We prove an analogue of Ananyan--Hochster's small subalgebra theorem in the context of sheaves on projective space, and deduce from this a version of Stillman's Conjecture for cohomology tables of sheaves. The main tools in the proof are Draisma's GL-noetherianity theorem and the BGG correspondence.
{"title":"Small projective spaces and Stillman uniformity for sheaves","authors":"D. Erman, Steven V. Sam, A. Snowden","doi":"10.14231/AG-2021-010","DOIUrl":"https://doi.org/10.14231/AG-2021-010","url":null,"abstract":"We prove an analogue of Ananyan--Hochster's small subalgebra theorem in the context of sheaves on projective space, and deduce from this a version of Stillman's Conjecture for cohomology tables of sheaves. The main tools in the proof are Draisma's GL-noetherianity theorem and the BGG correspondence.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48131932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a one-parameter degeneration of reduced compact complex analytic spaces of dimension n , we prove the invariance of the frontier Hodge numbers h p,q (that is, those with pq ( n − p )( n − q ) = 0) for the intersection cohomology of the fibers and also for the cohomology of their desingularizations, assuming that the central fiber is reduced, projective, and has only rational singularities. This can be shown to be equivalent to the invariance of the dimension of the cohomology of the structure sheaf since we can prove the Hodge symmetry for all the Hodge numbers h p,q together with E 1 -degeneration of the Hodge-to-de Rham spectral sequence for nearby fibers, assuming only the projectivity of the central fiber. For the proof of the main theorem, we calculate the graded pieces of the induced V -filtration for the first non-zero member of the Hodge filtration on the intersection complex Hodge module of the total space, which coincides with the direct image of the dualizing sheaf of a desingularization. This calculation also implies that the order of nilpotence of the local monodromy is smaller than in the general singularity case by 2 in the situation of the main theorem assuming further smoothness of general fibers.
{"title":"Deformation of rational singularities and Hodge structure","authors":"M. Kerr, R. Laza, M. Saito","doi":"10.14231/ag-2022-014","DOIUrl":"https://doi.org/10.14231/ag-2022-014","url":null,"abstract":"For a one-parameter degeneration of reduced compact complex analytic spaces of dimension n , we prove the invariance of the frontier Hodge numbers h p,q (that is, those with pq ( n − p )( n − q ) = 0) for the intersection cohomology of the fibers and also for the cohomology of their desingularizations, assuming that the central fiber is reduced, projective, and has only rational singularities. This can be shown to be equivalent to the invariance of the dimension of the cohomology of the structure sheaf since we can prove the Hodge symmetry for all the Hodge numbers h p,q together with E 1 -degeneration of the Hodge-to-de Rham spectral sequence for nearby fibers, assuming only the projectivity of the central fiber. For the proof of the main theorem, we calculate the graded pieces of the induced V -filtration for the first non-zero member of the Hodge filtration on the intersection complex Hodge module of the total space, which coincides with the direct image of the dualizing sheaf of a desingularization. This calculation also implies that the order of nilpotence of the local monodromy is smaller than in the general singularity case by 2 in the situation of the main theorem assuming further smoothness of general fibers.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43629632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a smooth complex projective variety such that the Albanese map of $X$ is generically finite onto its image. Here we study the so-called eventual $m$-paracanonical map of $X$ (when $m=1$ we also assume $chi(K_X)>0$). We show that for $m=1$ this map behaves in a similar way to the canonical map of a surface of general type, while it is birational for $m>1$. We also describe it explicitly in several examples.
{"title":"The eventual paracanonical map of a variety of maximal Albanese dimension","authors":"G. Farkas","doi":"10.14231/AG-2019-014","DOIUrl":"https://doi.org/10.14231/AG-2019-014","url":null,"abstract":"Let $X$ be a smooth complex projective variety such that the Albanese map of $X$ is generically finite onto its image. Here we study the so-called eventual $m$-paracanonical map of $X$ (when $m=1$ we also assume $chi(K_X)>0$). We show that for $m=1$ this map behaves in a similar way to the canonical map of a surface of general type, while it is birational for $m>1$. We also describe it explicitly in several examples.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45324427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that (K, $nu$) is a valued field, f (z) $in$ K[z] is a unitary and irreducible polynomial and (L, $omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $nu$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $omega$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $omega$ as an extension of the associated graded ring of A for the filtration defined by $nu$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $omega$ A[z]/(f (z)) generate it as a gr $nu$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $nu$) $subset$ (L, $omega$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $nu$ if and only if there is no defect. In this case, gr $omega$ A1[z]/(f (z)) is a finitely presented gr $nu$ A1-module.
