Pub Date : 2020-07-28DOI: 10.2422/2036-2145.202010_055
M. Iwai
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $mathcal{F}$: $X$ admits a smooth morphism $f: X rightarrow Y$ with rationally connected fibers such that $mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X rightarrow Y$. We also study foliations with nef anti-canonical bundles.
在本文中，我们研究了几乎非正则叶。我们给出了一个光滑射影簇$X$具有几乎网状正则叶理$mathcal{F}$的结构定理:$X$承认具有合理连接纤维的光滑态射$ F: X 右向Y$，使得$mathcal{F}$是$Y$上一个数值平面正则叶理的回拉。此外，$f$被表征为$mathcal{f}$的代数部分的相对MRC纤维。作为一个推论，一个合理连接的变种的几乎净切线束一般是充足的。为了证明，我们推广了Fujita分解定理。作为一个副产品，我们证明了$f_{*}(mK_{X/Y})$的自反体是任意代数纤维空间$f: X 右转Y$的厄米平面向量束和一般样本自反束的直接和。我们还研究了具有网络反正则束的叶。
{"title":"Almost nef regular foliations and Fujita's decomposition of reflexive sheaves","authors":"M. Iwai","doi":"10.2422/2036-2145.202010_055","DOIUrl":"https://doi.org/10.2422/2036-2145.202010_055","url":null,"abstract":"In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $mathcal{F}$: $X$ admits a smooth morphism $f: X rightarrow Y$ with rationally connected fibers such that $mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X rightarrow Y$. We also study foliations with nef anti-canonical bundles.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134192959","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 K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.
{"title":"Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces","authors":"Victoria Cantoral Farf'an","doi":"10.14288/1.0396004","DOIUrl":"https://doi.org/10.14288/1.0396004","url":null,"abstract":"We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114580688","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 : 2020-07-17DOI: 10.1142/S0219199721500140
Richárd Rimányi, Andrzej Weber
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
{"title":"Elliptic classes on Langlands dual flag varieties","authors":"Richárd Rimányi, Andrzej Weber","doi":"10.1142/S0219199721500140","DOIUrl":"https://doi.org/10.1142/S0219199721500140","url":null,"abstract":"Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121828020","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 define certain closed subvarieties of the flag variety, Hessenberg ideal fibers, and prove that they are paved by affines. Hessenberg ideal fibers are a natural generalization of Springer fibers. In type $G_2$, we give explicit descriptions of all Hessenberg ideal fibers, study some of their geometric properties and use them to completely classify Tymoczko's dot actions of the Weyl group on the cohomology of regular semisimple Hessenberg varieties.
{"title":"Affine Pavings of Hessenberg Ideal Fibers","authors":"Ke Xue","doi":"10.13016/EVQM-0Y32","DOIUrl":"https://doi.org/10.13016/EVQM-0Y32","url":null,"abstract":"We define certain closed subvarieties of the flag variety, Hessenberg ideal fibers, and prove that they are paved by affines. Hessenberg ideal fibers are a natural generalization of Springer fibers. In type $G_2$, we give explicit descriptions of all Hessenberg ideal fibers, study some of their geometric properties and use them to completely classify Tymoczko's dot actions of the Weyl group on the cohomology of regular semisimple Hessenberg varieties.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132825011","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 $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $alpha$-semistable pairs on $mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincare polynomial of the compactified space.
{"title":"Sheaf Theoretic Compactifications of the Space of Rational Quartic \u0000 Plane Curves","authors":"Kiryong Chung","doi":"10.11650/TJM/210103","DOIUrl":"https://doi.org/10.11650/TJM/210103","url":null,"abstract":"Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $alpha$-semistable pairs on $mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincare polynomial of the compactified space.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121880931","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 : 2020-07-04DOI: 10.1142/S0219199721500309
P. Frediani, J. Naranjo, I. Spelta
We study the ramified Prym map $mathcal P_{g,r} longrightarrow mathcal A_{g-1+frac r2}^{delta}$ which assigns to a ramified double cover of a smooth irreducible curve of genus $g$ ramified in $r$ points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fibre. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fibre of the Prym map ${mathcal P}_{1,2}$.
