We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e. they are never contained in non-trivial analytic families of equisingular invariant curves. In this case we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.
{"title":"The Poincaré problem for reducible curves","authors":"Pedro Fortuny Ayuso, Javier Ribón","doi":"10.4171/rmi/1451","DOIUrl":"https://doi.org/10.4171/rmi/1451","url":null,"abstract":"We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are isolated, i.e. they are never contained in non-trivial analytic families of equisingular invariant curves. In this case we show that the multiplicity of an invariant curve is at most twice the multiplicity of the foliation. Finally, we apply the local methods to foliations in the complex projective plane.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"26 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a method to calculate the action of the Mordell–Weil group of an elliptic $K3$ surface on the numerical Néron–Severi lattice of the $K3$ surface. As an application, we compute a finite generating set of the automorphism group of a $K3$ surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type.
{"title":"Mordell–Weil groups and automorphism groups of elliptic $K3$ surfaces","authors":"Ichiro Shimada","doi":"10.4171/rmi/1449","DOIUrl":"https://doi.org/10.4171/rmi/1449","url":null,"abstract":"We present a method to calculate the action of the Mordell–Weil group of an elliptic $K3$ surface on the numerical Néron–Severi lattice of the $K3$ surface. As an application, we compute a finite generating set of the automorphism group of a $K3$ surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135266428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona–Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group ${mathcal S}_5$. From this construction, we are able to describe the Chow ring explicitly.
{"title":"A four-dimensional cousin of the Segre cubic","authors":"Laurent Manivel","doi":"10.4171/rmi/1448","DOIUrl":"https://doi.org/10.4171/rmi/1448","url":null,"abstract":"This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona–Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group ${mathcal S}_5$. From this construction, we are able to describe the Chow ring explicitly.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"24 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.
{"title":"Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane","authors":"Joshua Flynn, Guozhen Lu, Qiaohua Yang","doi":"10.4171/rmi/1444","DOIUrl":"https://doi.org/10.4171/rmi/1444","url":null,"abstract":"The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An asymptotic formula for the number of $n$-dimensional representations of SU(3)","authors":"K. Bringmann, J. Franke","doi":"10.4171/rmi/1415","DOIUrl":"https://doi.org/10.4171/rmi/1415","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45825015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Positive solutions of the $p$-Laplacian with potential terms on weighted Riemannian manifolds with linear diameter growth","authors":"A. Kasue","doi":"10.4171/rmi/1432","DOIUrl":"https://doi.org/10.4171/rmi/1432","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47152345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a complete equisingular deformation classification of simple spatial quartic surfaces which are in fact $K3$-surfaces.
本文给出了简单空间四次曲面(实际上是$K3$-曲面)的完整等奇异变形分类。
{"title":"Deformation classification of quartic surfaces with simple singularities","authors":"cCisem Gunecs Aktacs","doi":"10.4171/rmi/1431","DOIUrl":"https://doi.org/10.4171/rmi/1431","url":null,"abstract":"We give a complete equisingular deformation classification of simple spatial quartic surfaces which are in fact $K3$-surfaces.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45294100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Factorization of the normalization of the Nash blowup of order $n$ of $mathcal{A}_{n}$ by the minimal resolution","authors":"E. Chávez-Martínez","doi":"10.4171/rmi/1421","DOIUrl":"https://doi.org/10.4171/rmi/1421","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41450538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms","authors":"Susmita Das, J. Sarkar","doi":"10.4171/rmi/1403","DOIUrl":"https://doi.org/10.4171/rmi/1403","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42681199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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