Pub Date : 2023-08-30DOI: 10.1515/crelle-2023-0048
I. Peeva
Abstract We prove a conjecture by Ene, Herzog, and Hibi (2011) that the Betti numbers of the binomial edge ideal J G {J_{G}} of a closed graph G coincide with the Betti numbers of its lex initial ideal M G {M_{G}} . We describe the Betti numbers of the ideal M G {M_{G}} .
摘要证明了Ene、Herzog和Hibi(2011)关于闭图G的二项式边理想J G {J_{G}}的Betti数与其lex初始理想M G {M_{G}}的Betti数重合的猜想。我们描述了理想M G {M_{G}}的贝蒂数。
{"title":"Closed binomial edge ideals","authors":"I. Peeva","doi":"10.1515/crelle-2023-0048","DOIUrl":"https://doi.org/10.1515/crelle-2023-0048","url":null,"abstract":"Abstract We prove a conjecture by Ene, Herzog, and Hibi (2011) that the Betti numbers of the binomial edge ideal J G {J_{G}} of a closed graph G coincide with the Betti numbers of its lex initial ideal M G {M_{G}} . We describe the Betti numbers of the ideal M G {M_{G}} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75107565","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}
Pub Date : 2023-07-22DOI: 10.1515/crelle-2023-0024
S. Spenko, Michel van den Bergh
Abstract Let a reductive group G act on a smooth variety X such that a good quotient X / / G {X/!!/G} exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X / / G {X/!!/G} , obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X / / G {X/!!/G} . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks.
摘要:令约化群G作用于光滑变量X，使得良商X/ / G {X/!!/ G}的存在。我们证明了X/ / G {X/!!/G}，由X上的G等变向量束得到，可以嵌入到X/ /G {X/!!/ G}。事实上，嵌入可以完成为半正交分解，其中其他部分都是光滑Deligne-Mumford叠上的Azumaya代数的派生范畴。
{"title":"Comparing the Kirwan and noncommutative resolutions of quotient varieties","authors":"S. Spenko, Michel van den Bergh","doi":"10.1515/crelle-2023-0024","DOIUrl":"https://doi.org/10.1515/crelle-2023-0024","url":null,"abstract":"Abstract Let a reductive group G act on a smooth variety X such that a good quotient X / / G {X/!!/G} exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X / / G {X/!!/G} , obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X / / G {X/!!/G} . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86559561","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}
Pub Date : 2023-06-27DOI: 10.1515/crelle-2023-0039
Wenkui Du, Robert Haslhofer
Abstract Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at - ∞ {-infty} is a cylinder ℝ k × S n - k {mathbb{R}^{k}times S^{n-k}} and that are rotating within the ℝ k {mathbb{R}^{k}} -factor. We note that while the ℝ k {mathbb{R}^{k}} -factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the ℝ k {mathbb{R}^{k}} -factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in ℝ 4 {mathbb{R}^{4}} .
{"title":"A nonexistence result for rotating mean curvature flows in ℝ4","authors":"Wenkui Du, Robert Haslhofer","doi":"10.1515/crelle-2023-0039","DOIUrl":"https://doi.org/10.1515/crelle-2023-0039","url":null,"abstract":"Abstract Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at - ∞ {-infty} is a cylinder ℝ k × S n - k {mathbb{R}^{k}times S^{n-k}} and that are rotating within the ℝ k {mathbb{R}^{k}} -factor. We note that while the ℝ k {mathbb{R}^{k}} -factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the ℝ k {mathbb{R}^{k}} -factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in ℝ 4 {mathbb{R}^{4}} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78299100","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}
Pub Date : 2023-06-27DOI: 10.1515/crelle-2023-0036
Noah Olander
Abstract We prove that if X is a regular quasi-projective variety of dimension d, the set of line bundles { 𝒪 X ( n ) } n ∈ ℤ {{mathcal{O}_{X}(n)}_{nin{mathbb{Z}}}} generates the bounded derived category of X in d steps. This proves new cases of a conjecture of Orlov as well as a conjecture of Elagin and Lunts.
