{"title":"Teissier's problem on the proportionality of big and nef classes over a compact K�hler manifold","authors":"Jian Xiao","doi":"10.14231/AG-2019-009","DOIUrl":"https://doi.org/10.14231/AG-2019-009","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45789066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves". A basic example is given by $X_{ban}$, the blowup along the diagonal of the fibered product of a generic rational elliptic surface $Sto mathbb{P}^{1}$ with itself. In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold $X_{ban }$ restricted to the 3-dimensional lattice $Gamma$ of curve classes supported in the fibers of $X_{ban}to mathbb{P}^{1}$. It is given by [ Z_{Gamma}(X_{ban}) = prod_{d_{1},d_{2},d_{3}geq 0} prod_{k} left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}right)^{-12c(||mathbf{d} ||,k)} ] where $||mathbf{d} || = 2d_{1}d_{2}+ 2d_{2}d_{3}+ 2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$, and the coefficients $c(a,k)$ have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely realated to the equivariant elliptic genera of $operatorname{Hilb} (mathbb{C}^{2})$. In an appendix with S. Pietromonaco, it is shown that the corresponding genus $g$ Gromov-Witten potential $F_{g}$ is a genus 2 Siegel modular form of weight $2g-2$ for $ggeq 2$; namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series: $frac{6|B_{2g}|}{g(2g-2)!} E_{2g}(tau )$.
{"title":"The Donaldson–Thomas partition function of the banana manifold n (with an appendix coauthored with Stephen Pietromonaco)","authors":"J. Bryan","doi":"10.14231/ag-2021-002","DOIUrl":"https://doi.org/10.14231/ag-2021-002","url":null,"abstract":"A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a \"banana configuration of curves\". A basic example is given by $X_{ban}$, the blowup along the diagonal of the fibered product of a generic rational elliptic surface $Sto mathbb{P}^{1}$ with itself. \u0000In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold $X_{ban }$ restricted to the 3-dimensional lattice $Gamma$ of curve classes supported in the fibers of $X_{ban}to mathbb{P}^{1}$. It is given by [ Z_{Gamma}(X_{ban}) = prod_{d_{1},d_{2},d_{3}geq 0} prod_{k} left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}right)^{-12c(||mathbf{d} ||,k)} ] where $||mathbf{d} || = 2d_{1}d_{2}+ 2d_{2}d_{3}+ 2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$, and the coefficients $c(a,k)$ have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely realated to the equivariant elliptic genera of $operatorname{Hilb} (mathbb{C}^{2})$. In an appendix with S. Pietromonaco, it is shown that the corresponding genus $g$ Gromov-Witten potential $F_{g}$ is a genus 2 Siegel modular form of weight $2g-2$ for $ggeq 2$; namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series: $frac{6|B_{2g}|}{g(2g-2)!} E_{2g}(tau )$.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41545663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a sequel to the paper cite{MO-mw} which identified maximally writhed algebraic links in $rp^3$ and classified them topologically. In this paper we prove that all maximally writhed links of the same topological type are rigidly isotopic, i.e. one can be deformed into another with a family of smooth real algebraic links of the same degree.
{"title":"Rigid isotopy of maximally writhed links","authors":"G. Mikhalkin, S. Orevkov","doi":"10.14231/AG-2021-006","DOIUrl":"https://doi.org/10.14231/AG-2021-006","url":null,"abstract":"This is a sequel to the paper cite{MO-mw} which identified maximally writhed algebraic links in $rp^3$ and classified them topologically. In this paper we prove that all maximally writhed links of the same topological type are rigidly isotopic, i.e. one can be deformed into another with a family of smooth real algebraic links of the same degree.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44491823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A filling-in problem and moderate degenerations of minimal algebraic varieties","authors":"S. Takayama","doi":"10.14231/AG-2019-002","DOIUrl":"https://doi.org/10.14231/AG-2019-002","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46837315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dynamical Mordell�Lang and automorphisms of blow-ups","authors":"Y. Tschinkel","doi":"10.14231/ag-2019-001","DOIUrl":"https://doi.org/10.14231/ag-2019-001","url":null,"abstract":"","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49199781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
{"title":"Global Prym–Torelli theorem for double coverings of elliptic curves","authors":"A. Ikeda","doi":"10.14231/ag-2020-019","DOIUrl":"https://doi.org/10.14231/ag-2020-019","url":null,"abstract":"The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43445294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X_0$ be a smooth geometrically connected variety defined over a finite field $mathbb F_q$ and let $mathcal E_0^{dagger}$ be an irreducible overconvergent $F$-isocrystal on $X_0$. We show that if a subobject of minimal slope of the underlying convergent F-isocrystal $mathcal E_0$ admits a non-zero morphism to $mathcal O_{X_0}$ as convergent isocrystal, then $mathcal E_0^{dagger}$ is isomorphic to $mathcal O^{dagger}_{X_0}$ as overconvergent isocrystal. This proves a special case of a conjecture of Kedlaya. The key ingredient in the proof is the study of the monodromy group of $mathcal E_0^{dagger}$ and the subgroup defined by $mathcal E_0$. The new input in this setting is that the subgroup contains a maximal torus of the entire monodromy group. This is a consequence of the existence of a Frobenius torus of maximal dimension. As an application, we prove a finiteness result for the torsion points of abelian varieties, which extends the previous theorem of Lang-N'eron and answers positively a question of Esnault.
