We give a lattice-theoretic characterization for a manifold of $operatorname {mathrm {OG10}}$ type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li–Pertusi–Zhao variety of $operatorname {mathrm {OG10}}$ type associated to any smooth cubic fourfold. Moreover, we determine when a birational transformation is induced by an automorphism of the K3 surface, and we use this to classify all induced birational symplectic involutions.
{"title":"O’Grady tenfolds as moduli spaces of sheaves","authors":"Camilla Felisetti, Franco Giovenzana, Annalisa Grossi","doi":"10.1017/fms.2024.46","DOIUrl":"https://doi.org/10.1017/fms.2024.46","url":null,"abstract":"We give a lattice-theoretic characterization for a manifold of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942400046X_inline1.png\"/> <jats:tex-math> $operatorname {mathrm {OG10}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li–Pertusi–Zhao variety of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942400046X_inline2.png\"/> <jats:tex-math> $operatorname {mathrm {OG10}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> type associated to any smooth cubic fourfold. Moreover, we determine when a birational transformation is induced by an automorphism of the K3 surface, and we use this to classify all induced birational symplectic involutions.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"22 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$ . For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$ -liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.
{"title":"Deformations of Calabi–Yau varieties with k-liminal singularities","authors":"Robert Friedman, Radu Laza","doi":"10.1017/fms.2024.44","DOIUrl":"https://doi.org/10.1017/fms.2024.44","url":null,"abstract":"The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000446_inline1.png\"/> <jats:tex-math> $4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000446_inline2.png\"/> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-liminal singularities (which are exactly the ordinary double points in dimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000446_inline3.png\"/> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous <jats:italic>k</jats:italic>-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"12 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a countable group G and a G-flow X, a probability measure $mu $ on X is called characteristic if it is $mathrm {Aut}(X, G)$ -invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups ${Delta _i: iin I}$ , when is there a faithful G-flow for which every $Delta _i$ acts minimally?
给定一个可数群 G 和一个 G 流 X，如果 X 上的概率度量 $mu $ 是 $mathrm {Aut}(X, G)$ -不变的，那么它就叫做特征度量。弗里施和塔穆兹提出了一个问题：对于任何群 G，是否存在一个最小的 G 流，它不允许特征度量？我们为每个可数群 G 构建了这样一个最小流。在此过程中，我们考虑了一系列我们称之为最小子动力学的问题：给定一个可数群 G 和一个无限子群的集合 ${Delta _i: iin I}$ ，什么时候存在一个忠实的 G 流，其中每个 $Delta _i$ 的作用都是最小的？
{"title":"Minimal subdynamics and minimal flows without characteristic measures","authors":"Joshua Frisch, Brandon Seward, Andy Zucker","doi":"10.1017/fms.2024.41","DOIUrl":"https://doi.org/10.1017/fms.2024.41","url":null,"abstract":"Given a countable group <jats:italic>G</jats:italic> and a <jats:italic>G</jats:italic>-flow <jats:italic>X</jats:italic>, a probability measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000410_inline1.png\"/> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:italic>X</jats:italic> is called characteristic if it is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000410_inline2.png\"/> <jats:tex-math> $mathrm {Aut}(X, G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-invariant. Frisch and Tamuz asked about the existence of a minimal <jats:italic>G</jats:italic>-flow, for any group <jats:italic>G</jats:italic>, which does not admit a characteristic measure. We construct for every countable group <jats:italic>G</jats:italic> such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group <jats:italic>G</jats:italic> and a collection of infinite subgroups <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000410_inline3.png\"/> <jats:tex-math> ${Delta _i: iin I}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, when is there a faithful <jats:italic>G</jats:italic>-flow for which every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000410_inline4.png\"/> <jats:tex-math> $Delta _i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> acts minimally?","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"29 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a short new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the simplicial case.
{"title":"A short proof of the Hanlon-Hicks-Lazarev Theorem","authors":"Michael K. Brown, Daniel Erman","doi":"10.1017/fms.2024.40","DOIUrl":"https://doi.org/10.1017/fms.2024.40","url":null,"abstract":"We give a short new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the simplicial case.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matteo Mucciconi, Makiko Sasada, Tomohiro Sasamoto, Hayate Suda
The box-ball system (BBS), which was introduced by Takahashi and Satsuma in 1990, is a soliton cellular automaton. Its dynamics can be linearized by a few methods, among which the best known is the Kerov–Kirillov–Reschetikhin (KKR) bijection using rigged partitions. Recently, a new linearization method in terms of ‘slot configurations’ was introduced by Ferrari–Nguyen–Rolla–Wang, but its relations to existing ones have not been clarified. In this paper, we investigate this issue and clarify the relation between the two linearizations. For this, we introduce a novel way of describing the BBS dynamics using a carrier with seat numbers. We show that the seat number configuration also linearizes the BBS and reveals explicit relations between the KKR bijection and the slot configuration. In addition, by using these explicit relations, we also show that even in case of finite carrier capacity the BBS can be linearized via the slot configuration.
