Let $X$ be a smooth variety over a field $K$ with function field $K(X)$. Using the interpretation of the torsion part of the 'etale cohomology group $H_{text{'et}}^2(K(X), mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $operatorname{Br}'(X)$ that are representable by Azumaya algebras on $X$. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck.
{"title":"Azumaya algebras over unramifed extensions of function fields","authors":"Mohammed Moutand","doi":"arxiv-2408.15893","DOIUrl":"https://doi.org/arxiv-2408.15893","url":null,"abstract":"Let $X$ be a smooth variety over a field $K$ with function field $K(X)$.\u0000Using the interpretation of the torsion part of the 'etale cohomology group\u0000$H_{text{'et}}^2(K(X), mathbb{G}_m)$ in terms of Milnor-Quillen algebraic\u0000$K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps\u0000along unramified extensions of $K(X)$ over $X$, there exist cohomological\u0000Brauer classes in $operatorname{Br}'(X)$ that are representable by Azumaya\u0000algebras on $X$. Theses conditions are almost satisfied in the case of number\u0000fields, providing then, a partial answer on a question of Grothendieck.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220298","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 two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to the Fundamental Theorem of Tropical Geometry. Additionally, under certain technical assumptions, we provide a real version of the Transverse Intersection Theorem. Building on these results, we propose an algorithm to compute a combinatorial bound on the number of positive real roots of a parametrized polynomial equations system. Furthermore, we discuss how this combinatorial bound can be applied to study the number of positive steady states in chemical reaction networks.
{"title":"Computing positive tropical varieties and lower bounds on the number of positive roots","authors":"Kemal Rose, Máté L. Telek","doi":"arxiv-2408.15719","DOIUrl":"https://doi.org/arxiv-2408.15719","url":null,"abstract":"We present two effective tools for computing the positive tropicalization of\u0000algebraic varieties. First, we outline conditions under which the initial ideal\u0000can be used to compute the positive tropicalization, offering a real analogue\u0000to the Fundamental Theorem of Tropical Geometry. Additionally, under certain\u0000technical assumptions, we provide a real version of the Transverse Intersection\u0000Theorem. Building on these results, we propose an algorithm to compute a\u0000combinatorial bound on the number of positive real roots of a parametrized\u0000polynomial equations system. Furthermore, we discuss how this combinatorial\u0000bound can be applied to study the number of positive steady states in chemical\u0000reaction networks.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220301","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 establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant function. A crucial step is the definition of the BPS sheaf as a complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf when the situation arises from a smooth affine weakly symplectic algebraic variety with a weak moment map. This situation gives local models for 1-Artin derived stacks with self-dual cotangent complex. We then apply these results to prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore homology of $0$-shifted symplectic stacks (or more generally, derived stacks with self-dual cotangent complex) having a proper good moduli space. One striking application is the purity of the Borel--Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group.
{"title":"Cohomological integrality for symmetric quotient stacks","authors":"Lucien Hennecart","doi":"arxiv-2408.15786","DOIUrl":"https://doi.org/arxiv-2408.15786","url":null,"abstract":"In this paper, we establish the sheafified version of the cohomological\u0000integrality conjecture for stacks obtained as a quotient of a smooth affine\u0000symmetric algebraic variety by a reductive algebraic group equipped with an\u0000invariant function. A crucial step is the definition of the BPS sheaf as a\u0000complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf\u0000when the situation arises from a smooth affine weakly symplectic algebraic\u0000variety with a weak moment map. This situation gives local models for 1-Artin\u0000derived stacks with self-dual cotangent complex. We then apply these results to\u0000prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore\u0000homology of $0$-shifted symplectic stacks (or more generally, derived stacks\u0000with self-dual cotangent complex) having a proper good moduli space. One\u0000striking application is the purity of the Borel--Moore homology of the moduli\u0000stack of principal Higgs bundles over a smooth projective curve for a reductive\u0000group.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220299","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 show that birational hyper-K"ahler varieties of $K3^{[n]}$-type are twisted derived equivalent with respect to some Brauer class. Furthermore, if a $K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm satisfies a congruence condition modulo 4, we show that any hyper-K"ahler variety birational to X is derived equivalent to X. This verifies new cases of the D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of our (twisted) derived equivalences are constructed from Markman's projectively hyperholomorphic bundles.
