Pub Date : 2020-03-19DOI: 10.46298/epiga.2022.6286
X. Roulleau
We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space. We study moreover the configurations of their finite set of $(-2)$-curves.
{"title":"An atlas of K3 surfaces with finite automorphism group","authors":"X. Roulleau","doi":"10.46298/epiga.2022.6286","DOIUrl":"https://doi.org/10.46298/epiga.2022.6286","url":null,"abstract":"We study the geometry of the K3 surfaces $X$ with a finite number\u0000automorphisms and Picard number $geq 3$. We describe these surfaces classified\u0000by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a\u0000projective space. We study moreover the configurations of their finite set of\u0000$(-2)$-curves.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48595110","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 : 2020-02-25DOI: 10.46298/epiga.2021.7020
S. Boissière, E. Floris
Let $G$ be a connected algebraic group. We study $G$-equivariant extremal contractions whose centre is a codimension three $G$-simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those contractions are weighted blow-ups. Comment: 22 pages, 2 figures
{"title":"Divisorial contractions to codimension three orbits","authors":"S. Boissière, E. Floris","doi":"10.46298/epiga.2021.7020","DOIUrl":"https://doi.org/10.46298/epiga.2021.7020","url":null,"abstract":"Let $G$ be a connected algebraic group. We study $G$-equivariant extremal\u0000contractions whose centre is a codimension three $G$-simply connected orbit. In\u0000the spirit of an important result by Kawakita in 2001, we prove that those\u0000contractions are weighted blow-ups.\u0000\u0000 Comment: 22 pages, 2 figures","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49365610","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 : 2020-02-17DOI: 10.46298/epiga.2022.6112
G. Boxer, V. Pilloni
We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve and we define a p-adic duality pairing between the theories in degree 0 and 1.
{"title":"Higher Hida and Coleman theories on the modular curve","authors":"G. Boxer, V. Pilloni","doi":"10.46298/epiga.2022.6112","DOIUrl":"https://doi.org/10.46298/epiga.2022.6112","url":null,"abstract":"We construct Hida and Coleman theories for the degree 0 and 1 cohomology of\u0000automorphic line bundles on the modular curve and we define a p-adic duality\u0000pairing between the theories in degree 0 and 1.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46993345","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 : 2020-02-13DOI: 10.46298/epiga.2020.volume4.6475
O. Debarre, A. Kuznetsov
We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds. Comment: 48 pages. Latest addition to our series of articles on the geometry of Gushel-Mukai varieties; v2: minor stylistic improvements, results unchanged; v3: minor improvements; v4: final version, published in EPIGA
{"title":"Gushel--Mukai varieties: intermediate Jacobians","authors":"O. Debarre, A. Kuznetsov","doi":"10.46298/epiga.2020.volume4.6475","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.6475","url":null,"abstract":"We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of\u0000dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove\u0000that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of\u0000the canonical double covering of any of the two dual Eisenbud-Popescu-Walter\u0000surfaces associated with $A$. As an application, we describe the period maps\u0000for Gushel-Mukai threefolds and fivefolds.\u0000\u0000 Comment: 48 pages. Latest addition to our series of articles on the geometry\u0000 of Gushel-Mukai varieties; v2: minor stylistic improvements, results\u0000 unchanged; v3: minor improvements; v4: final version, published in EPIGA","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484593","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 : 2020-01-21DOI: 10.46298/epiga.2021.volume5.6085
Alastair Craw, Liana Heuberger, Jesus Tapia Amador
Reid's recipe for a finite abelian subgroup $Gsubset text{SL}(3,mathbb{C})$ is a combinatorial procedure that marks the toric fan of the $G$-Hilbert scheme with irreducible representations of $G$. The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of $Gtext{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-V'{e}lez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models. Comment: 29 pages, published version
{"title":"Combinatorial Reid's recipe for consistent dimer models","authors":"Alastair Craw, Liana Heuberger, Jesus Tapia Amador","doi":"10.46298/epiga.2021.volume5.6085","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.6085","url":null,"abstract":"Reid's recipe for a finite abelian subgroup $Gsubset\u0000text{SL}(3,mathbb{C})$ is a combinatorial procedure that marks the toric fan\u0000of the $G$-Hilbert scheme with irreducible representations of $G$. The\u0000geometric McKay correspondence conjecture of Cautis--Logvinenko that describes\u0000certain objects in the derived category of $Gtext{-Hilb}$ in terms of Reid's\u0000recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any\u0000consistent dimer model by marking the toric fan of a crepant resolution of the\u0000vaccuum moduli space in a manner that is compatible with the geometric\u0000correspondence of Bocklandt--Craw--Quintero-V'{e}lez. Our main tool\u0000generalises the jigsaw transformations of Nakamura to consistent dimer models.\u0000\u0000 Comment: 29 pages, published version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47341301","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 : 2020-01-13DOI: 10.46298/epiga.2023.volume7.8393
Ya Deng
In this paper, we study various hyperbolicity properties for a quasi-compact K"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite 'etale cover $tilde{U}$ of $U$ from a quasi-projective manifold $tilde{U}$ such that any projective compactification $X$ of $tilde{U}$ is Picard hyperbolic modulo the boundary $X-tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.
