Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110100
Zhiyuan Wang , Chenglang Yang , Qingsheng Zhang
Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected n-point functions. Finally, we show that the quantum spectral curve of type B in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.
{"title":"BKP-affine coordinates and emergent geometry of generalized Brézin-Gross-Witten tau-functions","authors":"Zhiyuan Wang , Chenglang Yang , Qingsheng Zhang","doi":"10.1016/j.aim.2024.110100","DOIUrl":"10.1016/j.aim.2024.110100","url":null,"abstract":"<div><div>Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected <em>n</em>-point functions. Finally, we show that the quantum spectral curve of type <em>B</em> in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110100"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110082
Thomas Brazelton
We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin–Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the -action on are given by , where and denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines.
{"title":"Equivariant enumerative geometry","authors":"Thomas Brazelton","doi":"10.1016/j.aim.2024.110082","DOIUrl":"10.1016/j.aim.2024.110082","url":null,"abstract":"<div><div>We formulate an <em>equivariant conservation of number</em>, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Pontryagin–Thom transfer in the equivariant setting. We leverage this result to commence a study of enumerative geometry in the presence of a group action. As an illustration of the power of this machinery, we prove that any smooth complex cubic surface defined by a symmetric polynomial has 27 lines whose orbit types under the <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-action on <span><math><mi>C</mi><msup><mrow><mtext>P</mtext></mrow><mrow><mn>3</mn></mrow></msup></math></span> are given by <span><math><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo><mo>+</mo><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>]</mo><mo>+</mo><mo>[</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>/</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> denote two non-conjugate cyclic subgroups of order two. As a consequence we demonstrate that a real symmetric cubic surface can only contain 3 or 27 real lines.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110082"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110087
Francesco Lin
A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere Y we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of Y. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its -equivariant refinement). We apply this to study the subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying certain natural geometric constraints.
{"title":"Homology cobordism and the geometry of hyperbolic three-manifolds","authors":"Francesco Lin","doi":"10.1016/j.aim.2024.110087","DOIUrl":"10.1016/j.aim.2024.110087","url":null,"abstract":"<div><div>A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this paper, for a hyperbolic homology sphere <em>Y</em> we derive explicit bounds on the relative grading between irreducible solutions to the Seiberg-Witten equations and the reducible one in terms of the spectral and Riemannian geometry of <em>Y</em>. Using this, we provide explicit bounds on some numerical invariants arising in monopole Floer homology (and its <span><math><mrow><mi>Pin</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span>-equivariant refinement). We apply this to study the subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying certain natural geometric constraints.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110087"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110085
Gabriele Bianchi, Andrea Cianchi, Paolo Gronchi
This work is concerned with a Pólya-Szegö type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach uncovering geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.
{"title":"Anisotropic symmetrization, convex bodies, and isoperimetric inequalities","authors":"Gabriele Bianchi, Andrea Cianchi, Paolo Gronchi","doi":"10.1016/j.aim.2024.110085","DOIUrl":"10.1016/j.aim.2024.110085","url":null,"abstract":"<div><div>This work is concerned with a Pólya-Szegö type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach uncovering geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110085"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110099
Nadir Fasola , Sergej Monavari
Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme on a singular threefold and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define K-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.
{"title":"Tetrahedron instantons in Donaldson-Thomas theory","authors":"Nadir Fasola , Sergej Monavari","doi":"10.1016/j.aim.2024.110099","DOIUrl":"10.1016/j.aim.2024.110099","url":null,"abstract":"<div><div>Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme on a singular threefold and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf à la Oh-Thomas, by which we define <em>K</em>-theoretic invariants. We show that the partition function of such invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly determine it, as a product of shifted partition functions of rank one Donaldson-Thomas invariants of the three-dimensional affine space. Our geometric construction answers a series of questions of Pomoni-Yan-Zhang on the geometry of the moduli space of tetrahedron instantons and the behaviour of its partition function, and provides a new application of the recent work of Oh-Thomas.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110099"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110062
Eduard Stefanescu
Let be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates modulo 1, in terms of N. For any lacunary sequence we prove the existence of a dilation factor α such that the maximal gap is of order at most , and we prove that for Lebesgue almost all α the maximal gap is of order at most . The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.
