{"title":"On the local étale fundamental group of KLT threefold singularities n (with an appendix by János Kollár)","authors":"Javier Carvajal-Rojas, Axel Stäbler","doi":"10.14231/ag-2023-025","DOIUrl":"https://doi.org/10.14231/ag-2023-025","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"45 3-4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135161035","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}
A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We investigate their rationality, and also provide an example of a Fano-Enriques threefold, whose pliability is 9, i.e. a Fano-Enriques threefold birationally equivalent to exactly 9 Mori fibre spaces in Sarkisov category.
{"title":"On the rationality of Fano–Enriques threefolds","authors":"Arman Sarikyan","doi":"10.14231/ag-2023-023","DOIUrl":"https://doi.org/10.14231/ag-2023-023","url":null,"abstract":"A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We investigate their rationality, and also provide an example of a Fano-Enriques threefold, whose pliability is 9, i.e. a Fano-Enriques threefold birationally equivalent to exactly 9 Mori fibre spaces in Sarkisov category.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"44 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135161038","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 dilatations of general schemes and study their basic properties. Dilatations of group schemes are -- in favorable cases -- again group schemes, called N'eron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $mathfrak g$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under N'eron blowups of $G$.
{"title":"Néron blowups and low-degree cohomological applications","authors":"Arnaud Mayeux, Timo Richarz, Matthieu Romagny","doi":"10.14231/ag-2023-026","DOIUrl":"https://doi.org/10.14231/ag-2023-026","url":null,"abstract":"We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are -- in favorable cases -- again group schemes, called N'eron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $mathfrak g$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under N'eron blowups of $G$.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"388 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103070","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 show that a very general quartic hypersurface in $mathbb P^6 $ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem, who showed stable irrationality over fields of characteristic 0. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for rationality in characteristic zero, introduced by Nicaise--Shinder and Kontsevich--Tschinkel.
{"title":"The diagonal of quartic fivefolds","authors":"Nebojsa Pavic, Stefan Schreieder","doi":"10.14231/ag-2023-027","DOIUrl":"https://doi.org/10.14231/ag-2023-027","url":null,"abstract":"We show that a very general quartic hypersurface in $mathbb P^6 $ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem, who showed stable irrationality over fields of characteristic 0. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for rationality in characteristic zero, introduced by Nicaise--Shinder and Kontsevich--Tschinkel.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"40 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103400","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":"Factorization centers in dimension 2 and the Grothendieck ring of varieties","authors":"Hsueh-Yung Lin, Evgeny Shinder, Susanna Zimmermann","doi":"10.14231/ag-2023-024","DOIUrl":"https://doi.org/10.14231/ag-2023-024","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135161037","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 behavior of the Kodaira dimension under smooth morphisms","authors":"M. Popa, C. Schnell","doi":"10.14231/ag-2023-021","DOIUrl":"https://doi.org/10.14231/ag-2023-021","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43358495","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":"Dense entire curves in rationally connected manifolds (with an appendix by János Kollár)","authors":"F. Campana, J. Winkelmann","doi":"10.14231/ag-2023-018","DOIUrl":"https://doi.org/10.14231/ag-2023-018","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45128084","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":"Erratum: On the monodromy of irreducible symplectic manifolds n (Algebraic Geometry 3 (2016), no. 3, 385–391)","authors":"Giovanni Mongardi","doi":"10.14231/ag-2023-008","DOIUrl":"https://doi.org/10.14231/ag-2023-008","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47641788","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":"Subvarieties of geometric genus zero of a very general hypersurface","authors":"T. Abe","doi":"10.14231/ag-2023-002","DOIUrl":"https://doi.org/10.14231/ag-2023-002","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45379274","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 explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.
{"title":"Generators for the cohomology ring of the moduli of 1-dimensional sheaves on $mathbb{P}^2$","authors":"Weite Pi, Junliang Shen","doi":"10.14231/ag-2023-017","DOIUrl":"https://doi.org/10.14231/ag-2023-017","url":null,"abstract":"We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48703229","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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