We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For varieties of dimension 4, we describe families of rational curves in terms of Fujita's $a$-invariant. Our results verify Lehmann and Tanimoto's Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the complex numbers.
{"title":"Rational Curves on Coindex 3 Fano Varieties","authors":"Eric Jovinelly, Fumiya Okamura","doi":"arxiv-2409.00834","DOIUrl":"https://doi.org/arxiv-2409.00834","url":null,"abstract":"We describe the moduli space of rational curves on smooth Fano varieties of\u0000coindex 3. For varieties of dimension 5 or greater, we prove the moduli space\u0000has a single irreducible component for each effective numerical class of\u0000curves. For varieties of dimension 4, we describe families of rational curves\u0000in terms of Fujita's $a$-invariant. Our results verify Lehmann and Tanimoto's\u0000Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the\u0000complex numbers.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220286","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}
Based on the recent progress in the irregular Riemann-Hilbert correspondence, we study the monodromies at infinity of the holomorphic solutions of Fourier transforms of holonomic D-modules in some situations. Formulas for their eigenvalues are obtained by applying the theory of monodromy zeta functions to our previous results on the enhanced solution complexes of the Fourier transforms. In particular, in dimension one we thus find a reciprocity law between the monodromies at infinity of holonomic D-modules and their Fourier transforms.
基于不规则黎曼-希尔伯特对应关系的最新进展，我们研究了在某些情况下整体性 D 模块的傅里叶变换全形解的无穷大处单色性。通过将单旋转zeta函数理论应用于我们之前关于傅里叶变换的增强解复数的结果，我们得到了特征值的公式。特别是，在维数一中，我们发现了整体 D 模块的无穷大处单色性与它们的傅里叶变换之间的互易律。
{"title":"On the monodromies at infinity of Fourier transforms of holonomic D-modules","authors":"Kazuki Kudomi, Kiyoshi Takeuchi","doi":"arxiv-2409.00423","DOIUrl":"https://doi.org/arxiv-2409.00423","url":null,"abstract":"Based on the recent progress in the irregular Riemann-Hilbert correspondence,\u0000we study the monodromies at infinity of the holomorphic solutions of Fourier\u0000transforms of holonomic D-modules in some situations. Formulas for their\u0000eigenvalues are obtained by applying the theory of monodromy zeta functions to\u0000our previous results on the enhanced solution complexes of the Fourier\u0000transforms. In particular, in dimension one we thus find a reciprocity law\u0000between the monodromies at infinity of holonomic D-modules and their Fourier\u0000transforms.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220293","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 paper explores the relationship between L-equivalence and D-equivalence for K3 surfaces and hyperk"ahler manifolds. Building on Efimov's approach using Hodge theory, we prove that very general L-equivalent K3 surfaces are D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperk"ahler fourfolds and moduli spaces of sheaves on K3 surfaces.
本文探讨了K3曲面和超（hyperk"ahler）流形的L等价和D等价之间的关系。在埃菲莫夫利用霍奇理论的方法基础上，我们利用 K3 曲面的衍生托雷利定理证明了非常一般的 L 等价 K3 曲面是 D 等价的。我们的主要技术贡献是，通过霍奇结构的有理内定形，在不可还原的整体霍奇结构上的两个不同晶格结构是相关的。我们将我们的结果部分地扩展到超（hyperk"ahler）四叠加和 K3 曲面上的模空间。
{"title":"On L-equivalence for K3 surfaces and hyperkähler manifolds","authors":"Reinder Meinsma","doi":"arxiv-2408.17203","DOIUrl":"https://doi.org/arxiv-2408.17203","url":null,"abstract":"This paper explores the relationship between L-equivalence and D-equivalence\u0000for K3 surfaces and hyperk\"ahler manifolds. Building on Efimov's approach\u0000using Hodge theory, we prove that very general L-equivalent K3 surfaces are\u0000D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main\u0000technical contribution is that two distinct lattice structures on an integral,\u0000irreducible Hodge structure are related by a rational endomorphism of the Hodge\u0000structure. We partially extend our results to hyperk\"ahler fourfolds and\u0000moduli spaces of sheaves on K3 surfaces.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220296","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 note, we give a motivic characterization of the integral cohomology of dual boundary complexes of smooth quasi-projective complex algebraic varieties. As a corollary, the dual boundary complex of any stably affine space (of positive dimension) is contractible. In a separate paper [Su23], this corollary has been used by the author in his proof of the weak geometric P=W conjecture for very generic $GL_n(mathbb{C})$-character varieties over any punctured Riemann surfaces.
