barkowski, Igor Fedorov, Alexey Sleptsov

评论： 54页。增加了更多细节，尤其是关于0点函数的部分

主题： 高能物理 - 理论 (hep-th) ; 数学物理 (math-ph) ; 代数几何 (math.AG)

我们提出了一个关于3类的RNS超弦测度的新公式。 我们的推导基于不变式理论。 我们遵循维滕使用模空间的代数参数化的思想（他将其应用于重新推导D'Hoker和Phong关于2类的RNS超弦测度的公式）；但我们使用的特定参数化以前从未应用于超弦理论。 我们证明了超弦测度是三个已知函数的线性组合（具有复系数）。 此外，我们猜测了这个线性组合的系数值，并提供了对此猜测的证据。 与Cacciatori、Dalla Piazza和van Geemen在2008年的假设不同，我们的公式在双曲椭圆轨迹上具有极点奇异性；这种奇异性存在的事实由维滕在2015年确立。 此外，我们的公式不是假设，而是从基本原理推导出来的，除了三个系数的值外。 显示英文摘要

We propose a new formula for the RNS supersting measure for genus 3. Our derivation is based on invariant theory. We follow Witten's idea of using an algebraic parametrization of the moduli space (which he applied to re-derive D'Hoker and Phong's formula for the RNS superstring measure for genus 2); but the particular parametrization that we use has not been applied to superstring theory before. We prove that the superstring measure is a linear combinaition (with complex coefficients) of three known functions. Furthermore, we conjecture the values of the coefficients of this linear combination and provide evidence for this conjecture. Unlike the Ansatz of Cacciatori, Dalla Piazza and van Geemen from 2008, our formula has a polar singularity along the hyperelliptic locus; the existence of this singularity was established by Witten in 2015. Moreover, our formula is not an Ansatz but follows from first principles, except for the values of the three coefficients.

[27] arXiv:2505.08752 (替换) [中文pdf, pdf, 其他]

标题： 三音网络和镶嵌 显示英文标题

标题： Three Tone Networks and a Tessellation

Jeffrey R. Boland, Lane P. Hughston

评论： 36页，16图

主题： 组合数学 (math.CO) ; 音频与语音处理 (eess.AS) ; 代数几何 (math.AG)

欧拉顿内茨将三个小和弦与每个大和弦相关联，并将三个大和弦与每个小和弦相关联，可以用一个二部图表示，其中十二个白色顶点表示大和弦，十二个黑色顶点表示小和弦。 这个所谓的莱维图唯一地决定了实射影平面上十二个点和十二条线的显著配置的组合几何，该配置具有每条线上有三个点且每一点上有三条线的性质。 顿内茨的一些有趣特征，例如四个主要六循环和三个主要八循环的存在，对于理解十九世纪的声部进行至关重要，这些特征可以相当直接地作为该配置的属性读取。 我们展示如何为五声音阶音乐和十二音音乐构建类似的音调网络。 显示英文摘要

The Eulerian tonnetz, which associates three minor chords to each major chord and three major chords to each minor chord, can be represented by a bipartite graph with twelve white vertices signifying major chords and twelve black vertices signifying minor chords. This so-called Levi graph uniquely determines the combinatorial geometry of a remarkable configuration of twelve points and twelve lines in the real projective plane with the property that three points lie on each line and three lines pass through each point. Interesting features of the tonnetz, such as the existence of the four principal hexacycles and the three principal octacycles, crucial for the understanding of nineteenth-century voice leading, can be read off rather directly as properties of the configuration. We show how analogous tone networks can be constructed for pentatonic music and twelve-tone music.

[28] arXiv:2507.05845 (替换) [中文pdf, pdf, html, 其他]

标题： 从有理顶点算子代数的共形块中得到的模 functor 显示英文标题

标题： Modular functors from conformal blocks of rational vertex operator algebras

Chiara Damiolini, Lukas Woike

评论： 31页；v2：小幅度修改

主题： 量子代数 (math.QA) ; 数学物理 (math-ph) ; 代数几何 (math.AG) ; 代数拓扑 (math.AT)

对于顶点算子代数$V$，可以自然地按照 Frenkel-Ben-Zvi 的构造并由 Damiolini-Gibney-Tarasca 推广来定义共形块的空间。如果$V$是强有理的，这些共形块的空间在适合的代数曲线模空间上形成向量丛。在本文中，在相同的假设下，我们建立了广泛预期的拓扑结果，即共形块的空间产生一个模 functor，即在曲面操作符的扩展上的模代数。这意味着，可接受的$V$-模的范畴$\mathcal{C}_V$从零亏格曲面的拓扑中继承了一个带状 Grothendieck-Verdier 结构，甚至导致了一个模融合范畴的结构，其结构直接来自$V$的共形块的空间。作为直接结果，我们证明了来自共形块的模 functor 可以扩展为一个三维拓扑场理论，并且可以用分解同调来描述。 显示英文摘要

For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology.