SEBASTIAN CASALAINA-MARTIN, SAMUEL GRUSHEVSKY, KLAUS HULEK, RADU LAZA
Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${mathcal {M}}^{operatorname {GIT}}$ , as a Baily–Borel compactification of a ball quotient ${(mathcal {B}_4/Gamma )^*}$ , and as a compactified K -moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${mathcal {M}}^{operatorname {K}}rightarrow {mathcal {M}}^{operatorname {GIT}}$ , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${overline {mathcal {B}_4/Gamma }}rightarrow {(mathcal {B}_4/Gamma )^*}$ . The spaces ${mathcal {M}}^{operatorname {K}}$ and ${overline {mathcal {B}_4/Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${mathcal {M}}^{operatorname {K}}$ and ${overline {mathcal {B}_4/Gamma }}$ are equivalent in the Grothendieck ring, but not K -equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.
{"title":"NON-ISOMORPHIC SMOOTH COMPACTIFICATIONS OF THE MODULI SPACE OF CUBIC SURFACES","authors":"SEBASTIAN CASALAINA-MARTIN, SAMUEL GRUSHEVSKY, KLAUS HULEK, RADU LAZA","doi":"10.1017/nmj.2023.27","DOIUrl":"https://doi.org/10.1017/nmj.2023.27","url":null,"abstract":"Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${mathcal {M}}^{operatorname {GIT}}$ , as a Baily–Borel compactification of a ball quotient ${(mathcal {B}_4/Gamma )^*}$ , and as a compactified K -moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${mathcal {M}}^{operatorname {K}}rightarrow {mathcal {M}}^{operatorname {GIT}}$ , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${overline {mathcal {B}_4/Gamma }}rightarrow {(mathcal {B}_4/Gamma )^*}$ . The spaces ${mathcal {M}}^{operatorname {K}}$ and ${overline {mathcal {B}_4/Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${mathcal {M}}^{operatorname {K}}$ and ${overline {mathcal {B}_4/Gamma }}$ are equivalent in the Grothendieck ring, but not K -equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135738766","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 $({cal{A}},{cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${operatorname{Ext}}_{cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules.
{"title":"EXACT SUBCATEGORIES, SUBFUNCTORS OF , AND SOME APPLICATIONS","authors":"HAILONG DAO, SOUVIK DEY, MONALISA DUTTA","doi":"10.1017/nmj.2023.29","DOIUrl":"https://doi.org/10.1017/nmj.2023.29","url":null,"abstract":"Abstract Let $({cal{A}},{cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${operatorname{Ext}}_{cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135538812","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 consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .
{"title":"A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES","authors":"EDUARD LOOIJENGA","doi":"10.1017/nmj.2023.28","DOIUrl":"https://doi.org/10.1017/nmj.2023.28","url":null,"abstract":"Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136155225","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 a one-dimensional family of Bridgeland stability conditions on $mathbb {P}^n$ , named “Euler” stability condition. We conjecture that the “Euler” stability condition converges to Gieseker stability for coherent sheaves. Here, we focus on ${mathbb P}^3$ , first identifying Euler stability conditions with double-tilt stability conditions, and then we consider moduli of one-dimensional sheaves, proving some asymptotic results, boundedness for walls, and then explicitly computing walls and wall-crossings for sheaves supported on rational curves of degrees $3$ and $4$ .
