Pub Date : 2023-11-07DOI: 10.46298/epiga.2023.11019
Robert Lazarsfeld, Olivier Martin
Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic curve.
{"title":"Measures of association between algebraic varieties, II: self-correspondences","authors":"Robert Lazarsfeld, Olivier Martin","doi":"10.46298/epiga.2023.11019","DOIUrl":"https://doi.org/10.46298/epiga.2023.11019","url":null,"abstract":"Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic curve.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475304","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 : 2023-10-27DOI: 10.46298/epiga.2023.9962
Elisabetta Colombo, Paola Frediani, Juan Carlos Naranjo, Gian Pietro Pirola
We study the second fundamental form of the Siegel metric in $mathcal A_5$ restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals.
{"title":"The second fundamental form of the moduli space of cubic threefolds in $mathcal A_5$","authors":"Elisabetta Colombo, Paola Frediani, Juan Carlos Naranjo, Gian Pietro Pirola","doi":"10.46298/epiga.2023.9962","DOIUrl":"https://doi.org/10.46298/epiga.2023.9962","url":null,"abstract":"We study the second fundamental form of the Siegel metric in $mathcal A_5$ restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318942","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 : 2023-10-25DOI: 10.46298/epiga.2023.10806
René Mboro
This note presents some properties of the variety of planes $F_2(X)subset G(3,7)$ of a cubic $5$-fold $Xsubset mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.
{"title":"Remarks on the geometry of the variety of planes of a cubic fivefold","authors":"René Mboro","doi":"10.46298/epiga.2023.10806","DOIUrl":"https://doi.org/10.46298/epiga.2023.10806","url":null,"abstract":"This note presents some properties of the variety of planes $F_2(X)subset G(3,7)$ of a cubic $5$-fold $Xsubset mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135217983","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 : 2023-10-03DOI: 10.46298/epiga.2023.10307
Jonas Bergström, Carel Faber
We use a classification result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with $ell$-adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of $overline{mathcal M}_{3,n}$ and $mathcal M_{3,n}$ for $n leq 14$ and of local systems $mathbb{V}_{lambda}$ on $mathcal{A}_3$ for $|lambda| leq 16$.
{"title":"Cohomology of moduli spaces via a result of Chenevier and Lannes","authors":"Jonas Bergström, Carel Faber","doi":"10.46298/epiga.2023.10307","DOIUrl":"https://doi.org/10.46298/epiga.2023.10307","url":null,"abstract":"We use a classification result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with $ell$-adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of $overline{mathcal M}_{3,n}$ and $mathcal M_{3,n}$ for $n leq 14$ and of local systems $mathbb{V}_{lambda}$ on $mathcal{A}_3$ for $|lambda| leq 16$.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689977","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 : 2023-09-27DOI: 10.46298/epiga.2023.8562
Alexander B. Ivanov
We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$ introduced in [Iva21] attached to an unramified reductive group ${bf G}$ over a non-archimedean local field. We prove that when ${bf G}$ is classical, $b$ basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms, and prove a loop version of Frobenius-twisted Steinberg's cross section.
{"title":"On a decomposition of $p$-adic Coxeter orbits","authors":"Alexander B. Ivanov","doi":"10.46298/epiga.2023.8562","DOIUrl":"https://doi.org/10.46298/epiga.2023.8562","url":null,"abstract":"We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$ introduced in [Iva21] attached to an unramified reductive group ${bf G}$ over a non-archimedean local field. We prove that when ${bf G}$ is classical, $b$ basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramified tori to the case of extended pure inner forms, and prove a loop version of Frobenius-twisted Steinberg's cross section.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476212","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 : 2023-05-30DOI: 10.46298/epiga.2023.8507
Lev Borisov
We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective planes with this automorphism group. This includes, in particular, the fake projective plane discovered by J. Keum.
{"title":"On equations of fake projective planes with automorphism group of order $21$","authors":"Lev Borisov","doi":"10.46298/epiga.2023.8507","DOIUrl":"https://doi.org/10.46298/epiga.2023.8507","url":null,"abstract":"We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective planes with this automorphism group. This includes, in particular, the fake projective plane discovered by J. Keum.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135692589","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 : 2022-12-06DOI: 10.46298/epiga.2023.10432
Yifei Chen, Baohua Fu, Qifeng Li
To each complex composition algebra $mathbb{A}$, there associates a projective symmetric manifold $X(mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${rm Lag}(3,6), {rm Gr}(3,6), mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(mathbb{A})$, then every fiber is isomorphic to $X(mathbb{A})$.
{"title":"Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras","authors":"Yifei Chen, Baohua Fu, Qifeng Li","doi":"10.46298/epiga.2023.10432","DOIUrl":"https://doi.org/10.46298/epiga.2023.10432","url":null,"abstract":"To each complex composition algebra $mathbb{A}$, there associates a\u0000projective symmetric manifold $X(mathbb{A})$ of Picard number one, which is\u0000just a smooth hyperplane section of the following varieties ${rm Lag}(3,6),\u0000{rm Gr}(3,6), mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these\u0000varieties are rigid, namely for any smooth family of projective manifolds over\u0000a connected base, if one fiber is isomorphic to $X(mathbb{A})$, then every\u0000fiber is isomorphic to $X(mathbb{A})$.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484972","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 : 2022-11-22DOI: 10.46298/epiga.2023.10425
D. Huybrechts
The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.
{"title":"Chow groups of surfaces of lines in cubic fourfolds","authors":"D. Huybrechts","doi":"10.46298/epiga.2023.10425","DOIUrl":"https://doi.org/10.46298/epiga.2023.10425","url":null,"abstract":"The surface of lines in a cubic fourfold intersecting a fixed line splits\u0000motivically into two parts, one of which resembles a K3 surface. We define the\u0000analogue of the Beauville-Voisin class and study the push-forward map to the\u0000Fano variety of all lines with respect to the natural splitting of the\u0000Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484914","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 : 2022-10-29DOI: 10.46298/epiga.2023.10231
Chenyang Xu
We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as the original pair.
{"title":"K-stability for varieties with a big anticanonical class","authors":"Chenyang Xu","doi":"10.46298/epiga.2023.10231","DOIUrl":"https://doi.org/10.46298/epiga.2023.10231","url":null,"abstract":"We extend the algebraic K-stability theory to projective klt pairs with a big\u0000anticanonical class. While in general such a pair could behave pathologically,\u0000it is observed in this note that K-semistability condition will force them to\u0000have a klt anticanonical model, whose stability property is the same as the\u0000original pair.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485128","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 : 2022-07-19DOI: 10.46298/epiga.2023.9960
F. Anella, A. Horing
K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S subset mathbb{P}(Omega_S)$ that plays a similar role to the surface of bitangents for a quartic in $mathbb{P}^3$.
{"title":"The cotangent bundle of K3 surfaces of degree two","authors":"F. Anella, A. Horing","doi":"10.46298/epiga.2023.9960","DOIUrl":"https://doi.org/10.46298/epiga.2023.9960","url":null,"abstract":"K3 surfaces have been studied from many points of view, but the positivity of\u0000the cotangent bundle is not well understood. In this paper we explore the\u0000surprisingly rich geometry of the projectivised cotangent bundle of a very\u0000general polarised K3 surface $S$ of degree two. In particular, we describe the\u0000geometry of a surface $D_S subset mathbb{P}(Omega_S)$ that plays a similar\u0000role to the surface of bitangents for a quartic in $mathbb{P}^3$.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485506","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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