我们证明，$K3^{[n]}$型的双向超K/"ahler "综关于某个布劳尔类是扭曲派生等价的。此外，如果$K3^{[n]}$型 variety X 承认一个可分性为 1 的因子类，其规范满足 modulo 4 的全等条件，我们证明了任何与 X 双向的超（hyper-K"ahl）ervariety 都与 X 派生等价。我们的（扭曲的）派生等价的傅里叶-穆凯核是由马克曼的投影超holomorphic束构造的。
{"title":"The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles","authors":"Davesh Maulik, Junliang Shen, Qizheng Yin","doi":"arxiv-2408.14775","DOIUrl":"https://doi.org/arxiv-2408.14775","url":null,"abstract":"We show that birational hyper-K\"ahler varieties of $K3^{[n]}$-type are\u0000twisted derived equivalent with respect to some Brauer class. Furthermore, if a\u0000$K3^{[n]}$-type variety X admits a divisor class of divisibility 1 whose norm\u0000satisfies a congruence condition modulo 4, we show that any hyper-K\"ahler\u0000variety birational to X is derived equivalent to X. This verifies new cases of\u0000the D-equivalence conjecture in higher dimension. The Fourier-Mukai kernels of\u0000our (twisted) derived equivalences are constructed from Markman's projectively\u0000hyperholomorphic bundles.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220304","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 prove that, over a smooth quasi-projective curve, the set of non-isotrivial, smooth and projective families of polarized varieties with a fixed Hilbert polynomial and semi-ample canonical bundle is bounded. This extends the boundedness results of Arakelov, Parshin, and Kov'acs--Lieblich beyond the canonically polarized case.
{"title":"Boundedness results for families of non-canonically polarized projective varieties","authors":"Kenneth Ascher, Behrouz Taji","doi":"arxiv-2408.15153","DOIUrl":"https://doi.org/arxiv-2408.15153","url":null,"abstract":"We prove that, over a smooth quasi-projective curve, the set of\u0000non-isotrivial, smooth and projective families of polarized varieties with a\u0000fixed Hilbert polynomial and semi-ample canonical bundle is bounded. This\u0000extends the boundedness results of Arakelov, Parshin, and Kov'acs--Lieblich\u0000beyond the canonically polarized case.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220324","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 the first part of this article, we give bounds on self-intersections $C^2$ of integral curves $C$ on blow-ups $Bl_nX$ of surfaces $X$ with the anti-cannonical divisor $-K_X$ effective. In the last part, we prove the weak bounded negativity for self-intersections $C^2$ of integral curves $C$ in a family of surfaces $f:Ylongrightarrow B$ where $B$ is a smooth curve.
{"title":"On Weak bounded negativity conjecture","authors":"Snehajit Misra, Nabanita Ray","doi":"arxiv-2408.15187","DOIUrl":"https://doi.org/arxiv-2408.15187","url":null,"abstract":"In the first part of this article, we give bounds on self-intersections $C^2$\u0000of integral curves $C$ on blow-ups $Bl_nX$ of surfaces $X$ with the\u0000anti-cannonical divisor $-K_X$ effective. In the last part, we prove the weak\u0000bounded negativity for self-intersections $C^2$ of integral curves $C$ in a\u0000family of surfaces $f:Ylongrightarrow B$ where $B$ is a smooth curve.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220302","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}
Baran Hashemi, Roderic G. Corominas, Alessandro Giacchetto
How can Transformers model and learn enumerative geometry? What is a robust procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a new paradigm in computational enumerative geometry in analyzing the $psi$-class intersection numbers on the moduli space of curves. By formulating the enumerative problem as a continuous optimization task, we develop a Transformer-based model for computing $psi$-class intersection numbers based on the underlying quantum Airy structure. For a finite range of genera, our model is capable of regressing intersection numbers that span an extremely wide range of values, from $10^{-45}$ to $10^{45}$. To provide a proper inductive bias for capturing the recursive behavior of intersection numbers, we propose a new activation function, Dynamic Range Activator (DRA). Moreover, given the severe heteroscedasticity of $psi$-class intersections and the required precision, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window that is aware of the number of marked points. Next, we go beyond merely computing intersection numbers and explore the enumerative "world-model" of the Transformers. Through a series of causal inference and correlational interpretability analyses, we demonstrate that Transformers are actually modeling Virasoro constraints in a purely data-driven manner. Additionally, we provide evidence for the comprehension of several values appearing in the large genus asymptotic of $psi$-class intersection numbers through abductive hypothesis testing.
