Pub Date : 2019-10-20DOI: 10.46298/epiga.2020.volume4.5890
A. Pukhlikov
We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${mathbb P}^{M+1}$ for $Mgeqslant 5$ there are no Galois rational covers $Xdashrightarrow V$ of degree $dgeqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $Xdashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties. Comment: the final journal version
{"title":"Rationally connected rational double covers of primitive Fano varieties","authors":"A. Pukhlikov","doi":"10.46298/epiga.2020.volume4.5890","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5890","url":null,"abstract":"We show that for a Zariski general hypersurface $V$ of degree $M+1$ in\u0000${mathbb P}^{M+1}$ for $Mgeqslant 5$ there are no Galois rational covers\u0000$Xdashrightarrow V$ of degree $dgeqslant 2$ with an abelian Galois group,\u0000where $X$ is a rationally connected variety. In particular, there are no\u0000rational maps $Xdashrightarrow V$ of degree 2 with $X$ rationally connected.\u0000This fact is true for many other families of primitive Fano varieties as well\u0000and motivates a conjecture on absolute rigidity of primitive Fano varieties.\u0000\u0000 Comment: the final journal version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47986157","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 : 2019-08-28DOI: 10.46298/epiga.2022.7047
Matteo Altavilla, Marina Petković, Franco Rota
General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$ and investigate their moduli spaces, using the stability condition constructed by Bayer, Lahoz, Macr`i, and Stellari, and the Abel--Jacobi map. We identify a subvariety of the moduli space isomorphic to $Y$ itself, and as an application we prove a (refined) categorical Torelli theorem for general quartic double solids.
{"title":"Moduli spaces on the Kuznetsov component of Fano threefolds of index 2","authors":"Matteo Altavilla, Marina Petković, Franco Rota","doi":"10.46298/epiga.2022.7047","DOIUrl":"https://doi.org/10.46298/epiga.2022.7047","url":null,"abstract":"General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard\u0000rank 1 are del Pezzo surfaces, and their Picard group is related to a root\u0000system. To the corresponding roots, we associate objects in the Kuznetsov\u0000component of $Y$ and investigate their moduli spaces, using the stability\u0000condition constructed by Bayer, Lahoz, Macr`i, and Stellari, and the\u0000Abel--Jacobi map. We identify a subvariety of the moduli space isomorphic to\u0000$Y$ itself, and as an application we prove a (refined) categorical Torelli\u0000theorem for general quartic double solids.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47834184","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 : 2019-08-27DOI: 10.46298/epiga.2020.volume4.5721
Kirti Joshi, C. Pauly
In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus. Comment: 17 pages; Final version Epijournal de G'eom'etrie Alg'ebrique, Volume 4 (2020), Article Nr. 17
{"title":"Opers of higher types, Quot-schemes and Frobenius instability loci","authors":"Kirti Joshi, C. Pauly","doi":"10.46298/epiga.2020.volume4.5721","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5721","url":null,"abstract":"In this paper we continue our study of the Frobenius instability locus in the\u0000coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$\u0000over a smooth projective curve defined over an algebraically closed field of\u0000characteristic $p>0$. In a previous paper we identified the \"maximal\" Frobenius\u0000instability strata with opers (more precisely as opers of type $1$ in the\u0000terminology of the present paper) and related them to certain Quot-schemes of\u0000Frobenius direct images of line bundles. The main aim of this paper is to\u0000describe for any integer $q geq 1$ a conjectural generalization of this\u0000correspondence between opers of type $q$ (which we introduce here) and\u0000Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also\u0000give a conjectural formula for the dimension of the Frobenius instability\u0000locus.\u0000\u0000 Comment: 17 pages; Final version Epijournal de G'eom'etrie Alg'ebrique, Volume\u0000 4 (2020), Article Nr. 17","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483832","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 : 2019-08-16DOI: 10.46298/epiga.2022.volume6.5762
Gr'egoire Menet
We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.
