Andriy Regeta, Christian Urech, Immanuel van Santen
We give a description of the algebraic families of birational transformations of an algebraic variety X. As an application, we show that the morphisms to Bir(X) given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of Bir(X) are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter, Hanamura, and Ramanujam.
{"title":"The Structure of Algebraic Families of Birational Transformations","authors":"Andriy Regeta, Christian Urech, Immanuel van Santen","doi":"arxiv-2409.06475","DOIUrl":"https://doi.org/arxiv-2409.06475","url":null,"abstract":"We give a description of the algebraic families of birational transformations\u0000of an algebraic variety X. As an application, we show that the morphisms to\u0000Bir(X) given by algebraic families satisfy a Chevalley type result and a\u0000certain fibre-dimension formula. Moreover, we show that the algebraic subgroups\u0000of Bir(X) are exactly the closed finite-dimensional subgroups with finitely\u0000many components. We also study algebraic families of birational transformations\u0000preserving a fibration. This builds on previous work of Blanc-Furter, Hanamura,\u0000and Ramanujam.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220349","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 classify smooth projective varieties of Picard rank 2 which has two structures of blow-up of projective space along smooth subvarieties of different dimensions. This gives a characterization of the so called quadro-cubic Cremona transformation.
{"title":"Varieties with two smooth blow up structures","authors":"Supravat Sarkar","doi":"arxiv-2409.10560","DOIUrl":"https://doi.org/arxiv-2409.10560","url":null,"abstract":"We classify smooth projective varieties of Picard rank 2 which has two\u0000structures of blow-up of projective space along smooth subvarieties of\u0000different dimensions. This gives a characterization of the so called\u0000quadro-cubic Cremona transformation.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253205","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}
Indranil Biswas, Chandranandan Gangopadhyay, Ronnie Sebastian
Let $C$ be an irreducible smooth complex projective curve of genus $g$, with $g_C geqslant 2$. Let $E$ be a vector bundle on $C$ of rank $r$, with $rgeqslant 2$. Let $mc Q:=mc Q(E,,d)$ be the Quot Scheme parameterizing torsion quotients of $E$ of degree $d$. We explicitly describe all deformations of $mc Q$.
{"title":"Infinitesimal deformations of some quot schemes, II","authors":"Indranil Biswas, Chandranandan Gangopadhyay, Ronnie Sebastian","doi":"arxiv-2409.06434","DOIUrl":"https://doi.org/arxiv-2409.06434","url":null,"abstract":"Let $C$ be an irreducible smooth complex projective curve of genus $g$, with\u0000$g_C geqslant 2$. Let $E$ be a vector bundle on $C$ of rank $r$, with\u0000$rgeqslant 2$. Let $mc Q:=mc Q(E,,d)$ be the Quot Scheme parameterizing\u0000torsion quotients of $E$ of degree $d$. We explicitly describe all deformations\u0000of $mc Q$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227411","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 compute genus 0 orbifold Gromov--Witten invariants of Calabi--Yau threefold complete intersections in weighted projective stacks, regardless of convexity conditions. The traditional quantumn Lefschetz principle may fail even for invariants with ambient insertions. Using quasimap wall-crossing, we are able to compute invariants with insertions from a specific subring of the Chen--Ruan cohomology, which contains all the ambient cohomology classes. Quasimap wall-crossing gives a mirror theorem expressing the I-function in terms of the J-function via a mirror map. The key of this paper is to find a suitable GIT presentation of the target space, so that the mirror map is invertible. An explicit formula for the I-function is given for all those target spaces and many examples with explicit computations of invariants are provided.
