In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada’s classification of connected components of the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is, strata which are both Shimada root strata and monodromy strata. We also discuss the relationship with moduli spaces of lattice-polarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata.
{"title":"Moduli of elliptic $K3$ surfaces: Monodromy and Shimada root lattice strata n (with an appendix by Markus Kirschmer)","authors":"K. Hulek, M. Lonne","doi":"10.14231/ag-2022-006","DOIUrl":"https://doi.org/10.14231/ag-2022-006","url":null,"abstract":"In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada’s classification of connected components of the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is, strata which are both Shimada root strata and monodromy strata. We also discuss the relationship with moduli spaces of lattice-polarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41440042","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 prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.
{"title":"Inversion of adjunction for quotient singularities","authors":"Yusuke Nakamura, K. Shibata","doi":"10.14231/ag-2022-007","DOIUrl":"https://doi.org/10.14231/ag-2022-007","url":null,"abstract":"We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48607912","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 polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci, that is, the polarization satisfies property $(N_p)$ if the invariant is small. In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer a question about $(N_p)$ on abelian varieties asked by the author and V. Lozovanu in the three dimensional case.
{"title":"Basepoint-freeness thresholds and higher syzygies of abelian threefolds","authors":"Atsushi Ito","doi":"10.14231/ag-2022-023","DOIUrl":"https://doi.org/10.14231/ag-2022-023","url":null,"abstract":"For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant and show that the polarization is basepoint free or projectively normal if the invariant is small. Their result is generalized to higher syzygies by F. Caucci, that is, the polarization satisfies property $(N_p)$ if the invariant is small. In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer a question about $(N_p)$ on abelian varieties asked by the author and V. Lozovanu in the three dimensional case.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49001987","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 $Z subset mathbb{T}_d$ be a non-degenerate hypersurface in $d$-dimensional torus $mathbb{T}_d cong (mathbb{C}^*)^d$ defined by a Laurent polynomial $f$ with a given $d$-dimensional Newton polytope $P$. It follows from a theorem of Ishii that $Z$ is birational to a smooth projective variety $X$ of Kodaira dimension $kappa geq 0$ if and only if the Fine interior $F(P)$ of $P$ is nonempty. We define a unique projective model $widetilde{Z}$ of $Z$ having at worst canonical singularities which allows us to obtain minimal models $widehat{Z}$ of $Z$ by crepant morphisms $widehat{Z} to widetilde{Z}$. Moreover, we show that $kappa = min { d-1, dim F(P) }$ and that general fibers in the Iitaka fibration of the canonical model $widetilde{Z}$ are non-degenerate $(d-1-kappa)$-dimensional toric hypersurfaces of Kodaira dimension $0$. Using the rational polytope $F(P)$, we compute the stringy $E$-function of minimal models $widehat{Z}$ and obtain a combinatorial formula for their stringy Euler numbers.
设$Z subset mathbb{T}_d$是由给定$d$维牛顿多面体$P$的劳伦多项式$f$定义的$d$维环面$mathbb{T}_d cong (mathbb{C}^*)^d$中的非简并超曲面。由Ishii的定理可知$Z$与Kodaira维$kappa geq 0$的光滑投影变项$X$是分形的当且仅当$P$的Fine interior $F(P)$是非空的。我们定义了一个唯一的投影模型$widetilde{Z}$ ($Z$)，它在最坏的情况下具有规范奇点，这使得我们可以通过蠕变态射$widehat{Z} to widetilde{Z}$获得$Z$的最小模型$widehat{Z}$。此外，我们证明了$kappa = min { d-1, dim F(P) }$和典型模型$widetilde{Z}$的Iitaka纤维中的一般纤维是非简并的$(d-1-kappa)$ - Kodaira维的环面超曲面$0$。利用有理多面体$F(P)$，我们计算了最小模型$widehat{Z}$的弦$E$ -函数，得到了它们的弦欧拉数的组合公式。
