Pub Date : 2021-06-06DOI: 10.46298/epiga.2022.7615
Carlos Rito
We give two examples of surfaces with canonical map of degree 4 onto a canonical surface.
我们给出了两个具有4阶正则映射到棘面上的曲面的例子。
{"title":"Examples of surfaces with canonical map of degree 4","authors":"Carlos Rito","doi":"10.46298/epiga.2022.7615","DOIUrl":"https://doi.org/10.46298/epiga.2022.7615","url":null,"abstract":"We give two examples of surfaces with canonical map of degree 4 onto a\u0000canonical surface.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48856675","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 : 2021-05-31DOI: 10.46298/epiga.2022.8653
Richard Larkang, Elizabeth Wulcan
Given a finite locally free resolution of a coherent analytic sheaf $mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $mathcal F$, that represents the Chern class of $mathcal F$ and has support on the support of $mathcal F$. If the connections are $(1,0)$-connections and $mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $mathcal F$. The proof of this goes through a generalized Poincar'e-Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.
{"title":"Chern currents of coherent sheaves","authors":"Richard Larkang, Elizabeth Wulcan","doi":"10.46298/epiga.2022.8653","DOIUrl":"https://doi.org/10.46298/epiga.2022.8653","url":null,"abstract":"Given a finite locally free resolution of a coherent analytic sheaf $mathcal\u0000F$, equipped with Hermitian metrics and connections, we construct an explicit\u0000current, obtained as the limit of certain smooth Chern forms of $mathcal F$,\u0000that represents the Chern class of $mathcal F$ and has support on the support\u0000of $mathcal F$. If the connections are $(1,0)$-connections and $mathcal F$\u0000has pure dimension, then the first nontrivial component of this Chern current\u0000coincides with (a constant times) the fundamental cycle of $mathcal F$. The\u0000proof of this goes through a generalized Poincar'e-Lelong formula, previously\u0000obtained by the authors, and a result that relates the Chern current to the\u0000residue current associated with the locally free resolution.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484270","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 : 2021-03-11DOI: 10.46298/epiga.2022.7340
Thibault Poiret
We work with a smooth relative curve $X_U/U$ with nodal reduction over an excellent and locally factorial scheme $S$. We show that blowing up a nodal model of $X_U$ in the ideal sheaf of a section yields a new nodal model, and describe how these models relate to each other. We construct a N'eron model for the Jacobian of $X_U$, and describe it locally on $S$ as a quotient of the Picard space of a well-chosen nodal model. We provide a combinatorial criterion for the N'eron model to be separated.
{"title":"N'eron models of Jacobians over bases of arbitrary dimension","authors":"Thibault Poiret","doi":"10.46298/epiga.2022.7340","DOIUrl":"https://doi.org/10.46298/epiga.2022.7340","url":null,"abstract":"We work with a smooth relative curve $X_U/U$ with nodal reduction over an\u0000excellent and locally factorial scheme $S$. We show that blowing up a nodal\u0000model of $X_U$ in the ideal sheaf of a section yields a new nodal model, and\u0000describe how these models relate to each other. We construct a N'eron model\u0000for the Jacobian of $X_U$, and describe it locally on $S$ as a quotient of the\u0000Picard space of a well-chosen nodal model. We provide a combinatorial criterion\u0000for the N'eron model to be separated.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484156","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 : 2021-03-11DOI: 10.46298/epiga.2021.7275
I. Biswas, P. O'Sullivan
Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite 'etale covering of X is trivial as an H-equivariant bundle.
{"title":"'Etale triviality of finite equivariant vector bundles","authors":"I. Biswas, P. O'Sullivan","doi":"10.46298/epiga.2021.7275","DOIUrl":"https://doi.org/10.46298/epiga.2021.7275","url":null,"abstract":"Let H be a complex Lie group acting holomorphically on a complex analytic\u0000space X such that the restriction to X_{mathrm{red}} of every H-invariant\u0000regular function on X is constant. We prove that an H-equivariant holomorphic\u0000vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant\u0000bundles for two distinct polynomials f_1 and f_2 whose coefficients are\u0000nonnegative integers, if and only if the pullback of E along some H-equivariant\u0000finite 'etale covering of X is trivial as an H-equivariant bundle.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484519","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 : 2021-01-25DOI: 10.46298/epiga.2022.7482
S. Lichtenbaum, N. Ramachandran, T. Suzuki
We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.
{"title":"The conjectures of Artin-Tate and Birch-Swinnerton-Dyer","authors":"S. Lichtenbaum, N. Ramachandran, T. Suzuki","doi":"10.46298/epiga.2022.7482","DOIUrl":"https://doi.org/10.46298/epiga.2022.7482","url":null,"abstract":"We provide two proofs that the conjecture of Artin-Tate for a fibered surface\u0000is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of\u0000the generic fibre. As a byproduct, we obtain a new proof of a theorem of\u0000Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49348119","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 : 2021-01-01DOI: 10.46298/epiga.2021.7174
Svetlana A. Makarova
The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.
