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":"250 1","pages":"533 - 548"},"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":"251 1","pages":"606 - 621"},"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":"248 1","pages":"1005 - 1033"},"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":"250 1","pages":"352 - 364"},"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":"248 1","pages":"980 - 989"},"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}
Abstract We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.
{"title":"ON A BERNSTEIN–SATO POLYNOMIAL OF A MEROMORPHIC FUNCTION","authors":"K. Takeuchi","doi":"10.1017/nmj.2023.10","DOIUrl":"https://doi.org/10.1017/nmj.2023.10","url":null,"abstract":"Abstract We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"251 1","pages":"715 - 733"},"PeriodicalIF":0.8,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48700619","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 Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.
{"title":"ON AN AVERAGE GOLDBACH REPRESENTATION FORMULA OF FUJII","authors":"D. Goldston, A. I. Suriajaya","doi":"10.1017/nmj.2022.44","DOIUrl":"https://doi.org/10.1017/nmj.2022.44","url":null,"abstract":"Abstract Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"250 1","pages":"511 - 532"},"PeriodicalIF":0.8,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41532803","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 their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$ . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $ell $ -adic cohomology groups over $overline {{mathbb Q}_{ell }}$ of these Deligne–Lusztig varieties, where $ell not = p$ . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.
{"title":"COHOMOLOGY OF THE BRUHAT–TITS STRATA IN THE UNRAMIFIED UNITARY RAPOPORT–ZINK SPACE OF SIGNATURE \u0000$(1,n-1)$","authors":"Joseph Muller","doi":"10.1017/nmj.2022.39","DOIUrl":"https://doi.org/10.1017/nmj.2022.39","url":null,"abstract":"Abstract In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature \u0000$(1,n-1)$\u0000 . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the \u0000$ell $\u0000 -adic cohomology groups over \u0000$overline {{mathbb Q}_{ell }}$\u0000 of these Deligne–Lusztig varieties, where \u0000$ell not = p$\u0000 . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"250 1","pages":"470 - 497"},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43740382","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 study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.
{"title":"TRIPLE COVERS OF K3 SURFACES","authors":"Alice Garbagnati, M. Penegini","doi":"10.1017/nmj.2022.15","DOIUrl":"https://doi.org/10.1017/nmj.2022.15","url":null,"abstract":"Abstract We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"248 1","pages":"939 - 979"},"PeriodicalIF":0.8,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48659634","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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