{"title":"Relationships between two linearizations of the box-ball system: Kerov–Kirillov–Reschetikhin bijection and slot configuration","authors":"Matteo Mucciconi, Makiko Sasada, Tomohiro Sasamoto, Hayate Suda","doi":"10.1017/fms.2024.39","DOIUrl":"https://doi.org/10.1017/fms.2024.39","url":null,"abstract":"The box-ball system (BBS), which was introduced by Takahashi and Satsuma in 1990, is a soliton cellular automaton. Its dynamics can be linearized by a few methods, among which the best known is the Kerov–Kirillov–Reschetikhin (KKR) bijection using rigged partitions. Recently, a new linearization method in terms of ‘slot configurations’ was introduced by Ferrari–Nguyen–Rolla–Wang, but its relations to existing ones have not been clarified. In this paper, we investigate this issue and clarify the relation between the two linearizations. For this, we introduce a novel way of describing the BBS dynamics using a carrier with seat numbers. We show that the seat number configuration also linearizes the BBS and reveals explicit relations between the KKR bijection and the slot configuration. In addition, by using these explicit relations, we also show that even in case of finite carrier capacity the BBS can be linearized via the slot configuration.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a new algebraic invariant of a graph G called the Ceresa–Zharkov class and show that it is trivial if and only if G is of hyperelliptic type, equivalently, G does not have as a minor the complete graph on four vertices or the loop of three loops. After choosing edge lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph G that is closely related to the Ceresa cycle for an algebraic curve defined over $mathbb {C}(!(t)!)$ .
我们定义了图 G 的一个新代数不变量，称为 Ceresa-Zharkov 类，并证明当且仅当 G 是超椭圆型时，它才是微不足道的。在选择边长之后，这个类特殊化为具有底图 G 的热带曲线的代数不变量，它与定义在 $mathbb {C}(!(t)!)$ 上的代数曲线的 Ceresa 循环密切相关。
{"title":"The Ceresa class and tropical curves of hyperelliptic type","authors":"Daniel Corey, Wanlin Li","doi":"10.1017/fms.2024.36","DOIUrl":"https://doi.org/10.1017/fms.2024.36","url":null,"abstract":"We define a new algebraic invariant of a graph <jats:italic>G</jats:italic> called the Ceresa–Zharkov class and show that it is trivial if and only if <jats:italic>G</jats:italic> is of hyperelliptic type, equivalently, <jats:italic>G</jats:italic> does not have as a minor the complete graph on four vertices or the loop of three loops. After choosing edge lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph <jats:italic>G</jats:italic> that is closely related to the Ceresa cycle for an algebraic curve defined over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000367_inline1.png\"/> <jats:tex-math> $mathbb {C}(!(t)!)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"48 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.
{"title":"Affine Bruhat order and Demazure products","authors":"Felix Schremmer","doi":"10.1017/fms.2024.33","DOIUrl":"https://doi.org/10.1017/fms.2024.33","url":null,"abstract":"We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"89 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP $_{3}$ theory is simple.
{"title":"Generic Stability Independence and Treeless Theories","authors":"Itay Kaplan, Nicholas Ramsey, Pierre Simon","doi":"10.1017/fms.2024.35","DOIUrl":"https://doi.org/10.1017/fms.2024.35","url":null,"abstract":"We initiate a systematic study of <jats:italic>generic stability independence</jats:italic> and introduce the class of <jats:italic>treeless theories</jats:italic> in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000355_inline1.png\" /> <jats:tex-math> $_{3}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> theory is simple.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $alpha colon X to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $alpha $ is semistable if the genus of Y is at least $1$ and stable if the genus of Y is at least $2$ . We prove this conjecture if the map $alpha $ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve Y.
让 $alpha colon X to Y$ 是光滑曲线的有限盖。博维尔猜想，如果 Y 的属至少为 1$，则一般向量束在 $alpha $ 下的前推是半稳定的；如果 Y 的属至少为 2$，则前推是稳定的。如果 $alpha $ 映射在任意光滑曲线 Y 的盖的赫维茨空间的任意分量中是一般的，我们就证明了这个猜想。
{"title":"Generic Beauville’s Conjecture","authors":"Izzet Coskun, Eric Larson, Isabel Vogt","doi":"10.1017/fms.2024.21","DOIUrl":"https://doi.org/10.1017/fms.2024.21","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000215_inline1.png\" /> <jats:tex-math> $alpha colon X to Y$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000215_inline2.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is semistable if the genus of <jats:italic>Y</jats:italic> is at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000215_inline3.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and stable if the genus of <jats:italic>Y</jats:italic> is at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000215_inline4.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove this conjecture if the map <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000215_inline5.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is general in any component of the Hurwitz space of covers of an arbitrary smooth curve <jats:italic>Y</jats:italic>.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"98 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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