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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"NMJ volume 245 Cover and Back matter","authors":"","doi":"10.1017/nmj.2022.4","DOIUrl":"https://doi.org/10.1017/nmj.2022.4","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49464655","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 Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.
{"title":"ANNIHILATORS AND DIMENSIONS OF THE SINGULARITY CATEGORY","authors":"Jian Liu","doi":"10.1017/nmj.2022.45","DOIUrl":"https://doi.org/10.1017/nmj.2022.45","url":null,"abstract":"Abstract Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46692011","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 prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${mathcal Enr}(X)to mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.
{"title":"ENRIQUES INVOLUTIONS AND BRAUER CLASSES","authors":"A. Skorobogatov, D. Valloni","doi":"10.1017/nmj.2022.43","DOIUrl":"https://doi.org/10.1017/nmj.2022.43","url":null,"abstract":"Abstract We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set \u0000${mathcal Enr}(X)$\u0000 of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank \u0000$20,$\u0000 we prove that the fibers of \u0000${mathcal Enr}(X)to mathrm {{Br}}(X)[2]$\u0000 above the nonzero points have the same cardinality.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41340314","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}
Oleksandra Gasanova, J. Herzog, T. Hibi, S. Moradi
Abstract Let R be a Cohen–Macaulay local K-algebra or a standard graded K-algebra over a field K with a canonical module $omega _R$ . The trace of $omega _R$ is the ideal $operatorname {tr}(omega _R)$ of R which is the sum of those ideals $varphi (omega _R)$ with ${varphi in operatorname {Hom}_R(omega _R,R)}$ . The smallest number s for which there exist $varphi _1, ldots , varphi _s in operatorname {Hom}_R(omega _R,R)$ with $operatorname {tr}(omega _R)=varphi _1(omega _R) + cdots + varphi _s(omega _R)$ is called the Teter number of R. We say that R is of Teter type if $s = 1$ . It is shown that R is not of Teter type if R is generically Gorenstein. In the present paper, we focus especially on zero-dimensional graded and monomial K-algebras and present various classes of such algebras which are of Teter type.
{"title":"RINGS OF TETER TYPE","authors":"Oleksandra Gasanova, J. Herzog, T. Hibi, S. Moradi","doi":"10.1017/nmj.2022.18","DOIUrl":"https://doi.org/10.1017/nmj.2022.18","url":null,"abstract":"Abstract Let R be a Cohen–Macaulay local K-algebra or a standard graded K-algebra over a field K with a canonical module \u0000$omega _R$\u0000 . The trace of \u0000$omega _R$\u0000 is the ideal \u0000$operatorname {tr}(omega _R)$\u0000 of R which is the sum of those ideals \u0000$varphi (omega _R)$\u0000 with \u0000${varphi in operatorname {Hom}_R(omega _R,R)}$\u0000 . The smallest number s for which there exist \u0000$varphi _1, ldots , varphi _s in operatorname {Hom}_R(omega _R,R)$\u0000 with \u0000$operatorname {tr}(omega _R)=varphi _1(omega _R) + cdots + varphi _s(omega _R)$\u0000 is called the Teter number of R. We say that R is of Teter type if \u0000$s = 1$\u0000 . It is shown that R is not of Teter type if R is generically Gorenstein. In the present paper, we focus especially on zero-dimensional graded and monomial K-algebras and present various classes of such algebras which are of Teter type.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46950211","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 short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.
{"title":"AN OBSERVATION ON THE DIRICHLET PROBLEM AT INFINITY IN RIEMANNIAN CONES","authors":"J. Cortissoz","doi":"10.1017/nmj.2022.31","DOIUrl":"https://doi.org/10.1017/nmj.2022.31","url":null,"abstract":"Abstract In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41612210","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 coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are Hörmander’s $L^2$ -estimate and a singular version of a Demailly–Skoda-type result.
{"title":"SINGULAR HERMITIAN METRICS WITH ISOLATED SINGULARITIES","authors":"Takahiro Inayama","doi":"10.1017/nmj.2022.16","DOIUrl":"https://doi.org/10.1017/nmj.2022.16","url":null,"abstract":"Abstract In this paper, we study the coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are Hörmander’s \u0000$L^2$\u0000 -estimate and a singular version of a Demailly–Skoda-type result.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46157190","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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