{"title":"Ample line bundles and generation time","authors":"Noah Olander","doi":"10.1515/crelle-2023-0036","DOIUrl":"https://doi.org/10.1515/crelle-2023-0036","url":null,"abstract":"Abstract We prove that if X is a regular quasi-projective variety of dimension d, the set of line bundles { 𝒪 X ( n ) } n ∈ ℤ {{mathcal{O}_{X}(n)}_{nin{mathbb{Z}}}} generates the bounded derived category of X in d steps. This proves new cases of a conjecture of Orlov as well as a conjecture of Elagin and Lunts.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76955671","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}
Pub Date : 2023-06-15DOI: 10.1515/crelle-2023-0029
M. Çiperiani
Abstract Let E / Q mathrm{E}/mathbb{Q} be an elliptic curve and 𝑝 a prime of supersingular reduction for E mathrm{E} . Consider a quadratic extension L / Q p L/mathbb{Q}_{p} and the corresponding anticyclotomic Z p mathbb{Z}_{p} -extension L ∞ / L L_{infty}/L . We analyze the structure of the points E ( L ∞ ) mathrm{E}(L_{infty}) and describe two global implications of our results.
{"title":"Supersingular elliptic curves over ℤ𝑝-extensions","authors":"M. Çiperiani","doi":"10.1515/crelle-2023-0029","DOIUrl":"https://doi.org/10.1515/crelle-2023-0029","url":null,"abstract":"Abstract Let E / Q mathrm{E}/mathbb{Q} be an elliptic curve and 𝑝 a prime of supersingular reduction for E mathrm{E} . Consider a quadratic extension L / Q p L/mathbb{Q}_{p} and the corresponding anticyclotomic Z p mathbb{Z}_{p} -extension L ∞ / L L_{infty}/L . We analyze the structure of the points E ( L ∞ ) mathrm{E}(L_{infty}) and describe two global implications of our results.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83151223","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}
Pub Date : 2023-04-29DOI: 10.1515/crelle-2023-0015
H. Upmeier
Abstract We show that the “eigenbundle” (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank 𝑟 is a “singular” vector bundle (linearly fibred complex analytic space) which decomposes as a stratified sum of homogeneous vector bundles along a canonical stratification of length r + 1 r+1 . The fibres are realized in terms of representation theory on the normal space of the strata.
{"title":"𝐾-invariant Hilbert modules and singular vector bundles on bounded symmetric domains","authors":"H. Upmeier","doi":"10.1515/crelle-2023-0015","DOIUrl":"https://doi.org/10.1515/crelle-2023-0015","url":null,"abstract":"Abstract We show that the “eigenbundle” (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank 𝑟 is a “singular” vector bundle (linearly fibred complex analytic space) which decomposes as a stratified sum of homogeneous vector bundles along a canonical stratification of length r + 1 r+1 . The fibres are realized in terms of representation theory on the normal space of the strata.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83131149","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}
Pub Date : 2023-04-27DOI: 10.1515/crelle-2023-0019
Mingchen Xia
Abstract Let X be a compact Kähler manifold. Fix a big ( 1 , 1 ) {(1,1)} -cohomology class α with smooth representative θ. We study the spaces ℰ p ( X , θ ) {mathcal{E}^{p}(X,theta)} of finite energy Kähler potentials for each p ≥ 1 {pgeq 1} . We define a metric d p {d_{p}} without using the Finsler geometry nor solving Monge–Ampère-type equations. This construction generalizes the usual d p {d_{p}} -metric defined for an ample class.
