We obtain a criterion for the automorphism group of an affine toric variety to be connected in combinatorial terms and in terms of the divisor class group of the variety. The component group of the automorphism group of a non-degenerate affine toric variety is described. In particular, we show that the number of connected components of the automorphism group is finite.
{"title":"On the connectedness of the automorphism group of an affine toric variety","authors":"Veronika Kikteva","doi":"arxiv-2409.10349","DOIUrl":"https://doi.org/arxiv-2409.10349","url":null,"abstract":"We obtain a criterion for the automorphism group of an affine toric variety\u0000to be connected in combinatorial terms and in terms of the divisor class group\u0000of the variety. The component group of the automorphism group of a\u0000non-degenerate affine toric variety is described. In particular, we show that\u0000the number of connected components of the automorphism group is finite.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253156","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}
We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which parametrizes $k$-dimensional subspaces of $mathbb{C}^n$. First, we establish a connection between this joint system for $G(k,n)$ and the corresponding system for the projective space $mathbb{P}^{n-1}$. Specifically, we show that, under suitable textit{Satake identifications} of the equivariant cohomologies of $G(k,n)$ and $mathbb{P}^{n-1}$, the joint system for $G(k,n)$ is gauge equivalent to a differential-difference system on the $k$-th exterior power of the cohomology of $mathbb{P}^{n-1}$. Secondly, we demonstrate that the textcyr{B}-theorem for Grassmannians, as stated in arXiv:1909.06582, arXiv:2203.03039, is compatible with the Satake identification. This implies that the textcyr{B}-theorem for $mathbb{P}^{n-1}$ extends to $G(k,n)$ through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for $G(k,n)$. Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with $G(k,n)$. We prove that the Stokes bases of solutions correspond to explicit $K$-theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on $G(k,n)$. Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler-Poincar'e-Grothendieck pairing with respect to these exceptional $K$-theoretical bases.
{"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":"116 1","pages":""},"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}
This is the final paper in the series of five, in which we prove the geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing that there exists a unique (up to tensoring up by a vector space) Hecke eigensheaf corresponding to an irreducible local system (hence, the title of the paper). We achieve this by analyzing the geometry of the stack of local systems.
{"title":"Proof of the geometric Langlands conjecture V: the multiplicity one theorem","authors":"Dennis Gaitsgory, Sam Raskin","doi":"arxiv-2409.09856","DOIUrl":"https://doi.org/arxiv-2409.09856","url":null,"abstract":"This is the final paper in the series of five, in which we prove the\u0000geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing\u0000that there exists a unique (up to tensoring up by a vector space) Hecke\u0000eigensheaf corresponding to an irreducible local system (hence, the title of\u0000the paper). We achieve this by analyzing the geometry of the stack of local\u0000systems.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"132 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253157","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}
In this paper we will propose a new method to investigate Seshadri constants, namely by means of (nested) Hilbert schemes. This will allow us to use the geometry of the latter spaces, for example the computations of the nef cone via Bridgeland stability conditions to gain new insights and bounds on Seshadri constants. Moreover, it turns out that many known Seshadri constants turn up in the wall and chamber decomposition of the movable cone of Hilbert schemes.
{"title":"Hilbert Schemes and Seshadri Constants","authors":"Jonas Baltes","doi":"arxiv-2409.09694","DOIUrl":"https://doi.org/arxiv-2409.09694","url":null,"abstract":"In this paper we will propose a new method to investigate Seshadri constants,\u0000namely by means of (nested) Hilbert schemes. This will allow us to use the\u0000geometry of the latter spaces, for example the computations of the nef cone via\u0000Bridgeland stability conditions to gain new insights and bounds on Seshadri\u0000constants. Moreover, it turns out that many known Seshadri constants turn up in\u0000the wall and chamber decomposition of the movable cone of Hilbert schemes.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253158","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}
We present a Julia package HypersurfaceRegions.jl for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in $mathbb{R}^n$.
我们提出了一个 Julia 软件包 HypersurfaceRegions.jl，用于计算 $mathbb{R}^n$ 中实代数超曲面排列的补集中的所有连通部分。
{"title":"Computing Arrangements of Hypersurfaces","authors":"Paul Breiding, Bernd Sturmfels, Kexin Wang","doi":"arxiv-2409.09622","DOIUrl":"https://doi.org/arxiv-2409.09622","url":null,"abstract":"We present a Julia package HypersurfaceRegions.jl for computing all connected\u0000components in the complement of an arrangement of real algebraic hypersurfaces\u0000in $mathbb{R}^n$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253165","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}
A homology class $d in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical intersection number computed on the flag variety $G/P'$. We also prove an $n$-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags $G/B$. Our formulas make it straightforward to compute the big quantum K-theory ring of $X$ modulo degrees larger than line degrees.
{"title":"K-theoretic Gromov-Witten invariants of line degrees on flag varieties","authors":"Anders S. Buch, Linda Chen, Weihong Xu","doi":"arxiv-2409.09580","DOIUrl":"https://doi.org/arxiv-2409.09580","url":null,"abstract":"A homology class $d in H_2(X)$ of a complex flag variety $X = G/P$ is called\u0000a line degree if the moduli space $overline{M}_{0,0}(X,d)$ of 0-pointed stable\u0000maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum\u0000equals classical formula stating that any $n$-pointed (equivariant,\u0000K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal\u0000to a classical intersection number computed on the flag variety $G/P'$. We also\u0000prove an $n$-pointed analogue of the Peterson comparison formula stating that\u0000these invariants coincide with Gromov-Witten invariants of the variety of\u0000complete flags $G/B$. Our formulas make it straightforward to compute the big\u0000quantum K-theory ring of $X$ modulo degrees larger than line degrees.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253161","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}
We settle the problem of K-stability of quasi-smooth Fano 3-fold hypersurfaces with Fano index 1 by providing lower bounds for their delta invariants. We use the method introduced by Abban and Zhuang for computing lower bounds of delta invariants on flags of hypersurfaces in the Fano 3-fold.
