We define a class of skeletons on Berkovich analytic spaces, which we call "accessible", which contains the standard skeleton of the n-dimensional torus for every n and is preserved by G-glueing, by taking the inverse image along a morphism of relative dimension zero, and by taking the direct image along a morphism whose restriction to the involved skeleton is topologically proper.
我们定义了一类伯克维奇解析空间上的骨架，称之为 "可访问的"，它包含了每 n 个 n 维环面的标准骨架，并且通过 G 胶合、沿相对维数为零的非定态取反像以及沿其对相关骨架的限制是拓扑适当的非定态取直像而得到保留。
{"title":"Les squelettes accessibles d'un espace de Berkovich","authors":"Antoine Ducros, Amaury Thuillier","doi":"arxiv-2409.08755","DOIUrl":"https://doi.org/arxiv-2409.08755","url":null,"abstract":"We define a class of skeletons on Berkovich analytic spaces, which we call\u0000\"accessible\", which contains the standard skeleton of the n-dimensional torus\u0000for every n and is preserved by G-glueing, by taking the inverse image along a\u0000morphism of relative dimension zero, and by taking the direct image along a\u0000morphism whose restriction to the involved skeleton is topologically proper.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253203","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}
Grigoriy Blekherman, Rainer Sinn, Gregory G. Smith, Mauricio Velasco
We introduce tools for transferring nonnegativity certificates for global sections between line bundles on real algebraic surfaces. As applications, we improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative ternary forms, give a complete characterization of nonnegative real forms of del Pezzo surfaces, and establish quadratic upper bounds for the degrees of sum-of-squares multipliers for nonnegative forms on real ruled surfaces.
{"title":"Nonnegativity certificates on real algebraic surfaces","authors":"Grigoriy Blekherman, Rainer Sinn, Gregory G. Smith, Mauricio Velasco","doi":"arxiv-2409.08834","DOIUrl":"https://doi.org/arxiv-2409.08834","url":null,"abstract":"We introduce tools for transferring nonnegativity certificates for global\u0000sections between line bundles on real algebraic surfaces. As applications, we\u0000improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative\u0000ternary forms, give a complete characterization of nonnegative real forms of\u0000del Pezzo surfaces, and establish quadratic upper bounds for the degrees of\u0000sum-of-squares multipliers for nonnegative forms on real ruled surfaces.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253169","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}
D. Arinkin, D. Beraldo, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin, N. Rozenblyum
This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is ambidextrous; (iii) We reduce GLC to the study of a certain classical vector bundle with connection on the stack of irreducible local systems; (iv) We prove that GLC is equivalent to the contractibility of the space of generic oper structures on irreducible local systems; (v) Using [BKS], we deduce GLC for classical groups.
{"title":"Proof of the geometric Langlands conjecture IV: ambidexterity","authors":"D. Arinkin, D. Beraldo, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin, N. Rozenblyum","doi":"arxiv-2409.08670","DOIUrl":"https://doi.org/arxiv-2409.08670","url":null,"abstract":"This paper performs the following steps toward the proof of GLC in the de\u0000Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when\u0000restricted to the cuspidal category, is ambidextrous; (iii) We reduce GLC to the study of a certain classical vector bundle with\u0000connection on the stack of irreducible local systems; (iv) We prove that GLC is equivalent to the contractibility of the space of\u0000generic oper structures on irreducible local systems; (v) Using [BKS], we deduce GLC for classical groups.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253170","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}
For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree $0$ is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component which is smooth with a non-trivial canonical bundle.
{"title":"A smooth but non-symplectic moduli of sheaves on a hyperkähler variety","authors":"Andreas Krug, Fabian Reede, Ziyu Zhang","doi":"arxiv-2409.08991","DOIUrl":"https://doi.org/arxiv-2409.08991","url":null,"abstract":"For an abelian surface $A$, we consider stable vector bundles on a\u0000generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected\u0000component of the moduli space which contains the tautological bundles\u0000associated to line bundles of degree $0$ is isomorphic to the blowup of the\u0000dual abelian surface in one point. We believe that this is the first explicit\u0000example of a component which is smooth with a non-trivial canonical bundle.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253167","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 result of Green and Griffiths states that for the generic curve $C$ of genus $g geq 4$ with the canonical divisor $K$, its Faber--Pandharipande 0-cycle $Ktimes K-(2g-2)K_Delta$ on $Ctimes C$ is nontorsion in the Chow group of rational equivalence classes. For Shimura curves, however, we show that their Faber--Pandharipande 0-cycles are rationally equivalent to 0. This is predicted by a conjecture of Beilinson and Bloch.