{"title":"NEW MODULI SPACES OF ONE-DIMENSIONAL SHEAVES ON","authors":"DAPENG MU","doi":"10.1017/nmj.2023.26","DOIUrl":"https://doi.org/10.1017/nmj.2023.26","url":null,"abstract":"Abstract We define a one-dimensional family of Bridgeland stability conditions on $mathbb {P}^n$ , named “Euler” stability condition. We conjecture that the “Euler” stability condition converges to Gieseker stability for coherent sheaves. Here, we focus on ${mathbb P}^3$ , first identifying Euler stability conditions with double-tilt stability conditions, and then we consider moduli of one-dimensional sheaves, proving some asymptotic results, boundedness for walls, and then explicitly computing walls and wall-crossings for sheaves supported on rational curves of degrees $3$ and $4$ .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135153570","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}
The Nevanlinna-type spaces $N_varphi $ of analytic functions on the disk in the complex plane generated by strongly convex functions $varphi $ in the sense of Rudin are studied. We show for some special class of strongly convex functions asymptotic bounds on the growth of the Taylor coefficients of a function in $N_varphi $ and use these to characterize the coefficient multipliers from $N_varphi $ into the Hardy spaces $H^p$ with $0. As a by-product, we prove a representation of continuous linear functionals on $N_varphi $ .
{"title":"MULTIPLIERS AND CHARACTERIZATION OF THE DUAL OF NEVANLINNA-TYPE SPACES","authors":"Mieczysław Mastyło, Bartosz Staniów","doi":"10.1017/nmj.2023.24","DOIUrl":"https://doi.org/10.1017/nmj.2023.24","url":null,"abstract":"\u0000\t <jats:p>The Nevanlinna-type spaces <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=\"S0027763023000247_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$N_varphi $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> of analytic functions on the disk in the complex plane generated by strongly convex functions <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000247_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$varphi $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> in the sense of Rudin are studied. We show for some special class of strongly convex functions asymptotic bounds on the growth of the Taylor coefficients of a function in <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000247_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$N_varphi $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and use these to characterize the coefficient multipliers from <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000247_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$N_varphi $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> into the Hardy spaces <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000247_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$H^p$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with <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=\"S0027763023000247_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$0<pleqslant infty $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. As a by-product, we prove a representation of continuous linear functionals on <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000247_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$N_varphi $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>.</jats:p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43168025","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}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"NMJ volume 251 Cover and Front matter","authors":"","doi":"10.1017/nmj.2023.21","DOIUrl":"https://doi.org/10.1017/nmj.2023.21","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135149902","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 $Lambda $ be a finite-dimensional algebra. A wide subcategory of $mathsf {mod}Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $mathsf {mod}Lambda $ arising from $tau $ -tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $tau $ -tilting reduction. This leads to a natural way to extend the definition of the “ $tau $ -cluster morphism category” of $Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $tau $ -tilting finite case and by Igusa–Todorov in the hereditary case.
{"title":"-PERPENDICULAR WIDE SUBCATEGORIES","authors":"A. B. Buan, Eric J. Hanson","doi":"10.1017/nmj.2023.16","DOIUrl":"https://doi.org/10.1017/nmj.2023.16","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=\"S0027763023000168_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$Lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a finite-dimensional algebra. A wide subcategory of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf {mod}Lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is called <jats:italic>left finite</jats:italic> if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf {mod}Lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> arising from <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$tau $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$tau $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-tilting reduction. This leads to a natural way to extend the definition of the “<jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$tau $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-cluster morphism category” of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$Lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000168_inline9.png\" />\u0000\t\t<jats:tex-math>\u0000$tau $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-tilting finite case and by Igusa–Todorov in the hereditary case.</jats:p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45146262","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 this erratum, we correct an erroneous result in [PV2] and prove that the affine algebraic hypersurfaces $xy^2=1$ and $z=xy^2$ are not interpolating with respect to the Gaussian weight.
{"title":"ERRATUM TO “NON-UNIFORMLY FLAT AFFINE ALGEBRAIC HYPERSURFACES”","authors":"A. Mandal, Vamsi Pingali, Dror Varolin","doi":"10.1017/nmj.2023.13","DOIUrl":"https://doi.org/10.1017/nmj.2023.13","url":null,"abstract":"\u0000\t <jats:p>In this erratum, we correct an erroneous result in [PV2] and prove that the affine algebraic hypersurfaces <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=\"S0027763023000132_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$xy^2=1$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and <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=\"S0027763023000132_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$z=xy^2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> are not interpolating with respect to the Gaussian weight.</jats:p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48838512","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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