我们解决了法诺指数为 1 的准光滑法诺 3 折叠超曲面的 K 稳定性问题，提供了它们的三角变量下界。我们使用阿班和庄引入的方法计算法诺 3 折叠超曲面旗上的三角变量下界。
{"title":"K-stablity of Fano threefold hypersurfaces of index 1","authors":"Livia Campo, Takuzo Okada","doi":"arxiv-2409.09492","DOIUrl":"https://doi.org/arxiv-2409.09492","url":null,"abstract":"We settle the problem of K-stability of quasi-smooth Fano 3-fold\u0000hypersurfaces with Fano index 1 by providing lower bounds for their delta\u0000invariants. We use the method introduced by Abban and Zhuang for computing\u0000lower bounds of delta invariants on flags of hypersurfaces in the Fano 3-fold.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253162","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}
Let $ (X,0) $ denote an isolated singularity defined by a weighted homogeneous polynomial $ f $. Let $ mathcal{O}$ be the local algebra of all holomorphic function germs at the origin with the maximal ideal $m $. We study the $k$-th Tjurina algebra, defined by $ A_k(f): = mathcal{O} / left( f , m^k J(f) right) $, where $J(f)$ denotes the Jacobi ideal of $ mathcal{O}$. The zeroth Tjurina algebra is well known to represent the tangent space of the base space of the semi-universal deformation of $(X, 0)$. Motivated by this observation, we explore the deformation of $(X,0)$ with respect to a fixed $k$-residue point. We show that the tangent space of the corresponding deformation functor is a subspace of the $k$-th Tjurina algebra. Explicitly calculating the $k$-th Tjurina numbers, which correspond to the dimensions of the Tjurina algebra, plays a crucial role in understanding these deformations. According to the results of Milnor and Orlik, the zeroth Tjurina number can be expressed explicitly in terms of the weights of the variables in $f$. However, we observe that for values of $k$ exceeding the multiplicity of $X$, the $k$-th Tjurina number becomes more intricate and is not solely determined by the weights of variables. In this paper, we introduce a novel complex derived from the classical Koszul complex and obtain a computable formula for the $k$-th Tjurina numbers for all $ k geqslant 0 $. As applications, we calculate the $k$-th Tjurina numbers for all weighted homogeneous singularities in three variables.
{"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":"27 1","pages":""},"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}
We give an explicit formula for the descendent stable pair invariants of all (absolute) local curves in terms of certain power series called Bethe roots, which also appear in the physics/representation theory literature. We derive new explicit descriptions for the Bethe roots which are of independent interest. From this we derive rationality, functional equation and a characterization of poles for the full descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. We also sketch how our methods give a new approach to the spectrum of quantum multiplication on $mathsf{Hilb}^n(mathbf{C}^2)$.
{"title":"Stable pairs on local curves and Bethe roots","authors":"Maximilian Schimpf","doi":"arxiv-2409.09508","DOIUrl":"https://doi.org/arxiv-2409.09508","url":null,"abstract":"We give an explicit formula for the descendent stable pair invariants of all\u0000(absolute) local curves in terms of certain power series called Bethe roots,\u0000which also appear in the physics/representation theory literature. We derive\u0000new explicit descriptions for the Bethe roots which are of independent\u0000interest. From this we derive rationality, functional equation and a\u0000characterization of poles for the full descendent stable pair theory of local\u0000curves as conjectured by Pandharipande and Pixton. We also sketch how our\u0000methods give a new approach to the spectrum of quantum multiplication on\u0000$mathsf{Hilb}^n(mathbf{C}^2)$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253166","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}
We introduce a geometric refinement of Gromov-Witten invariants for $mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore we prove a refinement of the degeneration formula keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation and the refined degeneration formula we follow floor diagrams techniques to prove regularity results for the generating series of the invariants in the case of $mathbb P^1$-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.
{"title":"Correlated Gromov-Witten invariants","authors":"Thomas Blomme, Francesca Carocci","doi":"arxiv-2409.09472","DOIUrl":"https://doi.org/arxiv-2409.09472","url":null,"abstract":"We introduce a geometric refinement of Gromov-Witten invariants for $mathbb\u0000P^1$-bundles relative to the natural fiberwise boundary structure. We call\u0000these refined invariant correlated Gromov-Witten invariants. Furthermore we\u0000prove a refinement of the degeneration formula keeping track of the\u0000correlation. Finally, combining certain invariance properties of the correlated\u0000invariant, a local computation and the refined degeneration formula we follow\u0000floor diagrams techniques to prove regularity results for the generating series\u0000of the invariants in the case of $mathbb P^1$-bundles over elliptic curves.\u0000Such invariants are expected to play a role in the degeneration formula for\u0000reduced Gromov-Witten invariants for abelian and K3 surfaces.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"211 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253163","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}
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