Let $M_X(r,xi)$ be the moduli space of stable vector bundles, on a smooth complex projective curve $X$, of rank $r$ and fixed determinant $xi$ such that $°(xi)$ is coprime to $r$. If $E$ is a vector bundle $M_X(r,xi)$ whose restriction to every Hecke curve in $M_X(r,xi)$ is trivial, we prove that $E$ is trivial.
{"title":"On vector bundles over moduli spaces trivial on Hecke curves","authors":"I. Biswas, T. Gómez","doi":"10.1090/PROC/15560","DOIUrl":"https://doi.org/10.1090/PROC/15560","url":null,"abstract":"Let $M_X(r,xi)$ be the moduli space of stable vector bundles, on a smooth complex projective curve $X$, of rank $r$ and fixed determinant $xi$ such that $°(xi)$ is coprime to $r$. If $E$ is a vector bundle $M_X(r,xi)$ whose restriction to every Hecke curve in $M_X(r,xi)$ is trivial, we prove that $E$ is trivial.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"236 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133565968","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 ideal sheaves of lines in a Fano threefold $X$ of Picard rank one and index two are stable objects in the Kuznetsov component $mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macri and Stellari, giving a modular description to the Hilbert scheme of lines in $X$. When $X$ is a cubic threefold, we show that the Serre functor of $mathsf{Ku}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in $mathsf{Ku}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on $mathsf{Ku}(X)$.
{"title":"Some Remarks on Fano Three-Folds of Index Two and Stability Conditions","authors":"L. Pertusi, Song Yang","doi":"10.1093/IMRN/RNAA387","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA387","url":null,"abstract":"We prove that ideal sheaves of lines in a Fano threefold $X$ of Picard rank one and index two are stable objects in the Kuznetsov component $mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macri and Stellari, giving a modular description to the Hilbert scheme of lines in $X$. When $X$ is a cubic threefold, we show that the Serre functor of $mathsf{Ku}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in $mathsf{Ku}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on $mathsf{Ku}(X)$.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133696045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
引入广义磁流变对数正则曲面的概念，建立了完全一般广义磁流变对数正则曲面的极小模型理论。
{"title":"On Minimal Model Theory for Algebraic Log Surfaces","authors":"O. Fujino","doi":"10.11650/TJM/210102","DOIUrl":"https://doi.org/10.11650/TJM/210102","url":null,"abstract":"We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126827391","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}
Pub Date : 2020-04-01DOI: 10.1215/00192082-8827663
I. Biswas, A. Dey, Mainak Poddar, S. Rayan
Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.
{"title":"Toric co-Higgs bundles on toric varieties","authors":"I. Biswas, A. Dey, Mainak Poddar, S. Rayan","doi":"10.1215/00192082-8827663","DOIUrl":"https://doi.org/10.1215/00192082-8827663","url":null,"abstract":"Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121850953","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}
Pub Date : 2020-04-01DOI: 10.2422/2036-2145.202005_019
O. Fujino
We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wiśniewski. Moreover, we establish a generalization for quasi-log canonical pairs.
{"title":"A relative spannedness for log canonical pairs and quasi-log canonical pairs","authors":"O. Fujino","doi":"10.2422/2036-2145.202005_019","DOIUrl":"https://doi.org/10.2422/2036-2145.202005_019","url":null,"abstract":"We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wiśniewski. Moreover, we establish a generalization for quasi-log canonical pairs.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"143 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124888692","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}
Andrea Di Lorenzo, Damiano Fulghesu, Angelo Vistoli
We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.
{"title":"The integral Chow ring of the stack of smooth non-hyperelliptic curves of genus three","authors":"Andrea Di Lorenzo, Damiano Fulghesu, Angelo Vistoli","doi":"10.1090/tran/8354","DOIUrl":"https://doi.org/10.1090/tran/8354","url":null,"abstract":"We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125489310","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}
Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.
{"title":"The Hilbert series of Hodge ideals of hyperplane arrangements","authors":"Bradley Dirks, M. Mustaţă","doi":"10.5427/jsing.2020.20j","DOIUrl":"https://doi.org/10.5427/jsing.2020.20j","url":null,"abstract":"Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"07 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123046110","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 describe spectral data for singular fibres of the $mathsf{SL}(2,mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibres and give a description of $mathsf{SL}(2,mathbb{R})$-Higgs bundles in terms of these semi-abelian spectral data.
{"title":"Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System","authors":"Jo Horn","doi":"10.1093/IMRN/RNAA273","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA273","url":null,"abstract":"We describe spectral data for singular fibres of the $mathsf{SL}(2,mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibres and give a description of $mathsf{SL}(2,mathbb{R})$-Higgs bundles in terms of these semi-abelian spectral data.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128890690","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 local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.
{"title":"Local Uniformization of Abhyankar Valuations","authors":"S. Cutkosky","doi":"10.1307/mmj/20205888","DOIUrl":"https://doi.org/10.1307/mmj/20205888","url":null,"abstract":"We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"23 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128269190","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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