{"title":"Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties","authors":"Emiliano Ambrosi, Marco d’Addezio","doi":"10.14231/AG-2022-019","DOIUrl":"https://doi.org/10.14231/AG-2022-019","url":null,"abstract":"Let $X_0$ be a smooth geometrically connected variety defined over a finite field $mathbb F_q$ and let $mathcal E_0^{dagger}$ be an irreducible overconvergent $F$-isocrystal on $X_0$. We show that if a subobject of minimal slope of the underlying convergent F-isocrystal $mathcal E_0$ admits a non-zero morphism to $mathcal O_{X_0}$ as convergent isocrystal, then $mathcal E_0^{dagger}$ is isomorphic to $mathcal O^{dagger}_{X_0}$ as overconvergent isocrystal. This proves a special case of a conjecture of Kedlaya. The key ingredient in the proof is the study of the monodromy group of $mathcal E_0^{dagger}$ and the subgroup defined by $mathcal E_0$. The new input in this setting is that the subgroup contains a maximal torus of the entire monodromy group. This is a consequence of the existence of a Frobenius torus of maximal dimension. As an application, we prove a finiteness result for the torsion points of abelian varieties, which extends the previous theorem of Lang-N'eron and answers positively a question of Esnault.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46168547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.
{"title":"Projecting syzygies of curves","authors":"Michael Kemeny","doi":"10.14231/ag-2020-020","DOIUrl":"https://doi.org/10.14231/ag-2020-020","url":null,"abstract":"We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45531189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $tilde{A}_0$, $tilde{D}_4$, $tilde{E}_6$, $tilde{E}_7$, and $tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.
{"title":"Perverse filtrations, Hilbert schemes, and the $P=W$ Conjecture for parabolic Higgs bundles","authors":"Junliang Shen, Zili Zhang","doi":"10.14231/AG-2021-014","DOIUrl":"https://doi.org/10.14231/AG-2021-014","url":null,"abstract":"We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $tilde{A}_0$, $tilde{D}_4$, $tilde{E}_6$, $tilde{E}_7$, and $tilde{E}_8$. Our proof relies on the study of the tautological classes on the Hilbert scheme of points on an elliptic surface with respect to the perverse filtration.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42466443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $dge 2$ whose completion is isomorphic to $C(k)[[x_1,ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $hin C(k)[[x_1,ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,ldots,x_d^p)+(x_1,ldots,x_d)^{2p-2}$ of $k[[x_1,ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $text{Frac}(R)$ which extends to every local ring of $text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $ele p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.
{"title":"Purity for Barsotti–Tate groups in some mixed characteristic situations","authors":"O. Gabber, A. Vasiu","doi":"10.14231/AG-2021-015","DOIUrl":"https://doi.org/10.14231/AG-2021-015","url":null,"abstract":"Let $p$ be a prime. Let $R$ be a regular local ring of dimension $dge 2$ whose completion is isomorphic to $C(k)[[x_1,ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $hin C(k)[[x_1,ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,ldots,x_d^p)+(x_1,ldots,x_d)^{2p-2}$ of $k[[x_1,ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $text{Frac}(R)$ which extends to every local ring of $text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $ele p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43846122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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