arXiv - MATH - Algebraic Geometry最新文献_第2页

On the connectedness of the automorphism group of an affine toric variety 论仿射环状变的自变群的连通性

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-16 DOI: arxiv-2409.10349

Veronika Kikteva

引用次数: 0

On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians 论格拉斯曼等变量子微分方程与 qKZ 差分方程的 Satake 对应关系

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09657

Giordano Cotti, Alexander Varchenko

{"title":"On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians","authors":"Giordano Cotti, Alexander Varchenko","doi":"arxiv-2409.09657","DOIUrl":"https://doi.org/arxiv-2409.09657","url":null,"abstract":"We consider the joint system of equivariant quantum differential equations\u0000(qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which\u0000parametrizes $k$-dimensional subspaces of $mathbb{C}^n$. First, we establish a\u0000connection between this joint system for $G(k,n)$ and the corresponding system\u0000for the projective space $mathbb{P}^{n-1}$. Specifically, we show that, under\u0000suitable textit{Satake identifications} of the equivariant cohomologies of\u0000$G(k,n)$ and $mathbb{P}^{n-1}$, the joint system for $G(k,n)$ is gauge\u0000equivalent to a differential-difference system on the $k$-th exterior power of\u0000the cohomology of $mathbb{P}^{n-1}$. Secondly, we demonstrate that the textcyr{B}-theorem for Grassmannians, as\u0000stated in arXiv:1909.06582, arXiv:2203.03039, is compatible with the Satake\u0000identification. This implies that the textcyr{B}-theorem for\u0000$mathbb{P}^{n-1}$ extends to $G(k,n)$ through the Satake identification. As a\u0000consequence, we derive determinantal formulas and new integral representations\u0000for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system\u0000for $G(k,n)$. Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ\u0000equations associated with $G(k,n)$. We prove that the Stokes bases of solutions\u0000correspond to explicit $K$-theoretical classes of full exceptional collections\u0000in the derived category of equivariant coherent sheaves on $G(k,n)$.\u0000Furthermore, we show that the Stokes matrices equal the Gram matrices of the\u0000equivariant Euler-Poincar'e-Grothendieck pairing with respect to these\u0000exceptional $K$-theoretical bases.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Proof of the geometric Langlands conjecture V: the multiplicity one theorem 朗兰兹几何猜想 V 的证明：乘数一定理

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09856

Dennis Gaitsgory, Sam Raskin

引用次数: 0

Hilbert Schemes and Seshadri Constants 希尔伯特方案和塞沙德里常数

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09694

Jonas Baltes

引用次数: 0

Computing Arrangements of Hypersurfaces 超曲面排列计算

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09622

Paul Breiding, Bernd Sturmfels, Kexin Wang

引用次数: 0

K-theoretic Gromov-Witten invariants of line degrees on flag varieties 旗变体上线度的 K 理论格罗莫夫-维滕不变式

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-15 DOI: arxiv-2409.09580

Anders S. Buch, Linda Chen, Weihong Xu

引用次数: 0

K-stablity of Fano threefold hypersurfaces of index 1 指数为 1 的法诺三折超曲面的 K 稳定性

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-14 DOI: arxiv-2409.09492

Livia Campo, Takuzo Okada

引用次数: 0

On the $k$-th Tjurina number of weighted homogeneous singularities 关于加权同质奇点的第 k $-th 特朱里纳数

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-14 DOI: arxiv-2409.09384

Chuangqiang Hu, Stephen S. -T. Yau, Huaiqing Zuo

{"title":"On the $k$-th Tjurina number of weighted homogeneous singularities","authors":"Chuangqiang Hu, Stephen S. -T. Yau, Huaiqing Zuo","doi":"arxiv-2409.09384","DOIUrl":"https://doi.org/arxiv-2409.09384","url":null,"abstract":"Let $ (X,0) $ denote an isolated singularity defined by a weighted\u0000homogeneous polynomial $ f $. Let $ mathcal{O}$ be the local algebra of all\u0000holomorphic function germs at the origin with the maximal ideal $m $. We study\u0000the $k$-th Tjurina algebra, defined by $ A_k(f): = mathcal{O} / left( f , m^k\u0000J(f) right) $, where $J(f)$ denotes the Jacobi ideal of $ mathcal{O}$. The\u0000zeroth Tjurina algebra is well known to represent the tangent space of the base\u0000space of the semi-universal deformation of $(X, 0)$. Motivated by this\u0000observation, we explore the deformation of $(X,0)$ with respect to a fixed\u0000$k$-residue point. We show that the tangent space of the corresponding\u0000deformation functor is a subspace of the $k$-th Tjurina algebra. Explicitly\u0000calculating the $k$-th Tjurina numbers, which correspond to the dimensions of\u0000the Tjurina algebra, plays a crucial role in understanding these deformations.\u0000According to the results of Milnor and Orlik, the zeroth Tjurina number can be\u0000expressed explicitly in terms of the weights of the variables in $f$. However,\u0000we observe that for values of $k$ exceeding the multiplicity of $X$, the $k$-th\u0000Tjurina number becomes more intricate and is not solely determined by the\u0000weights of variables. In this paper, we introduce a novel complex derived from\u0000the classical Koszul complex and obtain a computable formula for the $k$-th\u0000Tjurina numbers for all $ k geqslant 0 $. As applications, we calculate the\u0000$k$-th Tjurina numbers for all weighted homogeneous singularities in three\u0000variables.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Stable pairs on local curves and Bethe roots 局部曲线上的稳定对和贝特根

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-14 DOI: arxiv-2409.09508

Maximilian Schimpf

引用次数: 0

Correlated Gromov-Witten invariants 相关格罗莫夫-维滕不变式

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-14 DOI: arxiv-2409.09472

Thomas Blomme, Francesca Carocci

引用次数: 0