We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points. 2020 Mathematics Subject Classification: 14J10, 14D15, 14J29.
{"title":"Smoothing semi-smooth stable Godeaux surfaces","authors":"B. Fantechi, M. Franciosi, R. Pardini","doi":"10.14231/ag-2022-015","DOIUrl":"https://doi.org/10.14231/ag-2022-015","url":null,"abstract":"We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points. 2020 Mathematics Subject Classification: 14J10, 14D15, 14J29.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46303612","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 start with a curve over an algebraically closed ground field of positive characteristic p > 0. By using specialization in cohomology techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new p-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of p. By coupling this p-periodicity in characteristic p with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.
我们从正特征为p >0 0的代数闭合地面场上的曲线开始。利用上同调技术的专门化，在适当的自然共序条件下，证明了希格斯束的模空间与曲线上的连接的模空间之间的上同调辛普森对应关系。我们还证明了阶差为p的Dolbeault模空间的上同环的一个新的p乘周期。通过将特征p上的p周期性与混合特征上的提升/专一化技术耦合，我们发现在任意特征上，不同阶差的Dolbeault模空间的上同环在秩上互素。有趣的是，最后一个结果被证明如下:我们证明了一个弱版本的正特征;我们提升和加强弱版本的结果特征为零;最后，我们将结果归结为正特征。我们处理的模空间承认某些自然同构(Hitchin, de Rham-Hitchin, Hodge-Hitchin)，并且我们发现的所有上同环同构都是由此产生的反常Leray滤波的过滤同构。
{"title":"A cohomological nonabelian Hodge Theorem in positive characteristic","authors":"M. A. Cataldo, Siqing Zhang","doi":"10.14231/ag-2022-018","DOIUrl":"https://doi.org/10.14231/ag-2022-018","url":null,"abstract":"We start with a curve over an algebraically closed ground field of positive characteristic p > 0. By using specialization in cohomology techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new p-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of p. By coupling this p-periodicity in characteristic p with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48482020","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 describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).
{"title":"Logarithmic intersections of double ramification cycles","authors":"D. Holmes, Rosa Schwarz","doi":"10.14231/ag-2022-017","DOIUrl":"https://doi.org/10.14231/ag-2022-017","url":null,"abstract":"We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46021901","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 study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.
{"title":"Deformations of rational curves on primitive symplectic varieties and applications","authors":"C. Lehn, Giovanni Mongardi, Gianluca Pacienza","doi":"10.14231/ag-2023-006","DOIUrl":"https://doi.org/10.14231/ag-2023-006","url":null,"abstract":"We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48752402","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 study a certain wormholing phenomenon that takes place in the Kollár–Shepherd-Barron–Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KBSA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.
{"title":"On wormholes in the moduli space of surfaces","authors":"G. Urz'ua, Nicol'as Vilches","doi":"10.14231/ag-2022-002","DOIUrl":"https://doi.org/10.14231/ag-2022-002","url":null,"abstract":"We study a certain wormholing phenomenon that takes place in the Kollár–Shepherd-Barron–Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KBSA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45153038","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}
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":"1 1","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
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