Algebra & Number Theory最新文献

{"title":"GKM-theory for torus actions on cyclic quiver Grassmannians","authors":"Martina Lanini, Alexander Pütz","doi":"10.2140/ant.2023.17.2055","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2055","url":null,"abstract":"<p>We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>. We show that these quiver Grassmannians equipped with our specific torus action are GKM-varieties and that their moment graph admits a combinatorial description in terms of the coefficient quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able to prove that moment graph techniques can be applied to construct module bases for the equivariant cohomology of the quiver Grassmannians listed above. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 19","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The behavior of essential dimension under specialization, II","authors":"Zinovy Reichstein, Federico Scavia","doi":"10.2140/ant.2023.17.1925","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1925","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-varieties over a base variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math>, the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> and stays constant away from this countable union. We give several applications of this result. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fitting ideals of class groups for CM abelian extensions","authors":"Mahiro Atsuta, Takenori Kataoka","doi":"10.2140/ant.2023.17.1901","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1901","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be a finite abelian CM-extension of a totally real field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>T</mi></math> a suitable finite set of finite primes of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We determine the Fitting ideal of the minus component of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>T</mi></math>-ray class group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, except for the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"11 23","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bézoutians and the 𝔸1-degree","authors":"Thomas Brazelton, Stephen McKean, Sabrina Pauli","doi":"10.2140/ant.2023.17.1985","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1985","url":null,"abstract":"<p>We prove that both the local and global <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave’s theorem in the univariate case, and our local theorem generalizes Kass–Wickelgren’s theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 47","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On self-correspondences on curves","authors":"Joël Bellaïche","doi":"10.2140/ant.2023.17.1867","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1867","url":null,"abstract":"<p>We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> over an algebraically closed field is the data of another curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math> and two nonconstant separable morphisms <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></math> from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math>. A subset <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> is <span>complete</span> if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>S</mi><mo stretchy=\"false\">)</mo></math>. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> is a union of finite complete sets. The latter ones are called <span>finitary</span>, and happen only when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> deg</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub>\u0000<mo>=</mo><mi> deg</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 45","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Differential operators, retracts, and toric face rings","authors":"Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, Janet Vassilev","doi":"10.2140/ant.2023.17.1959","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1959","url":null,"abstract":"<p>We give explicit descriptions of rings of differential operators of toric face rings in characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math>. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators. </p><p> Our main technique is to realize the k-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any k-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 8","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Axiomatizing the existential theory of 𝔽q((t))","authors":"Sylvy Anscombe, Philip Dittmann, Arno Fehm","doi":"10.2140/ant.2023.17.2013","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2013","url":null,"abstract":"<p>We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of — and therefore an algorithm to decide — the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 9","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The diagonal coinvariant ring of a complex reflection group","authors":"Stephen Griffeth","doi":"10.2140/ant.2023.17.2033","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2033","url":null,"abstract":"<p>For an irreducible complex reflection group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math> of rank <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> containing <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> reflections, we put <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>=</mo> <mn>2</mn><mi>N</mi><mo>∕</mo><mi>n</mi></math> and construct a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math>-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math>. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal coinvariant ring, and that further corrections to this estimate of the dimension of the diagonal coinvariant ring by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> should be orders of magnitude smaller. A crucial ingredient in the construction is the existence of a dot action of a certain product of symmetric groups (the Namikawa–Weyl group) acting on the parameter space of the rational Cherednik algebra and leaving invariant both the finite Hecke algebra and the spherical subalgebra; this fact is a consequence of ideas of Berest and Chalykh on the relationship between the Cherednik algebra and quasiinvariants. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 48","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Separation of periods of quartic surfaces","authors":"Pierre Lairez, Emre Can Sertöz","doi":"10.2140/ant.2023.17.1753","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1753","url":null,"abstract":"<p>We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 14","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hybrid subconvexity bounds for twists of GL(3) × GL(2) L-functions","authors":"Bingrong Huang, Zhao Xu","doi":"10.2140/ant.2023.17.1715","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1715","url":null,"abstract":"<p>We prove hybrid subconvexity bounds for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo>\u0000<mo>×</mo><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions twisted by a primitive Dirichlet character modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> (prime) in the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>- and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>t</mi></math>-aspects. We also improve hybrid subconvexity bounds for twists of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions in the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>- and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>t</mi></math>-aspects. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 16","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}