Sextic double solids, double covers of $mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $mathbb Q$ -factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n leq 8$ , describe models for each n explicitly, and prove that sextic double solids with $n> 3$ are birationally nonrigid.
{"title":"BIRATIONAL GEOMETRY OF SEXTIC DOUBLE SOLIDS WITH A COMPOUND SINGULARITY","authors":"ERIK PAEMURRU","doi":"10.1017/nmj.2024.17","DOIUrl":"https://doi.org/10.1017/nmj.2024.17","url":null,"abstract":"Sextic double solids, double covers of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline2.png\"/> <jats:tex-math> $mathbb P^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline3.png\"/> <jats:tex-math> $mathbb Q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline4.png\"/> <jats:tex-math> $A_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> singularity. We prove a sharp bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline5.png\"/> <jats:tex-math> $n leq 8$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, describe models for each <jats:italic>n</jats:italic> explicitly, and prove that sextic double solids with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline6.png\"/> <jats:tex-math> $n> 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are birationally nonrigid.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268627","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}
Using the technique of Gabor analysis, we characterize the boundedness of $e^{iDelta }: W^{p_1,q_1}_mrightarrow W^{p_2,q_2}$ with modulation and translation operators, where and m is a v-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.
{"title":"SCHRÖDINGER PROPAGATOR ON WIENER AMALGAM SPACES IN THE FULL RANGE","authors":"GUOPING ZHAO, WEICHAO GUO","doi":"10.1017/nmj.2024.14","DOIUrl":"https://doi.org/10.1017/nmj.2024.14","url":null,"abstract":"Using the technique of Gabor analysis, we characterize the boundedness of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S002776302400014X_inline1.png\"/> <jats:tex-math> $e^{iDelta }: W^{p_1,q_1}_mrightarrow W^{p_2,q_2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with modulation and translation operators, where <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S002776302400014X_inline2.png\"/> and <jats:italic>m</jats:italic> is a <jats:italic>v</jats:italic>-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226655","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 a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.
{"title":"CONSTANCY OF THE HILBERT–SAMUEL FUNCTION","authors":"VINCENT COSSART, OLIVIER PILTANT, BERND SCHOBER","doi":"10.1017/nmj.2024.13","DOIUrl":"https://doi.org/10.1017/nmj.2024.13","url":null,"abstract":"We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152397","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}
Let $mathbf {D}$ be a bounded homogeneous domain in ${mathbb {C}}^n$ . In this note, we give a characterization of the Stein domains in $mathbf {D}$ which are invariant under a maximal unipotent subgroup N of $Aut(mathbf {D})$ . We also exhibit an N-invariant potential of the Bergman metric of $mathbf {D}$ , expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case.
{"title":"A REMARK ON THE N-INVARIANT GEOMETRY OF BOUNDED HOMOGENEOUS DOMAINS","authors":"L. Geatti, A. Iannuzzi","doi":"10.1017/nmj.2024.12","DOIUrl":"https://doi.org/10.1017/nmj.2024.12","url":null,"abstract":"\u0000\t <jats:p>Let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000126_inline1.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbf {D}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a bounded homogeneous domain in <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000126_inline2.png\"/>\u0000\t\t<jats:tex-math>\u0000${mathbb {C}}^n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. In this note, we give a characterization of the Stein domains in <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000126_inline3.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbf {D}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> which are invariant under a maximal unipotent subgroup <jats:italic>N</jats:italic> of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000126_inline4.png\"/>\u0000\t\t<jats:tex-math>\u0000$Aut(mathbf {D})$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. We also exhibit an <jats:italic>N</jats:italic>-invariant potential of the Bergman metric of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000126_inline5.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbf {D}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case.</jats:p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141106304","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 254 Cover and Front matter","authors":"","doi":"10.1017/nmj.2024.9","DOIUrl":"https://doi.org/10.1017/nmj.2024.9","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141010205","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 254 Cover and Back matter","authors":"","doi":"10.1017/nmj.2024.10","DOIUrl":"https://doi.org/10.1017/nmj.2024.10","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141010418","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 explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.
{"title":"WHEN IS THE SILTING-DISCRETENESS INHERITED?","authors":"TAKUMA AIHARA, TAKAHIRO HONMA","doi":"10.1017/nmj.2024.8","DOIUrl":"https://doi.org/10.1017/nmj.2024.8","url":null,"abstract":"<p>We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.</p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566575","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 invoke the Bernstein–Gel $'$ fand–Gel $'$ fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case.
{"title":"SUBCOMPLEXES OF CERTAIN FREE RESOLUTIONS","authors":"MAYA BANKS, ALEKSANDRA SOBIESKA","doi":"10.1017/nmj.2024.7","DOIUrl":"https://doi.org/10.1017/nmj.2024.7","url":null,"abstract":"We invoke the Bernstein–Gel<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000072_inline1.png\" /> <jats:tex-math> $'$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>fand–Gel<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000072_inline2.png\" /> <jats:tex-math> $'$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302662","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}
In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.
{"title":"TILTING COMPLEXES AND CODIMENSION FUNCTIONS OVER COMMUTATIVE NOETHERIAN RINGS","authors":"MICHAL HRBEK, TSUTOMU NAKAMURA, JAN ŠŤOVÍČEK","doi":"10.1017/nmj.2024.1","DOIUrl":"https://doi.org/10.1017/nmj.2024.1","url":null,"abstract":"In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150357","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 253 Cover and Back matter","authors":"","doi":"10.1017/nmj.2024.5","DOIUrl":"https://doi.org/10.1017/nmj.2024.5","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140409389","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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