{"title":"Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures","authors":"Ya Deng","doi":"10.46298/epiga.2023.volume7.8393","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.8393","url":null,"abstract":"In this paper, we study various hyperbolicity properties for a quasi-compact\u0000K\"ahler manifold $U$ which admits a complex polarized variation of Hodge\u0000structures so that each fiber of the period map is zero-dimensional. In the\u0000first part, we prove that $U$ is algebraically hyperbolic and that the\u0000generalized big Picard theorem holds for $U$. In the second part, we prove that\u0000there is a finite 'etale cover $tilde{U}$ of $U$ from a quasi-projective\u0000manifold $tilde{U}$ such that any projective compactification $X$ of\u0000$tilde{U}$ is Picard hyperbolic modulo the boundary $X-tilde{U}$, and any\u0000irreducible subvariety of $X$ not contained in $X-tilde{U}$ is of general\u0000type. This result coarsely incorporates previous works by Nadel, Rousseau,\u0000Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients\u0000of bounded symmetric domains by torsion-free lattices.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485331","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 : 2019-11-15DOI: 10.46298/epiga.2022.5971
K. Hulek, Carsten Liese
In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in degree $2$ as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans.
{"title":"The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$","authors":"K. Hulek, Carsten Liese","doi":"10.46298/epiga.2022.5971","DOIUrl":"https://doi.org/10.46298/epiga.2022.5971","url":null,"abstract":"In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in\u0000degree $2$ as well as the associated secondary fan. The main result is an\u0000enumeration of all maximal dimensional cones of the two fans.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48086266","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 : 2019-11-13DOI: 10.46298/epiga.2020.volume4.6060
B. Toen
The objective of this work is to reconsider the schematization problem of [6], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type. We deduce from this several results on the behaviour of the schematization functor, which we propose as a solution to the schematization problem. Comment: 21 pages, french
{"title":"Le probl`eme de la sch'ematisation de Grothendieck revisit'e","authors":"B. Toen","doi":"10.46298/epiga.2020.volume4.6060","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.6060","url":null,"abstract":"The objective of this work is to reconsider the schematization problem of\u0000[6], with a particular focus on the global case over Z. For this, we prove the\u0000conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of\u0000the schematization of a simply connected homotopy type. We deduce from this\u0000several results on the behaviour of the schematization functor, which we\u0000propose as a solution to the schematization problem.\u0000\u0000 Comment: 21 pages, french","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484388","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 : 2019-11-06DOI: 10.46298/epiga.2021.volume5.6581
Tom Bachmann, E. Elmanto, Marc Hoyois, Adeel A. Khan, V. Sosnilo, Maria Yakerson
We obtain geometric models for the infinite loop spaces of the motivic spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are motivically equivalent to $mathbb{Z}times mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{Z}times mathrm{Hilb}_infty^mathrm{or}(mathbb{A}^infty)^+$, and $mathbb{Z}times mathrm{Hilb}_infty^mathrm{fr}(mathbb{A}^infty)^+$, respectively, where $mathrm{Hilb}_d^mathrm{lci}(mathbb{A}^n)$ (resp. $mathrm{Hilb}_d^mathrm{or}(mathbb{A}^n)$, $mathrm{Hilb}_d^mathrm{fr}(mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic. Comment: 13 pages. v5: published version; v4: final version, to appear in 'Epijournal G'eom. Alg'ebrique; v3: minor corrections; v2: added details in the moving lemma over finite fields
{"title":"On the infinite loop spaces of algebraic cobordism and the motivic\u0000 sphere","authors":"Tom Bachmann, E. Elmanto, Marc Hoyois, Adeel A. Khan, V. Sosnilo, Maria Yakerson","doi":"10.46298/epiga.2021.volume5.6581","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.6581","url":null,"abstract":"We obtain geometric models for the infinite loop spaces of the motivic\u0000spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are\u0000motivically equivalent to $mathbb{Z}times\u0000mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{Z}times\u0000mathrm{Hilb}_infty^mathrm{or}(mathbb{A}^infty)^+$, and $mathbb{Z}times\u0000mathrm{Hilb}_infty^mathrm{fr}(mathbb{A}^infty)^+$, respectively, where\u0000$mathrm{Hilb}_d^mathrm{lci}(mathbb{A}^n)$ (resp.\u0000$mathrm{Hilb}_d^mathrm{or}(mathbb{A}^n)$,\u0000$mathrm{Hilb}_d^mathrm{fr}(mathbb{A}^n)$) is the Hilbert scheme of lci\u0000points (resp. oriented points, framed points) of degree $d$ in $mathbb{A}^n$,\u0000and $+$ is Quillen's plus construction. Moreover, we show that the plus\u0000construction is redundant in positive characteristic.\u0000\u0000 Comment: 13 pages. v5: published version; v4: final version, to appear in\u0000 'Epijournal G'eom. Alg'ebrique; v3: minor corrections; v2: added details\u0000 in the moving lemma over finite fields","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48966705","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 : 2019-10-31DOI: 10.46298/epiga.2021.volume5.5980
B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki
We develop a theory of sheaves and cohomology on the category of proper modulus pairs. This complements [KMSY21], where a theory of sheaves and cohomology on the category of non-proper modulus pairs has been developed. Comment: 31 pages
{"title":"Motives with modulus, II: Modulus sheaves with transfers for proper\u0000 modulus pairs","authors":"B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki","doi":"10.46298/epiga.2021.volume5.5980","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.5980","url":null,"abstract":"We develop a theory of sheaves and cohomology on the category of proper\u0000modulus pairs. This complements [KMSY21], where a theory of sheaves and\u0000cohomology on the category of non-proper modulus pairs has been developed.\u0000\u0000 Comment: 31 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484565","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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