{"title":"Integral cohomology of quotients via toric geometry","authors":"Gr'egoire Menet","doi":"10.46298/epiga.2022.volume6.5762","DOIUrl":"https://doi.org/10.46298/epiga.2022.volume6.5762","url":null,"abstract":"We describe the integral cohomology of $X/G$ where $X$ is a compact complex\u0000manifold and $G$ a cyclic group of prime order with only isolated fixed points.\u0000As a preliminary step, we investigate the integral cohomology of toric blow-ups\u0000of quotients of $mathbb{C}^n$. We also provide necessary and sufficient\u0000conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to\u0000degenerate at the second page. As an application, we compute the\u0000Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a\u0000K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484770","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 : 2019-08-08DOI: 10.46298/epiga.2021.volume5.5979
B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki
We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20]. Comment: 64 pages
{"title":"Motives with modulus, I: Modulus sheaves with transfers for non-proper\u0000 modulus pairs","authors":"B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki","doi":"10.46298/epiga.2021.volume5.5979","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.5979","url":null,"abstract":"We develop a theory of modulus sheaves with transfers, which generalizes\u0000Voevodsky's theory of sheaves with transfers. This paper and its sequel are\u0000foundational for the theory of motives with modulus, which is developed in\u0000[KMSY20].\u0000\u0000 Comment: 64 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484560","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 : 2019-07-29DOI: 10.46298/epiga.2022.6819
M. Jardim, A. Maciocia
We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macr`i-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parametrizes geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use all of these facts to prove that Gieseker semistability is equivalent to asymptotic semistability along a class of paths in the upper half plane, and to show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane.
{"title":"Walls and asymptotics for Bridgeland stability conditions on 3-folds","authors":"M. Jardim, A. Maciocia","doi":"10.46298/epiga.2022.6819","DOIUrl":"https://doi.org/10.46298/epiga.2022.6819","url":null,"abstract":"We consider Bridgeland stability conditions for three-folds conjectured by\u0000Bayer-Macr`i-Toda in the case of Picard rank one. We study the differential\u0000geometry of numerical walls, characterizing when they are bounded, discussing\u0000possible intersections, and showing that they are essentially regular. Next, we\u0000prove that walls within a certain region of the upper half plane that\u0000parametrizes geometric stability conditions must always intersect the curve\u0000given by the vanishing of the slope function and, for a fixed value of s, have\u0000a maximum turning point there. We then use all of these facts to prove that\u0000Gieseker semistability is equivalent to asymptotic semistability along a class\u0000of paths in the upper half plane, and to show how to find large families of\u0000walls. We illustrate how to compute all of the walls and describe the\u0000Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex\u0000projective 3-space in a suitable region of the upper half plane.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484628","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 : 2019-07-02DOI: 10.46298/epiga.2022.6986
J. Bryan, 'Ad'am Gyenge
Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = sum_{n=0}^{infty} eleft(operatorname{Hilb}^{n}(X)^{G} right), q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $frac{1}{2} e(X/G)$ for the congruence subgroup $Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.
