Pub Date : 2019-04-16DOI: 10.46298/epiga.2020.volume4.5733
M. Wibmer
Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic. Comment: 36 pages, final accepted version
{"title":"Free Proalgebraic Groups","authors":"M. Wibmer","doi":"10.46298/epiga.2020.volume4.5733","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5733","url":null,"abstract":"Replacing finite groups by linear algebraic groups, we study an\u0000algebraic-geometric counterpart of the theory of free profinite groups. In\u0000particular, we introduce free proalgebraic groups and characterize them in\u0000terms of embedding problems. The main motivation for this endeavor is a\u0000differential analog of a conjecture of Shafarevic.\u0000\u0000 Comment: 36 pages, final accepted version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483948","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 : 2019-04-01DOI: 10.46298/epiga.2021.volume5.5573
K. Heinrich, R. Skjelnes, J. Stevens
We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space. Comment: 22 pages. Final version
{"title":"The space of twisted cubics","authors":"K. Heinrich, R. Skjelnes, J. Stevens","doi":"10.46298/epiga.2021.volume5.5573","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.5573","url":null,"abstract":"We consider the Cohen-Macaulay compactification of the space of twisted\u0000cubics in projective n-space. This compactification is the fine moduli scheme\u0000representing the functor of CM-curves with Hilbert polynomial 3t+1. We show\u0000that the moduli scheme of CM-curves in projective 3-space is isomorphic to the\u0000twisted cubic component of the Hilbert scheme. We also describe the\u0000compactification for twisted cubics in n-space.\u0000\u0000 Comment: 22 pages. Final version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484546","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 : 2019-03-14DOI: 10.46298/epiga.2020.volume3.5728
Stefan Schreieder
Let k be an uncountable algebraically closed field and let Y be a smooth projective k-variety which does not admit a decomposition of the diagonal. We prove that Y is not stably birational to a very general hypersurface of any given degree and dimension. We use this to study the variation of the stable birational types of Fano hypersurfaces over fields of arbitrary characteristic. This had been initiated by Shinder, whose method works in characteristic zero. Comment: 14 pages; final version, published in EPIGA
{"title":"Variation of stable birational types in positive characteristic","authors":"Stefan Schreieder","doi":"10.46298/epiga.2020.volume3.5728","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume3.5728","url":null,"abstract":"Let k be an uncountable algebraically closed field and let Y be a smooth\u0000projective k-variety which does not admit a decomposition of the diagonal. We\u0000prove that Y is not stably birational to a very general hypersurface of any\u0000given degree and dimension. We use this to study the variation of the stable\u0000birational types of Fano hypersurfaces over fields of arbitrary characteristic.\u0000This had been initiated by Shinder, whose method works in characteristic zero.\u0000\u0000 Comment: 14 pages; final version, published in EPIGA","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47198007","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 : 2019-03-09DOI: 10.46298/epiga.2020.volume4.5282
L. Gottsche
We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.
{"title":"Refined Verlinde formulas for Hilbert schemes of points and moduli\u0000 spaces of sheaves on K3 surfaces","authors":"L. Gottsche","doi":"10.46298/epiga.2020.volume4.5282","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5282","url":null,"abstract":"We compute generating functions for elliptic genera with values in line\u0000bundles on Hilbert schemes of points on surfaces. As an application we also\u0000compute generating functions for elliptic genera with values in determinant\u0000line bundles on moduli spaces of sheaves on K3 surfaces.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483934","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 : 2019-02-17DOI: 10.46298/epiga.2019.volume3.5541
Mattias Hemmig
In this article, we study isomorphisms between complements of irreducible curves in the projective plane $mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C subset mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $mathbb{P}^2 setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.
{"title":"Isomorphisms between complements of projective plane curves","authors":"Mattias Hemmig","doi":"10.46298/epiga.2019.volume3.5541","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.5541","url":null,"abstract":"In this article, we study isomorphisms between complements of irreducible\u0000curves in the projective plane $mathbb{P}^2$, over an arbitrary algebraically\u0000closed field. Of particular interest are rational unicuspidal curves. We prove\u0000that if there exists a line that intersects a unicuspidal curve $C subset\u0000mathbb{P}^2$ only in its singular point, then any other curve whose complement\u0000is isomorphic to $mathbb{P}^2 setminus C$ must be projectively equivalent to\u0000$C$. This generalizes a result of H. Yoshihara who proved this result over the\u0000complex numbers. Moreover, we study properties of multiplicity sequences of\u0000irreducible curves that imply that any isomorphism between the complements of\u0000these curves extends to an automorphism of $mathbb{P}^2$. Using these results,\u0000we show that two irreducible curves of degree $leq 7$ have isomorphic\u0000complements if and only if they are projectively equivalent. Finally, we\u0000describe new examples of irreducible projectively non-equivalent curves of\u0000degree $8$ that have isomorphic complements.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483700","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 : 2018-12-05DOI: 10.46298/EPIGA.2020.VOLUME4.5090
B. Pasquier
International audience We classify all smooth projective horospherical varieties of Picard group $mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.