{"title":"The fibers of the ramified Prym map","authors":"P. Frediani, J. Naranjo, I. Spelta","doi":"10.1142/S0219199721500309","DOIUrl":"https://doi.org/10.1142/S0219199721500309","url":null,"abstract":"We study the ramified Prym map $mathcal P_{g,r} longrightarrow mathcal A_{g-1+frac r2}^{delta}$ which assigns to a ramified double cover of a smooth irreducible curve of genus $g$ ramified in $r$ points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fibre. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fibre of the Prym map ${mathcal P}_{1,2}$.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123870122","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 tensor product surface $mathcal{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $phi$ from $mathbb{P}^1times mathbb{P}^1$ to $mathbb{P}^3$. We provide new determinantal representations of $mathcal{S}$ under the assumptions that $phi$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $phi$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.
{"title":"Determinantal tensor product surfaces and the method of moving quadrics","authors":"Laurent Bus'e, Falai Chen","doi":"10.1090/tran/8358","DOIUrl":"https://doi.org/10.1090/tran/8358","url":null,"abstract":"A tensor product surface $mathcal{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $phi$ from $mathbb{P}^1times mathbb{P}^1$ to $mathbb{P}^3$. We provide new determinantal representations of $mathcal{S}$ under the assumptions that $phi$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $phi$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121310917","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 : 2020-06-30DOI: 10.1017/9781108877831.005
A. Corti, Matej Filip, Andrea Petracci
We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.
{"title":"Mirror Symmetry and smoothing Gorenstein toric affine 3-folds","authors":"A. Corti, Matej Filip, Andrea Petracci","doi":"10.1017/9781108877831.005","DOIUrl":"https://doi.org/10.1017/9781108877831.005","url":null,"abstract":"We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126190316","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 : 2020-06-30DOI: 10.1142/S0129167X21500324
Purnaprajna Bangere, Jayan Mukherjee, D. Raychaudhury
In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = mathbb{F}_e$ parametrized by $mathbb P^1$, countably many of which are projective. For $Y = mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Drezet's article in arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet on $mathbb F_e$ with $e<3$ arises as a flat limit of embeddings degenerating to $2:1$ morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on $mathbb{F}_e$, embedded by a complete linear series are smooth points if and only if $0leq eleq 2$. In contrast, Hilbert points corresponding to projective K3 carpets supported on $mathbb{P}^2$ and embedded by a complete linear series are always smooth. The results in an a recent paper of Bangere-Gallego-Gonzalez show that there are no higher dimensional analogues of the results in this article.
{"title":"K3 carpets on minimal rational surfaces and their smoothings","authors":"Purnaprajna Bangere, Jayan Mukherjee, D. Raychaudhury","doi":"10.1142/S0129167X21500324","DOIUrl":"https://doi.org/10.1142/S0129167X21500324","url":null,"abstract":"In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = mathbb{F}_e$ parametrized by $mathbb P^1$, countably many of which are projective. For $Y = mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Drezet's article in arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet on $mathbb F_e$ with $e<3$ arises as a flat limit of embeddings degenerating to $2:1$ morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on $mathbb{F}_e$, embedded by a complete linear series are smooth points if and only if $0leq eleq 2$. In contrast, Hilbert points corresponding to projective K3 carpets supported on $mathbb{P}^2$ and embedded by a complete linear series are always smooth. The results in an a recent paper of Bangere-Gallego-Gonzalez show that there are no higher dimensional analogues of the results in this article.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123819062","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 : 2020-06-18DOI: 10.4007/annals.2021.194.1.5
Nguyen-Bac Dang, C. Favre
We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimension of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption that the square of the first dynamical degree is strictly larger than the second dynamical degree. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine 3-space are all algebraic numbers.
{"title":"Spectral interpretations of dynamical degrees and\u0000 applications","authors":"Nguyen-Bac Dang, C. Favre","doi":"10.4007/annals.2021.194.1.5","DOIUrl":"https://doi.org/10.4007/annals.2021.194.1.5","url":null,"abstract":"We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimension of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption that the square of the first dynamical degree is strictly larger than the second dynamical degree. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine 3-space are all algebraic numbers.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128782935","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：