在本文中，我们不考虑凸性条件，计算了加权投影堆栈中Calabi--Yau 三折完全相交的0 属轨道Gromov--Witten不变式。传统的量柱拉夫谢茨原理甚至可能对有环境插入的不变式失效。利用准映射穿墙术，我们可以从陈-阮同构的特定子环计算有插入的不变量，该子环包含所有环境同构类。准映射穿墙给出了一个镜像定理，通过镜像映射表达了 I 函数与 J 函数之间的关系。本文的关键在于找到目标空间的合适 GIT 呈现，从而使镜像映射是可逆的。本文给出了所有目标空间的 I 函数的明确公式，并提供了许多明确计算不变式的例子。
{"title":"Gromov--Witten Invariants of Non-Convex Complete Intersections in Weighted Projective Stacks","authors":"Felix Janda, Nawaz Sultani, Yang Zhou","doi":"arxiv-2409.06193","DOIUrl":"https://doi.org/arxiv-2409.06193","url":null,"abstract":"In this paper we compute genus 0 orbifold Gromov--Witten invariants of\u0000Calabi--Yau threefold complete intersections in weighted projective stacks,\u0000regardless of convexity conditions. The traditional quantumn Lefschetz\u0000principle may fail even for invariants with ambient insertions. Using quasimap\u0000wall-crossing, we are able to compute invariants with insertions from a\u0000specific subring of the Chen--Ruan cohomology, which contains all the ambient\u0000cohomology classes. Quasimap wall-crossing gives a mirror theorem expressing the I-function in\u0000terms of the J-function via a mirror map. The key of this paper is to find a\u0000suitable GIT presentation of the target space, so that the mirror map is\u0000invertible. An explicit formula for the I-function is given for all those\u0000target spaces and many examples with explicit computations of invariants are\u0000provided.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220374","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 the moduli of anti-invariant Higgs bundles as introduced by Zelaci. Using recent existence results of Alper, Halpern-Leistner and Heinloth we establish the existence of a separated good moduli space for semistable anti-invariant Higgs bundles. Along the way this produces a non-GIT proof of the existence of a separated good moduli space for semistable Higgs bundles. We also prove the properness of the Hitchin system in this setting.
{"title":"Moduli of Anti-Invariant Higgs Bundles","authors":"Karim Réga","doi":"arxiv-2409.05793","DOIUrl":"https://doi.org/arxiv-2409.05793","url":null,"abstract":"We study the moduli of anti-invariant Higgs bundles as introduced by Zelaci.\u0000Using recent existence results of Alper, Halpern-Leistner and Heinloth we\u0000establish the existence of a separated good moduli space for semistable\u0000anti-invariant Higgs bundles. Along the way this produces a non-GIT proof of\u0000the existence of a separated good moduli space for semistable Higgs bundles. We\u0000also prove the properness of the Hitchin system in this setting.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220375","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 give a complete classification, up to birational equivalence, of all fibrations by plane projective rational quartic curves in characteristic two.
我们给出了特性二中平面射影有理四分曲线的所有振型的完整分类，直至双等价。
{"title":"Fibrations by plane projective rational quartic curves in characteristic two","authors":"Cesar Hilario, Karl-Otto Stöhr","doi":"arxiv-2409.05464","DOIUrl":"https://doi.org/arxiv-2409.05464","url":null,"abstract":"We give a complete classification, up to birational equivalence, of all\u0000fibrations by plane projective rational quartic curves in characteristic two.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227412","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 parahoric reduction theory of formal connections (or Higgs fields) on a formal principal bundle with parahoric structures, which generalizes Babbitt-Varadarajan's result for the case without parahoric structures [5] and Boalch's result for the case of regular singularity [9]. As applications, we prove the equivalence between extrinsic definition and intrinsic definition of regular singularity and provide a criterion of relative regularity for formal connections, and also demonstrate a parahoric version of Frenkel-Zhu's Borel reduction theorem of formal connections [23].
{"title":"Parahoric reduction theory of formal connections (or Higgs fields)","authors":"Zhi Hu, Pengfei Huang, Ruiran Sun, Runhong Zong","doi":"arxiv-2409.05073","DOIUrl":"https://doi.org/arxiv-2409.05073","url":null,"abstract":"In this paper, we establish the parahoric reduction theory of formal\u0000connections (or Higgs fields) on a formal principal bundle with parahoric\u0000structures, which generalizes Babbitt-Varadarajan's result for the case without\u0000parahoric structures [5] and Boalch's result for the case of regular\u0000singularity [9]. As applications, we prove the equivalence between extrinsic\u0000definition and intrinsic definition of regular singularity and provide a\u0000criterion of relative regularity for formal connections, and also demonstrate a\u0000parahoric version of Frenkel-Zhu's Borel reduction theorem of formal\u0000connections [23].","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220436","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}
Charlie Bruggemann, Vera Choi, Brian Freidin, Jaedon Whyte
We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in $n$-dimensional projective space that are tangent to every $k$-dimensional subspace, for some value of $n$ and $k$. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces we construct curves tangent to each of the lowest degree curves defined over the base field.