{"title":"Smooth projective horospherical varieties of Picard group $mathbb{Z}^2$","authors":"B. Pasquier","doi":"10.46298/EPIGA.2020.VOLUME4.5090","DOIUrl":"https://doi.org/10.46298/EPIGA.2020.VOLUME4.5090","url":null,"abstract":"International audience\u0000 \u0000 We classify all smooth projective horospherical varieties of Picard group $mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483882","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 : 2018-12-03DOI: 10.46298/epiga.2019.volume3.5032
C. Sabbah, Jeng-Daw Yu
We give a formula computing the irregular Hodge numbers for a confluent hypergeometric differential equation. Comment: 9 pages. V2: typos corrected
给出了合流超几何微分方程的不规则霍奇数的计算公式。评论:9页。2:拼写错误纠正
{"title":"Irregular Hodge numbers of confluent hypergeometric differential\u0000 equations","authors":"C. Sabbah, Jeng-Daw Yu","doi":"10.46298/epiga.2019.volume3.5032","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.5032","url":null,"abstract":"We give a formula computing the irregular Hodge numbers for a confluent\u0000hypergeometric differential equation.\u0000\u0000 Comment: 9 pages. V2: typos corrected","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484075","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 : 2018-11-23DOI: 10.46298/epiga.2020.volume4.5550
Cong Xue
In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism. Comment: published version
{"title":"Finiteness of cohomology groups of stacks of shtukas as modules over\u0000 Hecke algebras, and applications","authors":"Cong Xue","doi":"10.46298/epiga.2020.volume4.5550","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5550","url":null,"abstract":"In this paper we prove that the cohomology groups with compact support of\u0000stacks of shtukas are modules of finite type over a Hecke algebra. As an\u0000application, we extend the construction of excursion operators, defined by V.\u0000Lafforgue on the space of cuspidal automorphic forms, to the space of\u0000automorphic forms with compact support. This gives the Langlands\u0000parametrization for some quotient spaces of the latter, which is compatible\u0000with the constant term morphism.\u0000\u0000 Comment: published version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483752","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 : 2018-11-05DOI: 10.46298/epiga.2021.volume5.5894
Nicolas Tholozan, J'er'emy Toulisse
We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $textrm{PU}(1,1) = mathrm{PSL}(2,mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.
{"title":"Compact connected components in relative character varieties of\u0000 punctured spheres","authors":"Nicolas Tholozan, J'er'emy Toulisse","doi":"10.46298/epiga.2021.volume5.5894","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.5894","url":null,"abstract":"We prove that some relative character varieties of the fundamental group of a\u0000punctured sphere into the Hermitian Lie groups $mathrm{SU}(p,q)$ admit compact\u0000connected components. The representations in these components have several\u0000counter-intuitive properties. For instance, the image of any simple closed\u0000curve is an elliptic element. These results extend a recent work of Deroin and\u0000the first author, which treated the case of $textrm{PU}(1,1) =\u0000mathrm{PSL}(2,mathbb{R})$. Our proof relies on the non-Abelian Hodge\u0000correspondance between relative character varieties and parabolic Higgs\u0000bundles. The examples we construct admit a rather explicit description as\u0000projective varieties obtained via Geometric Invariant Theory.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48400859","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 : 2018-11-01DOI: 10.46298/epiga.2022.6820
K. Kedlaya
Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $ell$-adic Weil cohomology: these are lisse Weil sheaves in 'etale cohomology when $ell neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $ell=p$. Using the Langlands correspondence for global function fields in both the 'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any 'etale coefficient object has 'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has 'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of 'etale coefficient objects; this subject will be pursued in a subsequent paper.
设$X$是特征$p$的有限域上的光滑格式。考虑$ well $-adicWeil上同调中$X$上的局部常秩系数对象:当$ well neq p$时，它们是$ well neq p$上同调中的lisse Weil束，当$ well =p$时，它们是刚性上同调中的过收敛$F$-同晶。利用在椭圆和晶体环境下的全局函数场的朗兰兹对应(分别是Lafforgue和Abe的工作)，人们看到在曲线上，一个类别中的任何系数对象在具有匹配的Frobeniusat闭点特征多项式的其他类别中都有“同伴”。一般$X$也有类似的语句;在Deligne的基础上，Drinfeld证明了任何一个具有固定系数的物体都有固定的伴体。我们采用了德林菲尔德的方法来证明任何晶体效率的物体都有其固定的伴星;这已经由abe -Esnault独立证明。我们还证明了一些与构造虚系数物体的晶体伴体有关的辅助结果;这个问题将在以后的论文中讨论。
{"title":"Etale and crystalline companions, I","authors":"K. Kedlaya","doi":"10.46298/epiga.2022.6820","DOIUrl":"https://doi.org/10.46298/epiga.2022.6820","url":null,"abstract":"Let $X$ be a smooth scheme over a finite field of characteristic $p$.\u0000Consider the coefficient objects of locally constant rank on $X$ in $ell$-adic\u0000Weil cohomology: these are lisse Weil sheaves in 'etale cohomology when $ell\u0000neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $ell=p$.\u0000Using the Langlands correspondence for global function fields in both the\u0000'etale and crystalline settings (work of Lafforgue and Abe, respectively), one\u0000sees that on a curve, any coefficient object in one category has \"companions\"\u0000in the other categories with matching characteristic polynomials of Frobenius\u0000at closed points. A similar statement is expected for general $X$; building on\u0000work of Deligne, Drinfeld showed that any 'etale coefficient object has\u0000'etale companions. We adapt Drinfeld's method to show that any crystalline\u0000coefficient object has 'etale companions; this has been shown independently by\u0000Abe--Esnault. We also prove some auxiliary results relevant for the\u0000construction of crystalline companions of 'etale coefficient objects; this\u0000subject will be pursued in a subsequent paper.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48313838","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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