Pub Date : 2020-09-15DOI: 10.46298/epiga.2021.6792
B. Totaro
We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group G_m acts on a quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to a closed subset Y in U, then the inclusion from Y to U should be an A^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine n-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.
{"title":"Torus actions, Morse homology, and the Hilbert scheme of points on\u0000 affine space","authors":"B. Totaro","doi":"10.46298/epiga.2021.6792","DOIUrl":"https://doi.org/10.46298/epiga.2021.6792","url":null,"abstract":"We formulate a conjecture on actions of the multiplicative group in motivic\u0000homotopy theory. In short, if the multiplicative group G_m acts on a\u0000quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to\u0000a closed subset Y in U, then the inclusion from Y to U should be an\u0000A^1-homotopy equivalence.\u0000 We prove several partial results. In particular, over the complex numbers,\u0000the inclusion is a homotopy equivalence on complex points. The proofs use an\u0000analog of Morse theory for singular varieties. Application: the Hilbert scheme\u0000of points on affine n-space is homotopy equivalent to the subspace consisting\u0000of schemes supported at the origin.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484196","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 : 2020-09-14DOI: 10.46298/epiga.2022.9959
Thibaut Delcroix
We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying the corresponding convex geometric problem. Thanks to recent work of Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable sufficient condition of existence of constant scalar curvature Kahler metrics. As a side effect, we show that, on several families of spherical varieties, G-uniform K-stability is equivalent to K-polystability with respect to G-equivariant test configurations for polarizations close to the anticanonical bundle.
{"title":"Uniform K-stability of polarized spherical varieties","authors":"Thibaut Delcroix","doi":"10.46298/epiga.2022.9959","DOIUrl":"https://doi.org/10.46298/epiga.2022.9959","url":null,"abstract":"We express notions of K-stability of polarized spherical varieties in terms\u0000of combinatorial data, vastly generalizing the case of toric varieties. We then\u0000provide a combinatorial sufficient condition of G-uniform K-stability by\u0000studying the corresponding convex geometric problem. Thanks to recent work of\u0000Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable\u0000sufficient condition of existence of constant scalar curvature Kahler metrics.\u0000As a side effect, we show that, on several families of spherical varieties,\u0000G-uniform K-stability is equivalent to K-polystability with respect to\u0000G-equivariant test configurations for polarizations close to the anticanonical\u0000bundle.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"33 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484714","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 : 2020-08-09DOI: 10.46298/epiga.2021.6715
Julia Schneider, Susanna Zimmermann
We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.
{"title":"Algebraic subgroups of the plane Cremona group over a perfect field","authors":"Julia Schneider, Susanna Zimmermann","doi":"10.46298/epiga.2021.6715","DOIUrl":"https://doi.org/10.46298/epiga.2021.6715","url":null,"abstract":"We show that any infinite algebraic subgroup of the plane Cremona group over\u0000a perfect field is contained in a maximal algebraic subgroup of the plane\u0000Cremona group. We classify the maximal groups, and their subgroups of rational\u0000points, up to conjugacy by a birational map.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48436961","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 : 2020-07-06DOI: 10.46298/epiga.2023.volume7.9818
S. Feyzbakhsh, Richard P. Thomas
We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda, such as $mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in $X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.
{"title":"Curve counting and S-duality","authors":"S. Feyzbakhsh, Richard P. Thomas","doi":"10.46298/epiga.2023.volume7.9818","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.9818","url":null,"abstract":"We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker\u0000conjecture of Bayer-Macr`i-Toda, such as $mathbb P^3$ or the quintic\u0000threefold.\u0000 We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are\u0000smooth bundles over Hilbert schemes of ideal sheaves of curves and points in\u0000$X$.\u0000 When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing\u0000curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of\u0000D4-D2-D0 branes. These latter invariants are predicted to have modular\u0000properties which we discuss from the point of view of S-duality and\u0000Noether-Lefschetz theory.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70486029","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 : 2020-05-26DOI: 10.46298/epiga.2022.6568
H. Esnault, M. Kerz
We propose a conjecture on the density of arithmetic points in the deformation space of representations of the 'etale fundamental group in positive characteristic. This? conjecture has applications to 'etale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $mathbb{P}^1setminus {0,1,infty}$. v2: very small typos corrected.v3: final. Publication in Epiga.
