Abstract The aim of this paper is to establish the boundedness of fractional type Marcinkiewicz integral $mathcal {M}_{iota ,rho ,m}$ and its commutator $mathcal {M}_{iota ,rho ,m,b}$ on generalized Morrey spaces and on Morrey spaces over nonhomogeneous metric measure spaces which satisfy the upper doubling and geometrically doubling conditions. Under the assumption that the dominating function $lambda $ satisfies $epsilon $ -weak reverse doubling condition, the author proves that $mathcal {M}_{iota ,rho ,m}$ is bounded on generalized Morrey space $L^{p,phi }(mu )$ and on Morrey space $M^{p}_{q}(mu )$ . Furthermore, the boundedness of the commutator $mathcal {M}_{iota ,rho ,m,b}$ generated by $mathcal {M}_{iota ,rho ,m}$ and regularized $mathrm {BMO}$ space with discrete coefficient (= $widetilde {mathrm {RBMO}}(mu )$ ) on space $L^{p,phi }(mu )$ and on space $M^{p}_{q}(mu )$ is also obtained.
{"title":"FRACTIONAL TYPE MARCINKIEWICZ INTEGRAL AND ITS COMMUTATOR ON NONHOMOGENEOUS SPACES","authors":"G. Lu","doi":"10.1017/nmj.2022.6","DOIUrl":"https://doi.org/10.1017/nmj.2022.6","url":null,"abstract":"Abstract The aim of this paper is to establish the boundedness of fractional type Marcinkiewicz integral \u0000$mathcal {M}_{iota ,rho ,m}$\u0000 and its commutator \u0000$mathcal {M}_{iota ,rho ,m,b}$\u0000 on generalized Morrey spaces and on Morrey spaces over nonhomogeneous metric measure spaces which satisfy the upper doubling and geometrically doubling conditions. Under the assumption that the dominating function \u0000$lambda $\u0000 satisfies \u0000$epsilon $\u0000 -weak reverse doubling condition, the author proves that \u0000$mathcal {M}_{iota ,rho ,m}$\u0000 is bounded on generalized Morrey space \u0000$L^{p,phi }(mu )$\u0000 and on Morrey space \u0000$M^{p}_{q}(mu )$\u0000 . Furthermore, the boundedness of the commutator \u0000$mathcal {M}_{iota ,rho ,m,b}$\u0000 generated by \u0000$mathcal {M}_{iota ,rho ,m}$\u0000 and regularized \u0000$mathrm {BMO}$\u0000 space with discrete coefficient (= \u0000$widetilde {mathrm {RBMO}}(mu )$\u0000 ) on space \u0000$L^{p,phi }(mu )$\u0000 and on space \u0000$M^{p}_{q}(mu )$\u0000 is also obtained.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"248 1","pages":"801 - 822"},"PeriodicalIF":0.8,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47020509","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 prove that the maximal number of conics in a smooth sextic K3-surface X ⊂ P is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible. §
{"title":"NMJ volume 246 Cover and Front matter","authors":"S. Rao, Quanting Zhao","doi":"10.1017/nmj.2022.9","DOIUrl":"https://doi.org/10.1017/nmj.2022.9","url":null,"abstract":"We prove that the maximal number of conics in a smooth sextic K3-surface X ⊂ P is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible. §","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"246 1","pages":"f1 - f3"},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41450607","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}
Abstract Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.
{"title":"SELMER GROUPS OF ELLIPTIC CURVES OVER THE \u0000$PGL(2)$\u0000 EXTENSION","authors":"Jishnu Ray, R. Sujatha","doi":"10.1017/nmj.2022.14","DOIUrl":"https://doi.org/10.1017/nmj.2022.14","url":null,"abstract":"Abstract Iwasawa theory of elliptic curves over noncommutative \u0000$GL(2)$\u0000 extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the \u0000$PGL(2)$\u0000 extension and connect it with Iwasawa theory over the \u0000$GL(2)$\u0000 extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the \u0000$GL(2)$\u0000 extension being torsion is related with that of the \u0000$PGL(2)$\u0000 extension.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"248 1","pages":"922 - 938"},"PeriodicalIF":0.8,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42441881","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}
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of $a^3b$ -quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.