{"title":"The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation","authors":"Eduard Stefanescu","doi":"10.1016/j.aim.2024.110062","DOIUrl":"10.1016/j.aim.2024.110062","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a lacunary sequence satisfying the Hadamard gap condition. We give upper bounds for the maximal gap of the set of dilates <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>α</mi><mo>}</mo></mrow><mrow><mi>n</mi><mo>≤</mo><mi>N</mi></mrow></msub></math></span> modulo 1, in terms of <em>N</em>. For any lacunary sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> we prove the existence of a dilation factor <em>α</em> such that the maximal gap is of order at most <span><math><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo><mo>/</mo><mi>N</mi></math></span>, and we prove that for Lebesgue almost all <em>α</em> the maximal gap is of order at most <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>/</mo><mi>N</mi></math></span>. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110062"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a convex body K in , , with the property that there is exactly one hyperplane H passing through , the centroid of K, such that the centroid of coincides with . This provides answers to questions of Grünbaum and Loewner for . The proof is based on the existence of non-intersection bodies in these dimensions.
{"title":"Answers to questions of Grünbaum and Loewner","authors":"Sergii Myroshnychenko , Kateryna Tatarko , Vladyslav Yaskin","doi":"10.1016/j.aim.2024.110081","DOIUrl":"10.1016/j.aim.2024.110081","url":null,"abstract":"<div><div>We construct a convex body <em>K</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, with the property that there is exactly one hyperplane <em>H</em> passing through <span><math><mi>c</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>, the centroid of <em>K</em>, such that the centroid of <span><math><mi>K</mi><mo>∩</mo><mi>H</mi></math></span> coincides with <span><math><mi>c</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. This provides answers to questions of Grünbaum and Loewner for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. The proof is based on the existence of non-intersection bodies in these dimensions.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110081"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110102
Oanh Nguyen , Allan Sly
We study the contact process on random graphs with low infection rate λ. For random d-regular graphs, it is known that the survival time is below the critical . By contrast, on the Erdős-Rényi random graphs , rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time . To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a sharper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.
{"title":"Subcritical epidemics on random graphs","authors":"Oanh Nguyen , Allan Sly","doi":"10.1016/j.aim.2024.110102","DOIUrl":"10.1016/j.aim.2024.110102","url":null,"abstract":"<div><div>We study the contact process on random graphs with low infection rate <em>λ</em>. For random <em>d</em>-regular graphs, it is known that the survival time is <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> below the critical <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>. By contrast, on the Erdős-Rényi random graphs <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>/</mo><mi>n</mi><mo>)</mo></math></span>, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time <span><math><msup><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></mrow></msup></math></span>. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a sharper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110102"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110086
Martin W. Liebeck , Cheryl E. Praeger
We determine all factorisations , where X is a finite almost simple group and are core-free subgroups such that is cyclic or dihedral. As a main application, we classify the graphs Γ admitting an almost simple arc-transitive group X of automorphisms, such that Γ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup G of X. We prove that apart from the case where X and G have socles and respectively, the only such graphs are the complete graphs with n a prime power, the Johnson graphs with a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings.
{"title":"Maps, simple groups, and arc-transitive graphs","authors":"Martin W. Liebeck , Cheryl E. Praeger","doi":"10.1016/j.aim.2024.110086","DOIUrl":"10.1016/j.aim.2024.110086","url":null,"abstract":"<div><div>We determine all factorisations <span><math><mi>X</mi><mo>=</mo><mi>A</mi><mi>B</mi></math></span>, where <em>X</em> is a finite almost simple group and <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> are core-free subgroups such that <span><math><mi>A</mi><mo>∩</mo><mi>B</mi></math></span> is cyclic or dihedral. As a main application, we classify the graphs Γ admitting an almost simple arc-transitive group <em>X</em> of automorphisms, such that Γ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup <em>G</em> of <em>X</em>. We prove that apart from the case where <em>X</em> and <em>G</em> have socles <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> respectively, the only such graphs are the complete graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> a prime power, the Johnson graphs <span><math><mi>J</mi><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> with <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110086"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110080
Viktor Balch Barth , William Hornslien , Gereon Quick , Glen Matthew Wilson
We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave's monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are very simple to describe.
{"title":"Making the motivic group structure on the endomorphisms of the projective line explicit","authors":"Viktor Balch Barth , William Hornslien , Gereon Quick , Glen Matthew Wilson","doi":"10.1016/j.aim.2024.110080","DOIUrl":"10.1016/j.aim.2024.110080","url":null,"abstract":"<div><div>We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave's monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are very simple to describe.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110080"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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