{"title":"Density of Arithmetic Representations of Function Fields","authors":"H. Esnault, M. Kerz","doi":"10.46298/epiga.2022.6568","DOIUrl":"https://doi.org/10.46298/epiga.2022.6568","url":null,"abstract":"We propose a conjecture on the density of arithmetic points in the\u0000deformation space of representations of the 'etale fundamental group in\u0000positive characteristic. This? conjecture has applications to 'etale\u0000cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove\u0000the density conjecture in tame degree two for the curve $mathbb{P}^1setminus\u0000{0,1,infty}$. v2: very small typos corrected.v3: final. Publication in\u0000Epiga.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49454744","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 : 2020-05-09DOI: 10.46298/epiga.2021.volume5.6573
C'edric Bonnaf'e, A. Sarti
We construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$--singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author. Comment: 26 pages
{"title":"Complex reflection groups and K3 surfaces I","authors":"C'edric Bonnaf'e, A. Sarti","doi":"10.46298/epiga.2021.volume5.6573","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.6573","url":null,"abstract":"We construct here many families of K3 surfaces that one can obtain as\u0000quotients of algebraic surfaces by some subgroups of the rank four complex\u0000reflection groups. We find in total 15 families with at worst\u0000$ADE$--singularities. In particular we classify all the K3 surfaces that can be\u0000obtained as quotients by the derived subgroup of the previous complex\u0000reflection groups. We prove our results by using the geometry of the weighted\u0000projective spaces where these surfaces are embedded and the theory of Springer\u0000and Lehrer-Springer on properties of complex reflection groups. This\u0000construction generalizes a previous construction by W. Barth and the second\u0000author.\u0000\u0000 Comment: 26 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48952919","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 : 2020-05-02DOI: 10.46298/epiga.2021.volume4.6456
Ana-Maria Castravet, J. Tevelev
We construct an $S_2times S_n$ invariant full exceptional collection on Hassett spaces of weighted stable rational curves with $n+2$ markings and weights $(frac{1}{2}+eta, frac{1}{2}+eta,epsilon,ldots,epsilon)$, for $0
对于$0
{"title":"Exceptional collections on certain Hassett spaces","authors":"Ana-Maria Castravet, J. Tevelev","doi":"10.46298/epiga.2021.volume4.6456","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume4.6456","url":null,"abstract":"We construct an $S_2times S_n$ invariant full exceptional collection on\u0000Hassett spaces of weighted stable rational curves with $n+2$ markings and\u0000weights $(frac{1}{2}+eta, frac{1}{2}+eta,epsilon,ldots,epsilon)$, for\u0000$0<epsilon, etall1$ and can be identified with symmetric GIT quotients of\u0000$(mathbb{P}^1)^n$ by the diagonal action of $mathbb{G}_m$ when $n$ is odd,\u0000and their Kirwan desingularization when $n$ is even. The existence of such an\u0000exceptional collection is one of the needed ingredients in order to prove the\u0000existence of a full $S_n$-invariant exceptional collection on\u0000$overline{mathcal{M}}_{0,n}$. To prove exceptionality we use the method of\u0000windows in derived categories. To prove fullness we use previous work on the\u0000existence of invariant full exceptional collections on Losev-Manin spaces.\u0000\u0000 Comment: At the request of the referee, the paper arXiv:1708.06340 has been\u0000 split into two parts. This is the second of those papers (submitted). 36\u0000 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484521","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 : 2020-04-13DOI: 10.46298/epiga.2023.volume7.6849
Dean Bisogno, Wanlin Li, Daniel Litt, P. Srinivasan
Let l be a prime and G a pro-l group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when l=2. We apply this to the pro-l etale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic Abel-Jacobi map.
{"title":"Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class","authors":"Dean Bisogno, Wanlin Li, Daniel Litt, P. Srinivasan","doi":"10.46298/epiga.2023.volume7.6849","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.6849","url":null,"abstract":"Let l be a prime and G a pro-l group with torsion-free abelianization. We\u0000produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the\u0000case of surface groups, these cocycles appear to refine existing constructions\u0000when l=2. We apply this to the pro-l etale fundamental groups of smooth curves\u0000to obtain Galois-cohomological analogues, and discuss their relationship to\u0000work of Hain and Matsumoto in the case the curve is proper. We analyze many of\u0000the fundamental properties of these classes and use them to give an example of\u0000a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic\u0000Abel-Jacobi map.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485276","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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