设X是一个紧的Kähler流形。固定一个具有光滑表示θ的大{(1,1)(1,1)}-上同调类α。我们研究了每一个p≥1 p {geq}{ 1的有限能量Kähler位能}{{的空间(}}{p}{²(X， θ) }{mathcal{E}}{ ^p(X, }{}{theta}{)}。我们{在不使用Finsler几何和不{求解}}monge - ampement - re型方程的情况下定义了度规d p d＿p。此构造泛化了{为示例类定义的通常的{pd＿p}} -度量。
{"title":"Mabuchi geometry of big cohomology classes","authors":"Mingchen Xia","doi":"10.1515/crelle-2023-0019","DOIUrl":"https://doi.org/10.1515/crelle-2023-0019","url":null,"abstract":"Abstract Let X be a compact Kähler manifold. Fix a big ( 1 , 1 ) {(1,1)} -cohomology class α with smooth representative θ. We study the spaces ℰ p ( X , θ ) {mathcal{E}^{p}(X,theta)} of finite energy Kähler potentials for each p ≥ 1 {pgeq 1} . We define a metric d p {d_{p}} without using the Finsler geometry nor solving Monge–Ampère-type equations. This construction generalizes the usual d p {d_{p}} -metric defined for an ample class.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80417184","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}
Pub Date : 2023-03-28DOI: 10.1515/crelle-2023-0006
A. Swaminathan
Abstract We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a “higher composition law” discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the 2-class group in families of number fields defined by binary n-ic forms, where n ≥ 3 {ngeq 3} is an arbitrary integer, odd or even; in the paper [A. Swaminathan, Most integral odd-degree binary forms fail to properly represent a square, preprint 2020], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [M. Bhargava, A. Shankar and A. Swaminathan, The second moment of the size of the 2-Selmer group of elliptic curves, preprint 2021], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the 2-Selmer group of elliptic curves.
{"title":"A new parametrization for ideal classes in rings defined by binary forms, and applications","authors":"A. Swaminathan","doi":"10.1515/crelle-2023-0006","DOIUrl":"https://doi.org/10.1515/crelle-2023-0006","url":null,"abstract":"Abstract We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a “higher composition law” discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the 2-class group in families of number fields defined by binary n-ic forms, where n ≥ 3 {ngeq 3} is an arbitrary integer, odd or even; in the paper [A. Swaminathan, Most integral odd-degree binary forms fail to properly represent a square, preprint 2020], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [M. Bhargava, A. Shankar and A. Swaminathan, The second moment of the size of the 2-Selmer group of elliptic curves, preprint 2021], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the 2-Selmer group of elliptic curves.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81604211","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}
Pub Date : 2023-03-25DOI: 10.1515/crelle-2023-0002
J. Bland, T. Duchamp
Abstract We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ( 1 ) mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as Q D / F Q_{mathbb{D}}/mathcal{F} for some holomorphic foliation ℱ, where Q D Q_{mathbb{D}} is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 Qsubsetmathbb{P}^{2}timesmathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X Y = Z n XY=Z^{n} .
摘要利用二维射影结构的Hitchin扭扭变换，得到了O(1) 数学{O}(1)有理曲线的拟凹邻域内的正坐标;在构造中，我们给出了对于某些全纯叶形(v)的每一个这样的邻域Q D / F Q_{mathbb{D}}/mathcal{F}，其中Q D Q_{mathbb{D}}是标准二次曲面Q∧p2 × p2 Q子集mathbb{P}^{2}乘以mathbb{P}^{2}中的一个开放邻域。作为标准坐标的结果，我们得到了三球上与标准结构同位素的柯西黎曼结构的一种新的标准形式。本文最后给出了由正常孤立奇点X¹Y=Z n Y=Z^{n}的变形引起的情况的显式计算。
{"title":"A twistor transform and normal forms for Cauchy Riemann structures","authors":"J. Bland, T. Duchamp","doi":"10.1515/crelle-2023-0002","DOIUrl":"https://doi.org/10.1515/crelle-2023-0002","url":null,"abstract":"Abstract We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ( 1 ) mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as Q D / F Q_{mathbb{D}}/mathcal{F} for some holomorphic foliation ℱ, where Q D Q_{mathbb{D}} is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 Qsubsetmathbb{P}^{2}timesmathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X Y = Z n XY=Z^{n} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74522351","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}
Pub Date : 2023-01-27DOI: 10.1515/crelle-2022-0082
L. Kühne
Abstract We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 {>1} in families of abelian varieties with trivial trace.
{"title":"The relative Bogomolov conjecture for fibered products of elliptic curves","authors":"L. Kühne","doi":"10.1515/crelle-2022-0082","DOIUrl":"https://doi.org/10.1515/crelle-2022-0082","url":null,"abstract":"Abstract We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 {>1} in families of abelian varieties with trivial trace.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87437477","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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