{"title":"Moduli spaces of stable sheaves over quasi-polarized surfaces, and the\u0000 relative Strange Duality morphism","authors":"Svetlana A. Makarova","doi":"10.46298/epiga.2021.7174","DOIUrl":"https://doi.org/10.46298/epiga.2021.7174","url":null,"abstract":"The main result of the present paper is a construction of relative moduli\u0000spaces of stable sheaves over the stack of quasipolarized projective surfaces.\u0000For this, we use the theory of good moduli spaces, whose study was initiated by\u0000Alper. As a corollary, we extend the relative Strange Duality morphism to the\u0000locus of quasipolarized K3 surfaces.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484482","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 : 2020-12-09DOI: 10.46298/epiga.2021.6969
Fumiaki Suzuki
As an application of the theory of Lawson homology and morphic cohomology, Walker proved that the Abel-Jacobi map factors through another regular homomorphism. In this note, we give a direct proof of the theorem.
{"title":"Factorization of the Abel-Jacobi maps","authors":"Fumiaki Suzuki","doi":"10.46298/epiga.2021.6969","DOIUrl":"https://doi.org/10.46298/epiga.2021.6969","url":null,"abstract":"As an application of the theory of Lawson homology and morphic cohomology,\u0000Walker proved that the Abel-Jacobi map factors through another regular\u0000homomorphism. In this note, we give a direct proof of the theorem.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484297","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 : 2020-11-25DOI: 10.46298/epiga.2023.volume7.7700
D. Pirozhkov
A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X times Y$ are induced from decompositions of $X$. We prove that any variety whose Albanese morphism is finite is NSSI, and that the total space of a fibration over NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C times mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus.
{"title":"Stably semiorthogonally indecomposable varieties","authors":"D. Pirozhkov","doi":"10.46298/epiga.2023.volume7.7700","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.7700","url":null,"abstract":"A triangulated category is said to be indecomposable if it admits no\u0000nontrivial semiorthogonal decompositions. We introduce a definition of a\u0000noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This\u0000propery implies, among other things, that each smooth proper subvariety has\u0000indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI,\u0000then for any variety $X$ all semiorthogonal decompositions of $X times Y$ are\u0000induced from decompositions of $X$. We prove that any variety whose Albanese\u0000morphism is finite is NSSI, and that the total space of a fibration over NSSI\u0000base with NSSI fibers is also NSSI. We apply this indecomposability to deduce\u0000that there are no phantom subcategories in some varieties, including surfaces\u0000$C times mathbb{P}^1$, where $C$ is any smooth proper curve of positive\u0000genus.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485285","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 : 2020-11-13DOI: 10.46298/epiga.2021.6908
S. Boucksom, Walter Gubler, Florent Martin
Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.
{"title":"Non-Archimedean volumes of metrized nef line bundles","authors":"S. Boucksom, Walter Gubler, Florent Martin","doi":"10.46298/epiga.2021.6908","DOIUrl":"https://doi.org/10.46298/epiga.2021.6908","url":null,"abstract":"Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a\u0000non-trivially valued non-Archimedean field $K$. Roughly speaking, the\u0000non-Archimedean volume of a continuous metric on the Berkovich analytification\u0000of $L$ measures the asymptotic growth of the space of small sections of tensor\u0000powers of $L$. For a continuous semipositive metric on $L$ in the sense of\u0000Zhang, we show first that the non-Archimedean volume agrees with the energy.\u0000The existence of such a semipositive metric yields that $L$ is nef. A second\u0000result is that the non-Archimedean volume is differentiable at any semipositive\u0000continuous metric. These results are known when $L$ is ample, and the purpose\u0000of this paper is to generalize them to the nef case. The method is based on a\u0000detailed study of the content and the volume of a finitely presented torsion\u0000module over the (possibly non-noetherian) valuation ring of $K$.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484214","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 : 2020-10-16DOI: 10.46298/epiga.2022.8987
P. Corvaja, F. Zucconi
Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.
{"title":"Quartic surface, its bitangents and rational points","authors":"P. Corvaja, F. Zucconi","doi":"10.46298/epiga.2022.8987","DOIUrl":"https://doi.org/10.46298/epiga.2022.8987","url":null,"abstract":"Let X be a smooth quartic surface not containing lines, defined over a number\u0000field K. We prove that there are only finitely many bitangents to X which are\u0000defined over K. This result can be interpreted as saying that a certain\u0000surface, having vanishing irregularity, contains only finitely many rational\u0000points. In our proof, we use the geometry of lines of the quartic double solid\u0000associated to X. In a somewhat opposite direction, we show that on any quartic\u0000surface X over a number field K, the set of algebraic points in X(overeline K)\u0000which are quadratic over a suitable finite extension K' of K is Zariski-dense.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484453","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}
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