Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${mathbb P}^3$ with the surprising property that their general projection to ${mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.
菲利克斯·克莱因在研究正二十面体及其对称性的过程中遇到了一个高度对称的构型$60$点在${mathbb P}^3$中。这种配置以各种形式出现，也许最引人注目的是在Shephard-Todd列表中$G_{31}$组中$60$反射面对偶点的配置。在本报告中，我们表明，从最近开始的两条研究路径的角度来看，$60$点显示出有趣的特性。首先，它们产生了两个完全不同的6次意想不到的曲面。Cook II, Harbourne, Migliore, Nagel在2018年引入了意想不到的超表面。与$60$点的配置相关的一个意想不到的曲面是一个具有单个多重奇点$6$的圆锥，另一个具有三个多重奇点$4,2$和$2$。其次，Chiantini和Migliore在2020年观察到${mathbb P}^3$中存在非平凡的点集，它们到${mathbb P}^2$的一般投影是一个完全相交。他们发现了一组这样的集合，他们称之为网格。他们论文的附录描述了${mathbb P}^3$中$24$点的奇异构型，它不是网格，但具有其一般投影是完全相交的显著性质。我们证明Klein构型也不是一个网格，它投射到一个完整的交叉点。我们还确定了它的固有子集，它们具有相同的性质。＼
{"title":"Unexpected Properties of the Klein Configuration of 60 Points in $mathbb{P}^3$","authors":"Piotr Pokora, T. Szemberg, J. Szpond","doi":"10.14760/OWP-2020-19","DOIUrl":"https://doi.org/10.14760/OWP-2020-19","url":null,"abstract":"Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. \u0000In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${mathbb P}^3$ with the surprising property that their general projection to ${mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property. ","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"9 19","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114085573","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 deals with some analytic aspects of GG system introduced by I.M.Gelfand and M.I.Graev: We compute the dimension of the solution space of GG system over the field of functions meromorphic and periodic with respect to a lattice. We describe the monodromy invariant subspace of the solution space. We give a connection formula between a pair of bases consisting of $Gamma$-series solutions of GG system associated to a pair of regular triangulations adjacent to each other in the secondary fan.
{"title":"Global Analysis of GG Systems","authors":"Saiei-Jaeyeong Matsubara-Heo","doi":"10.1093/IMRN/RNAB144","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB144","url":null,"abstract":"This paper deals with some analytic aspects of GG system introduced by I.M.Gelfand and M.I.Graev: We compute the dimension of the solution space of GG system over the field of functions meromorphic and periodic with respect to a lattice. We describe the monodromy invariant subspace of the solution space. We give a connection formula between a pair of bases consisting of $Gamma$-series solutions of GG system associated to a pair of regular triangulations adjacent to each other in the secondary fan.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116871613","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 the boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi-Brauer varieties and quadrics over such fields.
{"title":"Boundedness for finite subgroups of linear algebraic groups","authors":"C. Shramov, V. Vologodsky","doi":"10.1090/tran/8511","DOIUrl":"https://doi.org/10.1090/tran/8511","url":null,"abstract":"We show the boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi-Brauer varieties and quadrics over such fields.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124819460","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 : 2020-09-29DOI: 10.46298/epiga.2021.7041
Claudia Stadlmayr
We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${rm char}(k)=p geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.
我们确定了代数闭域上任意次Picard秩的RDP del Pezzo曲面上出现的所有有理双点构型 $k$ 具有任意特性 ${rm char}(k)=p geq 0$，将杜瓦尔的经典著作推广到正特征。此外，我们还给出了所有RDP del Pezzo次曲面的简化方程 $1$ 包含非紧致有理双点的。
{"title":"Which rational double points occur on del Pezzo surfaces","authors":"Claudia Stadlmayr","doi":"10.46298/epiga.2021.7041","DOIUrl":"https://doi.org/10.46298/epiga.2021.7041","url":null,"abstract":"We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${rm char}(k)=p geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"200 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133839742","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}
O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic $pneq 2$, called OG6 varieties. Assuming $pgeq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.
{"title":"Supersingular O’Grady Varieties of Dimension Six","authors":"L. Fu, Zhiyuan Li, Haitao Zou","doi":"10.1093/IMRN/RNAA349","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA349","url":null,"abstract":"O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic $pneq 2$, called OG6 varieties. Assuming $pgeq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121218394","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 is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.
{"title":"On the Weak Lefschetz Principle in Birational Geometry","authors":"C'esar Lozano Huerta, Alex Massarenti","doi":"10.1090/noti2205","DOIUrl":"https://doi.org/10.1090/noti2205","url":null,"abstract":"This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124768419","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 prove an infinitesimal version of the classical Terracini Lemma for 3--secant planes to a variety. Precisely we prove that if $Xsubseteq PP^r$ is an irreducible, non--degenerate, projective complex variety of dimension $n$ with $rgeq 3n+2$, such that the variety of osculating planes to curves in $X$ has the expected dimension $3n$ and for every $0$--dimensional, curvilinear scheme $gamma$ of length 3 contained in $X$ the family of hyperplanes sections of $X$ which are singular along $gamma$ has dimension larger that $r-3(n+1)$, then $X$ is $2$--secant defective.
{"title":"On the infinitesimal Terracini Lemma","authors":"C. Ciliberto","doi":"10.4171/rlm/926","DOIUrl":"https://doi.org/10.4171/rlm/926","url":null,"abstract":"In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3--secant planes to a variety. Precisely we prove that if $Xsubseteq PP^r$ is an irreducible, non--degenerate, projective complex variety of dimension $n$ with $rgeq 3n+2$, such that the variety of osculating planes to curves in $X$ has the expected dimension $3n$ and for every $0$--dimensional, curvilinear scheme $gamma$ of length 3 contained in $X$ the family of hyperplanes sections of $X$ which are singular along $gamma$ has dimension larger that $r-3(n+1)$, then $X$ is $2$--secant defective.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"131 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123518080","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 certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops.
{"title":"A categorical sl_2 action on some moduli spaces of sheaves","authors":"N. Addington, R. Takahashi","doi":"10.1090/tran/8779","DOIUrl":"https://doi.org/10.1090/tran/8779","url":null,"abstract":"We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133953382","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 : 2020-09-10DOI: 10.1142/s0129167x21500075
Fabian Reede
In this note we prove that the Fourier-Mukai transform $Phi_{mathcal{U}}$ induced by the universal family of the moduli space $mathcal{M}_{mathbb{P}^2}(4,1,3)$ is not fully faithful.
{"title":"The Fourier–Mukai transform of a universal family of stable vector bundles","authors":"Fabian Reede","doi":"10.1142/s0129167x21500075","DOIUrl":"https://doi.org/10.1142/s0129167x21500075","url":null,"abstract":"In this note we prove that the Fourier-Mukai transform $Phi_{mathcal{U}}$ induced by the universal family of the moduli space $mathcal{M}_{mathbb{P}^2}(4,1,3)$ is not fully faithful.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128884981","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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