{"title":"Canonical models of toric hypersurfaces","authors":"V. Batyrev","doi":"10.14231/ag-2023-013","DOIUrl":"https://doi.org/10.14231/ag-2023-013","url":null,"abstract":"Let $Z subset mathbb{T}_d$ be a non-degenerate hypersurface in $d$-dimensional torus $mathbb{T}_d cong (mathbb{C}^*)^d$ defined by a Laurent polynomial $f$ with a given $d$-dimensional Newton polytope $P$. It follows from a theorem of Ishii that $Z$ is birational to a smooth projective variety $X$ of Kodaira dimension $kappa geq 0$ if and only if the Fine interior $F(P)$ of $P$ is nonempty. We define a unique projective model $widetilde{Z}$ of $Z$ having at worst canonical singularities which allows us to obtain minimal models $widehat{Z}$ of $Z$ by crepant morphisms $widehat{Z} to widetilde{Z}$. Moreover, we show that $kappa = min { d-1, dim F(P) }$ and that general fibers in the Iitaka fibration of the canonical model $widetilde{Z}$ are non-degenerate $(d-1-kappa)$-dimensional toric hypersurfaces of Kodaira dimension $0$. Using the rational polytope $F(P)$, we compute the stringy $E$-function of minimal models $widehat{Z}$ and obtain a combinatorial formula for their stringy Euler numbers.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44633498","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}
Generalizing the Moret-Bailly pencil of supersingular abelian surfaces to higher dimensions, we construct for each field of characteristic p>0 a smooth projective variety with trivial dualizing sheaf that does not formally lift to characteristic zero. Our approach heavily relies on local unipotent group schemes, the Beauville--Bogomolov Decomposition for Kahler manifolds with $c_1=0$, and equivariant deformation theory
{"title":"Moret-Bailly families and non-liftable schemes","authors":"D. Roessler, Stefan Schroer","doi":"10.14231/ag-2022-004","DOIUrl":"https://doi.org/10.14231/ag-2022-004","url":null,"abstract":"Generalizing the Moret-Bailly pencil of supersingular abelian surfaces to higher dimensions, we construct for each field of characteristic p>0 a smooth projective variety with trivial dualizing sheaf that does not formally lift to characteristic zero. Our approach heavily relies on local unipotent group schemes, the Beauville--Bogomolov Decomposition for Kahler manifolds with $c_1=0$, and equivariant deformation theory","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44552043","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 an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.
{"title":"Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces","authors":"T. Beckmann, G. Oberdieck","doi":"10.14231/ag-2022-012","DOIUrl":"https://doi.org/10.14231/ag-2022-012","url":null,"abstract":"Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47773935","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 $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results of Boe and Fu about the characteristic cycles of the intersection homology sheaves of Schubert varieties. We also conjecture that certain `Mather polynomials' are unimodal in general Lie type, and log concave in type A.
{"title":"Mather classes and conormal spaces of Schubert varieties in cominuscule spaces","authors":"L. Mihalcea, R. Singh","doi":"10.14231/ag-2023-019","DOIUrl":"https://doi.org/10.14231/ag-2023-019","url":null,"abstract":"Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results of Boe and Fu about the characteristic cycles of the intersection homology sheaves of Schubert varieties. We also conjecture that certain `Mather polynomials' are unimodal in general Lie type, and log concave in type A.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41949523","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":"On the boundedness of $n$-folds with $kappa(X)=n-1$","authors":"Stefano Filipazzi","doi":"10.14231/AG-2024-011","DOIUrl":"https://doi.org/10.14231/AG-2024-011","url":null,"abstract":"In this note we study certain sufficient conditions for a set of minimal klt pairs $(X,Delta)$ with $kappa(X,Delta)=dim(X)-1$ to be bounded.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141205500","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 define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]$ of the zero-sections of the map $Higgs_{r,d} to Coh_{r,d}$, for $r geq 0, d in Z$).
在任意取向Borel-Moore同调理论的背景下，我们定义了光滑投影曲线$X$上($2$维)Calabi-Yau类希格斯束的上同调霍尔代数${AHA}_{Higgs(X)}$，以及它的幂零和半稳定变体。在通常的Borel-Moore同调的情况下，${AHA}_{Higgs(X)}$是$X$上相干束堆叠的(普遍)上同调环$mathbb{H}$上的一个模。我们证明了它是一个无扭转的$mathbb{H}$ -模块，并且我们提供了一个显式的生成器集合(对于$r geq 0, d in Z$，映射$Higgs_{r,d} to Coh_{r,d}$的零截面的基本类集合$[Coh_{r,d}]$)。
{"title":"Cohomological Hall algebra of Higgs sheaves on a curve","authors":"G. Farkas","doi":"10.14231/AG-2020-010","DOIUrl":"https://doi.org/10.14231/AG-2020-010","url":null,"abstract":"We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]$ of the zero-sections of the map $Higgs_{r,d} to Coh_{r,d}$, for $r geq 0, d in Z$).","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66815974","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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