{"title":"$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta\u0000 products","authors":"J. Bryan, 'Ad'am Gyenge","doi":"10.46298/epiga.2022.6986","DOIUrl":"https://doi.org/10.46298/epiga.2022.6986","url":null,"abstract":"Let $X$ be a complex $K3$ surface with an effective action of a group $G$\u0000which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) =\u0000sum_{n=0}^{infty} eleft(operatorname{Hilb}^{n}(X)^{G} right), q^{n-1} $$\u0000be the generating function for the Euler characteristics of the Hilbert schemes\u0000of $G$-invariant length $n$ subschemes. We show that its reciprocal,\u0000$Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight\u0000$frac{1}{2} e(X/G)$ for the congruence subgroup $Gamma_{0}(|G|)$. We give an\u0000explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82\u0000possible $(X,G)$. The key intermediate result we prove is of independent\u0000interest: it establishes an eta product identity for a certain shifted theta\u0000function of the root lattice of a simply laced root system. We extend our\u0000results to various refinements of the Euler characteristic, namely the Elliptic\u0000genus, the Chi-$y$ genus, and the motivic class.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484632","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 : 2019-05-11DOI: 10.46298/epiga.2020.volume4.6151
Andrea Fanelli, Stefan Schroer
We introduce and study the maximal unipotent finite quotient for algebraic group schemes in positive characteristics. Applied to Picard schemes, this quotient encodes unusual torsion. We construct integral Fano threefolds where such unusual torsion actually appears. The existence of such threefolds is surprising, because the torsion vanishes for del Pezzo surfaces. Our construction relies on the theory of exceptional Enriques surfaces, as developed by Ekedahl and Shepherd-Barron. Comment: 29 pages; minor changes
{"title":"The maximal unipotent finite quotient, unusual torsion in Fano\u0000 threefolds, and exceptional Enriques surfaces","authors":"Andrea Fanelli, Stefan Schroer","doi":"10.46298/epiga.2020.volume4.6151","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.6151","url":null,"abstract":"We introduce and study the maximal unipotent finite quotient for algebraic\u0000group schemes in positive characteristics. Applied to Picard schemes, this\u0000quotient encodes unusual torsion. We construct integral Fano threefolds where\u0000such unusual torsion actually appears. The existence of such threefolds is\u0000surprising, because the torsion vanishes for del Pezzo surfaces. Our\u0000construction relies on the theory of exceptional Enriques surfaces, as\u0000developed by Ekedahl and Shepherd-Barron.\u0000\u0000 Comment: 29 pages; minor changes","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484529","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 : 2019-05-02DOI: 10.46298/epiga.2020.volume4.5601
R. Pandharipande, Johannes Schmitt
While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed. Comment: Published version
{"title":"Zero cycles on the moduli space of curves","authors":"R. Pandharipande, Johannes Schmitt","doi":"10.46298/epiga.2020.volume4.5601","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5601","url":null,"abstract":"While the Chow groups of 0-dimensional cycles on the moduli spaces of\u0000Deligne-Mumford stable pointed curves can be very complicated, the span of the\u00000-dimensional tautological cycles is always of rank 1. The question of whether\u0000a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is\u0000subtle. Our main results address the question for curves on rational and K3\u0000surfaces. If C is a nonsingular curve on a nonsingular rational surface of\u0000positive degree with respect to the anticanonical class, we prove\u0000[C,p_1,...,p_n] is tautological if the number of markings does not exceed the\u0000virtual dimension in Gromov-Witten theory of the moduli space of stable maps.\u0000If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is\u0000tautological if the number of markings does not exceed the genus of C and every\u0000marking is a Beauville-Voisin point. The latter result provides a connection\u0000between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1\u0000tautological 0-cycles on K3 surfaces. Several further results related to\u0000tautological 0-cycles on the moduli spaces of curves are proven. Many open\u0000questions concerning the moduli points of curves on other surfaces (Abelian,\u0000Enriques, general type) are discussed.\u0000\u0000 Comment: Published version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47207174","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 : 2019-04-18DOI: 10.46298/epiga.2019.volume3.5395
Kazuya Kato, Takeshi Saito
There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide. Comment: 16 pages. Formatted using epigamath.sty
{"title":"Coincidence of two Swan conductors of abelian characters","authors":"Kazuya Kato, Takeshi Saito","doi":"10.46298/epiga.2019.volume3.5395","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.5395","url":null,"abstract":"There are two ways to define the Swan conductor of an abelian character of\u0000the absolute Galois group of a complete discrete valuation field. We prove that\u0000these two Swan conductors coincide.\u0000\u0000 Comment: 16 pages. Formatted using epigamath.sty","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483651","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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