{"title":"TILINGS OF THE SPHERE BY CONGRUENT QUADRILATERALS II: EDGE COMBINATION WITH RATIONAL ANGLES","authors":"Yixi Liao, Erxiao Wang","doi":"10.1017/nmj.2023.20","DOIUrl":"https://doi.org/10.1017/nmj.2023.20","url":null,"abstract":"\u0000 Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of \u0000 \u0000 \u0000 \u0000$a^3b$\u0000\u0000 \u0000 -quadrilaterals with all angles being rational degrees. There are \u0000 \u0000 \u0000 \u0000$12$\u0000\u0000 \u0000 sporadic and \u0000 \u0000 \u0000 \u0000$3$\u0000\u0000 \u0000 infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and \u0000 \u0000 \u0000 \u0000$3$\u0000\u0000 \u0000 sporadic quadrilaterals admitting \u0000 \u0000 \u0000 \u0000$4$\u0000\u0000 \u0000 exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43638524","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}
Abstract We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
摘要我们发展了等变对偶几何中普遍扭子的形式，并将其应用于有限群的非对偶但稳定对偶作用的新例子。
{"title":"TORSORS AND STABLE EQUIVARIANT BIRATIONAL GEOMETRY","authors":"B. Hassett, Y. Tschinkel","doi":"10.1017/nmj.2022.29","DOIUrl":"https://doi.org/10.1017/nmj.2022.29","url":null,"abstract":"Abstract We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"250 1","pages":"275 - 297"},"PeriodicalIF":0.8,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45178195","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}
Abstract In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $mathbb {P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $mathbb {P}^n.$ We show that if $mathcal {M}$ is a collection of m lines in general linear position in $mathbb {P}^n$ with $2m leq n+1$ and R is the coordinate ring of $mathcal {M},$ then R is Koszul. Furthermore, if $mathcal {M}$ is a generic collection of m lines in $mathbb {P}^n$ and R is the coordinate ring of $mathcal {M}$ with m even and $m +1leq n$ or m is odd and $m +2leq n,$ then R is Koszul. Lastly, we show that if $mathcal {M}$ is a generic collection of m lines such that $$ begin{align*} m> frac{1}{72}left(3(n^2+10n+13)+sqrt{3(n-1)^3(3n+5)}right),end{align*} $$ then R is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for $n leq 6$ or $m leq 6$ . We also determine the Castelnuovo–Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.
{"title":"GENERIC LINES IN PROJECTIVE SPACE AND THE KOSZUL PROPERTY","authors":"J. Rice","doi":"10.1017/nmj.2022.42","DOIUrl":"https://doi.org/10.1017/nmj.2022.42","url":null,"abstract":"Abstract In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in \u0000$mathbb {P}^n$\u0000 and the homogeneous coordinate ring of a collection of lines in general linear position in \u0000$mathbb {P}^n.$\u0000 We show that if \u0000$mathcal {M}$\u0000 is a collection of m lines in general linear position in \u0000$mathbb {P}^n$\u0000 with \u0000$2m leq n+1$\u0000 and R is the coordinate ring of \u0000$mathcal {M},$\u0000 then R is Koszul. Furthermore, if \u0000$mathcal {M}$\u0000 is a generic collection of m lines in \u0000$mathbb {P}^n$\u0000 and R is the coordinate ring of \u0000$mathcal {M}$\u0000 with m even and \u0000$m +1leq n$\u0000 or m is odd and \u0000$m +2leq n,$\u0000 then R is Koszul. Lastly, we show that if \u0000$mathcal {M}$\u0000 is a generic collection of m lines such that \u0000$$ begin{align*} m> frac{1}{72}left(3(n^2+10n+13)+sqrt{3(n-1)^3(3n+5)}right),end{align*} $$\u0000 then R is not Koszul. We give a complete characterization of the Koszul property of the coordinate ring of a generic collection of lines for \u0000$n leq 6$\u0000 or \u0000$m leq 6$\u0000 . We also determine the Castelnuovo–Mumford regularity of the coordinate ring for a generic collection of lines and the projective dimension of the coordinate ring of collection of lines in general linear position.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"251 1","pages":"576 - 605"},"PeriodicalIF":0.8,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46758139","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}
Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.
{"title":"ANALYTIC SPREAD OF FILTRATIONS ON TWO-DIMENSIONAL NORMAL LOCAL RINGS","authors":"S. Cutkosky","doi":"10.1017/nmj.2022.35","DOIUrl":"https://doi.org/10.1017/nmj.2022.35","url":null,"abstract":"Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"249 1","pages":"239 - 268"},"PeriodicalIF":0.8,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43452240","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 establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in [1] specialized at q = 1.
{"title":"NMJ volume 245 Cover and Front matter","authors":"","doi":"10.1017/nmj.2022.3","DOIUrl":"https://doi.org/10.1017/nmj.2022.3","url":null,"abstract":"We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in [1] specialized at q = 1.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"245 1","pages":"f1 - f4"},"PeriodicalIF":0.8,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47447840","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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