{"title":"Integral cohomology of dual boundary complexes is motivic","authors":"Tao Su","doi":"arxiv-2408.17301","DOIUrl":"https://doi.org/arxiv-2408.17301","url":null,"abstract":"In this note, we give a motivic characterization of the integral cohomology\u0000of dual boundary complexes of smooth quasi-projective complex algebraic\u0000varieties. As a corollary, the dual boundary complex of any stably affine space\u0000(of positive dimension) is contractible. In a separate paper [Su23], this\u0000corollary has been used by the author in his proof of the weak geometric P=W\u0000conjecture for very generic $GL_n(mathbb{C})$-character varieties over any\u0000punctured Riemann surfaces.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220291","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}
Nathan W. Henry, Giovanni Luca Marchetti, Kathlén Kohn
We consider function spaces defined by self-attention networks without normalization, and theoretically analyze their geometry. Since these networks are polynomial, we rely on tools from algebraic geometry. In particular, we study the identifiability of deep attention by providing a description of the generic fibers of the parametrization for an arbitrary number of layers and, as a consequence, compute the dimension of the function space. Additionally, for a single-layer model, we characterize the singular and boundary points. Finally, we formulate a conjectural extension of our results to normalized self-attention networks, prove it for a single layer, and numerically verify it in the deep case.
{"title":"Geometry of Lightning Self-Attention: Identifiability and Dimension","authors":"Nathan W. Henry, Giovanni Luca Marchetti, Kathlén Kohn","doi":"arxiv-2408.17221","DOIUrl":"https://doi.org/arxiv-2408.17221","url":null,"abstract":"We consider function spaces defined by self-attention networks without\u0000normalization, and theoretically analyze their geometry. Since these networks\u0000are polynomial, we rely on tools from algebraic geometry. In particular, we\u0000study the identifiability of deep attention by providing a description of the\u0000generic fibers of the parametrization for an arbitrary number of layers and, as\u0000a consequence, compute the dimension of the function space. Additionally, for a\u0000single-layer model, we characterize the singular and boundary points. Finally,\u0000we formulate a conjectural extension of our results to normalized\u0000self-attention networks, prove it for a single layer, and numerically verify it\u0000in the deep case.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220292","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 develop an approach to study the irreducibility of generic complete intersections in the algebraic torus defined by equations with fixed monomials and fixed linear relations on coefficients. Using our approach we generalize the irreducibility theorems of Khovanskii to fields of arbitrary characteristic. Also we get a combinatorial sufficient conditions for irreducibility of engineered complete intersections. As an application we give a combinatorial condition of irreducibility for some critical loci and Thom-Bordmann strata: $f = f'_x = 0$, $f'_x = f'_y = 0$, $f = f'_x = f'_{xx} = 0$, etc.
{"title":"Irreducibility of toric complete intersections","authors":"Andrey Zhizhin","doi":"arxiv-2409.00188","DOIUrl":"https://doi.org/arxiv-2409.00188","url":null,"abstract":"We develop an approach to study the irreducibility of generic complete\u0000intersections in the algebraic torus defined by equations with fixed monomials\u0000and fixed linear relations on coefficients. Using our approach we generalize\u0000the irreducibility theorems of Khovanskii to fields of arbitrary\u0000characteristic. Also we get a combinatorial sufficient conditions for\u0000irreducibility of engineered complete intersections. As an application we give\u0000a combinatorial condition of irreducibility for some critical loci and\u0000Thom-Bordmann strata: $f = f'_x = 0$, $f'_x = f'_y = 0$, $f = f'_x = f'_{xx} =\u00000$, etc.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220290","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 study algebraic varieties that encode the kinematic data for $n$ massless particles in $d$-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for $d=5$ in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices.