{"title":"Faber--Pandharipande Cycles vanish for Shimura curves","authors":"Congling Qiu","doi":"arxiv-2409.08989","DOIUrl":"https://doi.org/arxiv-2409.08989","url":null,"abstract":"A result of Green and Griffiths states that for the generic curve $C$ of\u0000genus $g geq 4$ with the canonical divisor $K$, its Faber--Pandharipande\u00000-cycle $Ktimes K-(2g-2)K_Delta$ on $Ctimes C$ is nontorsion in the Chow\u0000group of rational equivalence classes. For Shimura curves, however, we show\u0000that their Faber--Pandharipande 0-cycles are rationally equivalent to 0. This\u0000is predicted by a conjecture of Beilinson and Bloch.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"192 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253168","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 a sufficient condition for a Brauer-Severi surface bundle over a rational 3-fold to not be stably rational. Additionally, we present an example that satisfies this condition and demonstrate the existence of families of Brauer-Severi surface bundles whose general members are smooth and not stably rational.
{"title":"Rationality of Brauer-Severi surface bundles over rational 3-folds","authors":"Shitan Xu","doi":"arxiv-2409.08504","DOIUrl":"https://doi.org/arxiv-2409.08504","url":null,"abstract":"We give a sufficient condition for a Brauer-Severi surface bundle over a\u0000rational 3-fold to not be stably rational. Additionally, we present an example\u0000that satisfies this condition and demonstrate the existence of families of\u0000Brauer-Severi surface bundles whose general members are smooth and not stably\u0000rational.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253171","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 determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng's 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.
{"title":"Cubic fourfolds with symplectic automorphisms","authors":"Kenji Koike","doi":"arxiv-2409.08448","DOIUrl":"https://doi.org/arxiv-2409.08448","url":null,"abstract":"We determine projective equations of smooth complex cubic fourfolds with\u0000symplectic automorphisms by classifying 6-dimensional projective\u0000representations of Laza and Zheng's 34 groups. In particular, we determine the\u0000number of irreducible components for moduli spaces of cubic fourfolds with\u0000symplectic actions by these groups. We also discuss the fields of definition of\u0000cubic fourfolds in six maximal cases.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"211 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253172","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 logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang. We focus then on the properties of the subfamily of log-Enriques varieties that admit a quasi-'etale cover by a singular symplectic variety and we give many examples.
{"title":"Logarithmic Enriques varieties","authors":"Samuel Boissiere, Chiara Camere, Alessandra Sarti","doi":"arxiv-2409.09160","DOIUrl":"https://doi.org/arxiv-2409.09160","url":null,"abstract":"We introduce logarithmic Enriques varieties as a singular analogue of\u0000Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced\u0000by Zhang. We focus then on the properties of the subfamily of log-Enriques\u0000varieties that admit a quasi-'etale cover by a singular symplectic variety and\u0000we give many examples.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253097","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}
Andriy Regeta, Christian Urech, Immanuel van Santen
Let X be an irreducible variety and Bir(X) its group of birational transformations. We show that the group structure of Bir(X) determines whether X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X, with equality occurring if and only if X is rational and the Borel subgroup is standard. We also provide examples of non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving conjectures by Popov and Furter-Poloni.
设 X 是不可还原 variety，Bir(X) 是其双变换群。我们证明，Bir(X) 的群结构决定了 X 是否有理以及 X 是否有规则。此外，我们还证明了 Bir(X) 的任何 Borel 子群的派生长度最多为 X 维数的两倍，只有当且仅当 X 是有理的且 Borel 子群是标准群时才会发生相等。我们还举例说明了 Bir(P^n) 和 Aut(A^n) 的非标准 Borel 子群，从而解决了 Popov 和 Furter-Poloni 的猜想。
{"title":"Group Theoretical Characterizations of Rationality","authors":"Andriy Regeta, Christian Urech, Immanuel van Santen","doi":"arxiv-2409.07864","DOIUrl":"https://doi.org/arxiv-2409.07864","url":null,"abstract":"Let X be an irreducible variety and Bir(X) its group of birational\u0000transformations. We show that the group structure of Bir(X) determines whether\u0000X is rational and whether X is ruled. Additionally, we prove that any Borel subgroup of Bir(X) has derived length\u0000at most twice the dimension of X, with equality occurring if and only if X is\u0000rational and the Borel subgroup is standard. We also provide examples of\u0000non-standard Borel subgroups of Bir(P^n) and Aut(A^n), thereby resolving\u0000conjectures by Popov and Furter-Poloni.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220342","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}
Stephen Coughlan, Marco Franciosi, Rita Pardini, Sönke Rollenske
The compactification $overline M_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $chi(X)=3$ in the moduli space of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.
{"title":"2-Gorenstein stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$","authors":"Stephen Coughlan, Marco Franciosi, Rita Pardini, Sönke Rollenske","doi":"arxiv-2409.07854","DOIUrl":"https://doi.org/arxiv-2409.07854","url":null,"abstract":"The compactification $overline M_{1,3}$ of the Gieseker moduli space of\u0000surfaces of general type with $K_X^2 =1 $ and $chi(X)=3$ in the moduli space\u0000of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a\u0000mix of algebraic and geometric techniques. We find a new divisor in the closure\u0000of the Gieseker component and a new irreducible component of the moduli space.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220341","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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