{"title":"Transverse-freeness in finite geometries","authors":"Charlie Bruggemann, Vera Choi, Brian Freidin, Jaedon Whyte","doi":"arxiv-2409.05248","DOIUrl":"https://doi.org/arxiv-2409.05248","url":null,"abstract":"We study projective curves and hypersurfaces defined over a finite field that\u0000are tangent to every member of a class of low-degree varieties. Extending\u00002-dimensional work of Asgarli, we first explore the lowest degrees attainable\u0000by smooth hypersurfaces in $n$-dimensional projective space that are tangent to\u0000every $k$-dimensional subspace, for some value of $n$ and $k$. We then study\u0000projective surfaces that serve as models of finite inversive and hyperbolic\u0000planes, finite analogs of spherical and hyperbolic geometries. In these\u0000surfaces we construct curves tangent to each of the lowest degree curves\u0000defined over the base field.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220376","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 article, we explore the extremal contractions of several projective bundles over smooth Fano varieties of Picard rank $1$. We provide a whole class of examples of projective bundles with smooth blow-up structures, derived from the notion of drums which was introduced by Occhetta-Romano-Conde-Wi'sniewski to study interaction with $mathbb{C}^*$-actions and birational geometry. By manipulating projective bundles, we give a simple geometric construction of the rooftop flip, which was introduced recently by Barban-Franceschini. Additionally, we obtain analogues of some recent results of Vats in higher dimensions. The list of projective bundles we consider includes all globally generated bundles over projective space with first Chern class $2$. For each of them, we compute the nef and pseudoeffective cones.
{"title":"Extremal Contraction of Projective Bundles","authors":"Ashima Bansal, Supravat Sarkar, Shivam Vats","doi":"arxiv-2409.05091","DOIUrl":"https://doi.org/arxiv-2409.05091","url":null,"abstract":"In this article, we explore the extremal contractions of several projective\u0000bundles over smooth Fano varieties of Picard rank $1$. We provide a whole class\u0000of examples of projective bundles with smooth blow-up structures, derived from\u0000the notion of drums which was introduced by Occhetta-Romano-Conde-Wi'sniewski\u0000to study interaction with $mathbb{C}^*$-actions and birational geometry. By\u0000manipulating projective bundles, we give a simple geometric construction of the\u0000rooftop flip, which was introduced recently by Barban-Franceschini.\u0000Additionally, we obtain analogues of some recent results of Vats in higher\u0000dimensions. The list of projective bundles we consider includes all globally\u0000generated bundles over projective space with first Chern class $2$. For each of\u0000them, we compute the nef and pseudoeffective cones.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220377","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}
Thaís M. Dalbelo, Daniel Duarte, Maria Aparecida Soares Ruas
We show that the Nash blowup of 2-generic determinantal varieties over fields of positive characteristic is non-singular. We prove this in two steps. Firstly, we explicitly describe the toric structure of such varieties. Secondly, we show that in this case the combinatorics of Nash blowups are free of characteristic. The result then follows from the analogous result in characteristic zero proved by W. Ebeling and S. M. Gusein-Zade.
我们证明了在正特征域上的二元行列式变种的纳什炸裂是非星形的。我们分两步证明这一点：首先，我们明确描述了此类变体的环状结构；其次，我们证明在这种情况下，纳什炸裂的组合学是无特征的。这一结果来自 W. Ebeling 和 S. M. Gusein-Zade 所证明的特性为零的类似结果。
{"title":"Nash blowups of 2-generic determinantal varieties in positive characteristic","authors":"Thaís M. Dalbelo, Daniel Duarte, Maria Aparecida Soares Ruas","doi":"arxiv-2409.04688","DOIUrl":"https://doi.org/arxiv-2409.04688","url":null,"abstract":"We show that the Nash blowup of 2-generic determinantal varieties over fields\u0000of positive characteristic is non-singular. We prove this in two steps.\u0000Firstly, we explicitly describe the toric structure of such varieties.\u0000Secondly, we show that in this case the combinatorics of Nash blowups are free\u0000of characteristic. The result then follows from the analogous result in\u0000characteristic zero proved by W. Ebeling and S. M. Gusein-Zade.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"172 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220380","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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