{"title":"On the construction of valuations and generating sequences on hypersurface singularities","authors":"S. Cutkosky, H. Mourtada, B. Teissier","doi":"10.14231/ag-2021-022","DOIUrl":"https://doi.org/10.14231/ag-2021-022","url":null,"abstract":"Suppose that (K, $nu$) is a valued field, f (z) $in$ K[z] is a unitary and irreducible polynomial and (L, $omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $nu$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $omega$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $omega$ as an extension of the associated graded ring of A for the filtration defined by $nu$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $omega$ A[z]/(f (z)) generate it as a gr $nu$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $nu$) $subset$ (L, $omega$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $nu$ if and only if there is no defect. In this case, gr $omega$ A1[z]/(f (z)) is a finitely presented gr $nu$ A1-module.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48819931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Qto X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Qto X$ has a reduction of structure group to a finite one.
给定一个Henselian和Japanese离散估值环$ a $和一个平面和投影的$ a $-方案$X$，我们遵循Biswas-dos Santos的方法，引入$X$上的相干模的完整子范畴，然后证明它是Tannakian的。然后证明了在一般纤维的正规性下，相关联的仿射平群在强意义上是亲有限的(因此它的函数环是一个Mittag-Leffler模)，并证明了它对有限环子$Q到X$进行分类。这建立了与Nori关于本质上有限基本群的理论的类比。此外，我们将我们的理论与Mehta-Subramanian和Antei-Emsalem-Gasbarri最近发展的理论进行了比较。通过与前者的比较，我们证明了任意拟有限扭量$Qto X$都有一个结构群约简为有限结构群。
{"title":"Finite torsors on projective schemes defined over a discrete valuation ring","authors":"P. H. Hai, J. Santos","doi":"10.14231/ag-2023-001","DOIUrl":"https://doi.org/10.14231/ag-2023-001","url":null,"abstract":"Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Qto X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Qto X$ has a reduction of structure group to a finite one.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42831702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. We show that the (transcendentally-defined) period map actually respects these algebraic structures, and thus that occult period maps are arithmetic. As an intermediate tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.
{"title":"Arithmetic occult period maps","authors":"Jeff Achter","doi":"10.14231/AG-2020-021","DOIUrl":"https://doi.org/10.14231/AG-2020-021","url":null,"abstract":"Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. We show that the (transcendentally-defined) period map actually respects these algebraic structures, and thus that occult period maps are arithmetic. As an intermediate tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42071194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.
{"title":"Logarithmic Gromov–Witten theory with expansions","authors":"Dhruv Ranganathan","doi":"10.14231/ag-2022-022","DOIUrl":"https://doi.org/10.14231/ag-2022-022","url":null,"abstract":"We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44338651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $bar{k}$-rational but not $k$-rational. When $k=mathbb{R}$, we can moreover ensure that their real locus is diffeomorphic to the real locus of a smooth projective $mathbb{R}$-rational variety and that all their unramified cohomology groups are trivial.
{"title":"The Clemens–Griffiths method over non-closed fields","authors":"Olivier Benoist, Olivier Benoist, Olivier Wittenberg, Olivier Wittenberg","doi":"10.14231/AG-2020-025","DOIUrl":"https://doi.org/10.14231/AG-2020-025","url":null,"abstract":"We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $bar{k}$-rational but not $k$-rational. When $k=mathbb{R}$, we can moreover ensure that their real locus is diffeomorphic to the real locus of a smooth projective $mathbb{R}$-rational variety and that all their unramified cohomology groups are trivial.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41318260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dualit� et principe local-global pour les anneaux locaux hens�liens de dimension 2 n (avec un appendice de Jo�l Riou)","authors":"Diego Izquierdo","doi":"10.14231/ag-2019-008","DOIUrl":"https://doi.org/10.14231/ag-2019-008","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42756578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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