{"title":"Kinematic Varieties for Massless Particles","authors":"Smita Rajan, Svala Sverrisdóttir, Bernd Sturmfels","doi":"arxiv-2408.16711","DOIUrl":"https://doi.org/arxiv-2408.16711","url":null,"abstract":"We study algebraic varieties that encode the kinematic data for $n$ massless\u0000particles in $d$-dimensional spacetime subject to momentum conservation. Their\u0000coordinates are spinor brackets, which we derive from the Clifford algebra\u0000associated to the Lorentz group. This was proposed for $d=5$ in the recent\u0000physics literature. Our kinematic varieties are given by polynomial constraints\u0000on tensors with both symmetric and skew symmetric slices.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220295","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 $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with an algebraic action of ${rm PSL}_3$; (2) $X$ is ${rm PSL}_3$-equivariantly birationally isomorphic to a homogeneous fiber space over ${rm PSL}_3/K$ with fiber $mathbb P^1$ for some subgroup $K$ isomorphic to the binary tetrahedral group ${rm SL}_2(mathbb F_3)$.
{"title":"The variety of flexes of plane cubics","authors":"Vladimir L. Popov","doi":"arxiv-2408.16488","DOIUrl":"https://doi.org/arxiv-2408.16488","url":null,"abstract":"Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an\u0000irreducible rational algebraic variety endowed with an algebraic action of\u0000${rm PSL}_3$; (2) $X$ is ${rm PSL}_3$-equivariantly birationally isomorphic\u0000to a homogeneous fiber space over ${rm PSL}_3/K$ with fiber $mathbb P^1$ for\u0000some subgroup $K$ isomorphic to the binary tetrahedral group ${rm\u0000SL}_2(mathbb F_3)$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"170 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220297","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}
Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety being singular along the last. We study invariants of the singularities for these varieties. In the case of an arbitrary curve, we compute the intersection cohomology in terms of the cohomology of the curve. We then turn our attention to rational normal curves. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincar'e duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.
{"title":"Invariants of the singularities of secant varieties of curves","authors":"Daniel Brogan","doi":"arxiv-2408.16736","DOIUrl":"https://doi.org/arxiv-2408.16736","url":null,"abstract":"Consider a smooth projective curve and a given embedding into projective\u0000space via a sufficiently positive line bundle. We can form the secant variety\u0000of $k$-planes through the curve. These are singular varieties, with each secant\u0000variety being singular along the last. We study invariants of the singularities\u0000for these varieties. In the case of an arbitrary curve, we compute the\u0000intersection cohomology in terms of the cohomology of the curve. We then turn\u0000our attention to rational normal curves. In this setting, we prove that all of\u0000the secant varieties are rational homology manifolds, meaning their singular\u0000cohomology satisfies Poincar'e duality. We then compute the nearby and\u0000vanishing cycles for the largest nontrivial secant variety, which is a\u0000projective hypersurface.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220294","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}
Daniele Agostini, Daniel Plaumann, Rainer Sinn, Jannik Lennart Wesner
Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic.
{"title":"Plane quartics and heptagons","authors":"Daniele Agostini, Daniel Plaumann, Rainer Sinn, Jannik Lennart Wesner","doi":"arxiv-2408.15759","DOIUrl":"https://doi.org/arxiv-2408.15759","url":null,"abstract":"Every polygon with n vertices in the complex projective plane is naturally\u0000associated with its adjoint curve of degree n-3. Hence the adjoint of a\u0000heptagon is a plane quartic. We prove that a general plane quartic is the\u0000adjoint of exactly 864 distinct complex heptagons. This number had been\u0000numerically computed by Kohn et al. We use intersection theory and the Scorza\u0000correspondence for quartics to show that 864 is an upper bound, complemented by\u0000a lower bound obtained through explicit analysis of the famous Klein quartic.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220300","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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