Pub Date : 2021-10-18DOI: 10.2422/2036-2145.202112_010
Quentin Posva
We develop a gluing theory in the sense of Kollár for slc surfaces and threefolds in positive characteristic. For surfaces we are able to deal with every positive characteristic p, while for threefolds we assume that p > 5. Along the way we study nodes in characteristic 2 and establish a theory of sources and springs à la Kollár for threefolds. We also give applications to the topology of lc centers on slc threefolds, and to the projectivity of the moduli space of stable surfaces in characteristic p > 5.
对于slc表面和正特性的三倍，我们发展了Kollár意义上的粘合理论。对于曲面，我们可以处理每一个正特征p，而对于三倍曲面，我们假设p > 5。在此过程中，我们研究了特征2中的节点，并建立了三倍的源和弹簧理论 la Kollár。我们也给出了在slc三折上的lc中心拓扑的应用，以及特征p > 5的稳定曲面模空间的投影。
{"title":"Gluing theory for slc surfaces and threefolds in positive characteristic","authors":"Quentin Posva","doi":"10.2422/2036-2145.202112_010","DOIUrl":"https://doi.org/10.2422/2036-2145.202112_010","url":null,"abstract":"We develop a gluing theory in the sense of Kollár for slc surfaces and threefolds in positive characteristic. For surfaces we are able to deal with every positive characteristic p, while for threefolds we assume that p > 5. Along the way we study nodes in characteristic 2 and establish a theory of sources and springs à la Kollár for threefolds. We also give applications to the topology of lc centers on slc threefolds, and to the projectivity of the moduli space of stable surfaces in characteristic p > 5.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88097525","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 : 2021-10-14DOI: 10.2422/2036-2145.202201_007
Iulia Gheorghita, Nicola Tarasca
Inside the projectivized $k$-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point.
在投影的$k$- h Hodge束内，我们构造了一个通过在Brill-Noether特殊点上施加消失而得到的除数集合。我们用两种方法计算这些因子闭包的类，使用入射几何和对各种族的限制，包括K3曲面上的曲线铅笔和Du Val曲线铅笔。我们还证明了由光滑的尖曲线和经过标记点的正则或2正则因子组成的入射因子闭合的极值性和刚性。
{"title":"k-canonical divisors through Brill-Noether special points","authors":"Iulia Gheorghita, Nicola Tarasca","doi":"10.2422/2036-2145.202201_007","DOIUrl":"https://doi.org/10.2422/2036-2145.202201_007","url":null,"abstract":"Inside the projectivized $k$-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83714598","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 : 2021-10-13DOI: 10.2422/2036-2145.202005_016
M. Winkler
{"title":"$L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation","authors":"M. Winkler","doi":"10.2422/2036-2145.202005_016","DOIUrl":"https://doi.org/10.2422/2036-2145.202005_016","url":null,"abstract":"","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"28 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89309896","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 : 2021-10-13DOI: 10.2422/2036-2145.202001_018
G. Goldstein, J. Goldstein, A. Kogoj, A. Rhandi, C. Tacelli
. In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group.
。本文将[3]和[15]的瞬时爆破结果推广到海森堡群上奇异势摄动的热方程。
{"title":"Instantaneous blowup and singular potentials on Heisenberg groups","authors":"G. Goldstein, J. Goldstein, A. Kogoj, A. Rhandi, C. Tacelli","doi":"10.2422/2036-2145.202001_018","DOIUrl":"https://doi.org/10.2422/2036-2145.202001_018","url":null,"abstract":". In this paper we generalize the instantaneous blowup result from [3] and [15] to the heat equation perturbed by singular potentials on the Heisenberg group.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81383869","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 : 2021-10-12DOI: 10.2422/2036-2145.202110_009
Idriss Mazari, Y. Privat
Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form −∆uβ = f(x, uβ) endowed with the Robin boundary conditions ∂νuβ+β(x)uβ = 0. The optimisation variable is the function β, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form J (β) = ́ Ω or ∂Ω j(uβ). We prove that, depending on the monotonicity of the function j, the optimisers may be of bang-bang type (in other words, the optimisers write 1Γ for some measurable subset Γ of ∂Ω) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann (∂νu = 0 ) and constant Robin conditions (∂νu+u = 0) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal β∗.