{"title":"Can Transformers Do Enumerative Geometry?","authors":"Baran Hashemi, Roderic G. Corominas, Alessandro Giacchetto","doi":"arxiv-2408.14915","DOIUrl":"https://doi.org/arxiv-2408.14915","url":null,"abstract":"How can Transformers model and learn enumerative geometry? What is a robust\u0000procedure for using Transformers in abductive knowledge discovery within a\u0000mathematician-machine collaboration? In this work, we introduce a new paradigm\u0000in computational enumerative geometry in analyzing the $psi$-class\u0000intersection numbers on the moduli space of curves. By formulating the\u0000enumerative problem as a continuous optimization task, we develop a\u0000Transformer-based model for computing $psi$-class intersection numbers based\u0000on the underlying quantum Airy structure. For a finite range of genera, our\u0000model is capable of regressing intersection numbers that span an extremely wide\u0000range of values, from $10^{-45}$ to $10^{45}$. To provide a proper inductive\u0000bias for capturing the recursive behavior of intersection numbers, we propose a\u0000new activation function, Dynamic Range Activator (DRA). Moreover, given the\u0000severe heteroscedasticity of $psi$-class intersections and the required\u0000precision, we quantify the uncertainty of the predictions using Conformal\u0000Prediction with a dynamic sliding window that is aware of the number of marked\u0000points. Next, we go beyond merely computing intersection numbers and explore\u0000the enumerative \"world-model\" of the Transformers. Through a series of causal\u0000inference and correlational interpretability analyses, we demonstrate that\u0000Transformers are actually modeling Virasoro constraints in a purely data-driven\u0000manner. Additionally, we provide evidence for the comprehension of several\u0000values appearing in the large genus asymptotic of $psi$-class intersection\u0000numbers through abductive hypothesis testing.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220305","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 prove the ACC conjecture for local volumes. Moreover, when the local volume is bounded away from zero, we prove Shokurov's ACC conjecture for minimal log discrepancies.
{"title":"ACC for local volumes","authors":"Jingjun Han, Jihao Liu, Lu Qi","doi":"arxiv-2408.15090","DOIUrl":"https://doi.org/arxiv-2408.15090","url":null,"abstract":"We prove the ACC conjecture for local volumes. Moreover, when the local\u0000volume is bounded away from zero, we prove Shokurov's ACC conjecture for\u0000minimal log discrepancies.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220303","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 real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights $-1$ and $-2$. A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its ``general fiber'' does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.
{"title":"Periods of Real Biextensions","authors":"Richard Hain","doi":"arxiv-2408.13997","DOIUrl":"https://doi.org/arxiv-2408.13997","url":null,"abstract":"A real biextension is a real mixed Hodge structure that is an extension of\u0000R(0) by a mixed Hodge structure with weights $-1$ and $-2$. A unipotent real\u0000biextension over an algebraic manifold is a variation of mixed Hodge structure\u0000over it, each of whose fibers is a real biextension and whose weight graded\u0000quotients are do not vary. We show that if a unipotent real biextension has non\u0000abelian monodromy, then its ``general fiber'' does not split. This result is a\u0000tool for investigating the boundary behaviour of normal functions and is\u0000applied in arXiv:2408.07809 to study the boundary behaviour of the normal\u0000function of the Ceresa cycle.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220321","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 study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational elliptic surface. We then study the property of bigness of the tangent bundle $T_S$ of a weak del Pezzo surface $S$. When the degree of $S$ is $4$, we completely determine the bigness of the tangent bundle through the configuration of $(-2)$-curves. When the degree $d$ of $S$ is less than or equal to $3$, we get a partial answer. In particular, we show that $T_S$ is not big when the number of $(-2)$-curves is less than or equal to $7-d$, and $T_S$ is big when $d=3$ and $S$ has the maximum number of $(-2)$-curves. The main ingredient of the proof is to produce irreducible effective divisors on $mathbb{P}(T_S)$, using Serrano's work on the relative tangent bundle when $S$ has a fibration, or the total dual VMRT associated to a conic fibration on $S$.
{"title":"Positivity of the tangent bundle of rational surfaces with nef anticanonical divisor","authors":"Hosung Kim, Jeong-Seop Kim, Yongnam Lee","doi":"arxiv-2408.14411","DOIUrl":"https://doi.org/arxiv-2408.14411","url":null,"abstract":"In this paper, we study the property of bigness of the tangent bundle of a\u0000smooth projective rational surface with nef anticanonical divisor. We first\u0000show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational\u0000elliptic surface. We then study the property of bigness of the tangent bundle\u0000$T_S$ of a weak del Pezzo surface $S$. When the degree of $S$ is $4$, we\u0000completely determine the bigness of the tangent bundle through the\u0000configuration of $(-2)$-curves. When the degree $d$ of $S$ is less than or\u0000equal to $3$, we get a partial answer. In particular, we show that $T_S$ is not\u0000big when the number of $(-2)$-curves is less than or equal to $7-d$, and $T_S$\u0000is big when $d=3$ and $S$ has the maximum number of $(-2)$-curves. The main\u0000ingredient of the proof is to produce irreducible effective divisors on\u0000$mathbb{P}(T_S)$, using Serrano's work on the relative tangent bundle when $S$\u0000has a fibration, or the total dual VMRT associated to a conic fibration on $S$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220306","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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