{"title":"Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients","authors":"Idriss Mazari, Y. Privat","doi":"10.2422/2036-2145.202110_009","DOIUrl":"https://doi.org/10.2422/2036-2145.202110_009","url":null,"abstract":"Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form −∆uβ = f(x, uβ) endowed with the Robin boundary conditions ∂νuβ+β(x)uβ = 0. The optimisation variable is the function β, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form J (β) = ́ Ω or ∂Ω j(uβ). We prove that, depending on the monotonicity of the function j, the optimisers may be of bang-bang type (in other words, the optimisers write 1Γ for some measurable subset Γ of ∂Ω) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann (∂νu = 0 ) and constant Robin conditions (∂νu+u = 0) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal β∗.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89466069","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 : 2021-10-11DOI: 10.2422/2036-2145.202101_001
Carl Lian, R. Pandharipande
Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $beta$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $beta$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $beta$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic.
{"title":"Enumerativity of virtual Tevelev degrees","authors":"Carl Lian, R. Pandharipande","doi":"10.2422/2036-2145.202101_001","DOIUrl":"https://doi.org/10.2422/2036-2145.202101_001","url":null,"abstract":"Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $beta$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $beta$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $beta$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86099556","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 : 2021-10-11DOI: 10.2422/2036-2145.202111_001
J. Andrade, Rayssa Caju, 'O JoaoMarcosdo, J. Ratzkin, Almir Silva Santos
We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.
{"title":"Constant Q-curvature metrics with Delaunay ends: the nondegenerate case","authors":"J. Andrade, Rayssa Caju, 'O JoaoMarcosdo, J. Ratzkin, Almir Silva Santos","doi":"10.2422/2036-2145.202111_001","DOIUrl":"https://doi.org/10.2422/2036-2145.202111_001","url":null,"abstract":"We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89501957","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 : 2021-10-10DOI: 10.2422/2036-2145.202110_011
Philippe Bouafia, T. Pauw
We give an integral representation formula for members of the dual ofSBV (R) in terms of functions that are defined on R̂, an appropriate fiber space that we introduce, consisting of pairs (x, [E ]x ) where [E ]x is an approximate germ of an (n−1)-rectifiable set E at x.
{"title":"A representation formula for members of SBV dual","authors":"Philippe Bouafia, T. Pauw","doi":"10.2422/2036-2145.202110_011","DOIUrl":"https://doi.org/10.2422/2036-2145.202110_011","url":null,"abstract":"We give an integral representation formula for members of the dual ofSBV (R) in terms of functions that are defined on R̂, an appropriate fiber space that we introduce, consisting of pairs (x, [E ]x ) where [E ]x is an approximate germ of an (n−1)-rectifiable set E at x.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73031419","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 : 2021-10-09DOI: 10.2422/2036-2145.202111_011
Mattheus Aguiar, P. Zalesski
The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We generalize the pro-$p$ version of it proved by Weigel and the second author to splittings over infinite pro-$p$ groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability.
{"title":"Generalized Stallings' decomposition theorems for pro-p groups","authors":"Mattheus Aguiar, P. Zalesski","doi":"10.2422/2036-2145.202111_011","DOIUrl":"https://doi.org/10.2422/2036-2145.202111_011","url":null,"abstract":"The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We generalize the pro-$p$ version of it proved by Weigel and the second author to splittings over infinite pro-$p$ groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91543212","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 : 2021-10-08DOI: 10.2422/2036-2145.202112_002
Michał Kijaczko
. We investigate the form of the closure of the smooth, compactly supported functions C ∞ c (Ω) in the weighted fractional Sobolev space W s,p ; w,v (Ω) for bounded Ω . We focus on the weights w, v being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of Ω . For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.
． 研究了加权分数Sobolev空间中光滑紧支撑函数C∞C (Ω)的闭包形式;W v (Ω)对于有界的Ω。我们关注权重w, v是到定义域边界距离的幂。我们的结果依赖于Ω边界的上下协维。对于这些权值，我们还证明了满权分数型Gagliardo半模与截断半模的可比性。
{"title":"Fractional Sobolev spaces with power weights","authors":"Michał Kijaczko","doi":"10.2422/2036-2145.202112_002","DOIUrl":"https://doi.org/10.2422/2036-2145.202112_002","url":null,"abstract":". We investigate the form of the closure of the smooth, compactly supported functions C ∞ c (Ω) in the weighted fractional Sobolev space W s,p ; w,v (Ω) for bounded Ω . We focus on the weights w, v being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of Ω . For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84450161","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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