Forum of Mathematics Sigma最新文献

{"title":"Relative rank and regularization","authors":"Amichai Lampert, Tamar Ziegler","doi":"10.1017/fms.2024.15","DOIUrl":"https://doi.org/10.1017/fms.2024.15","url":null,"abstract":"<p>We introduce a new concept of rank – <span>relative rank</span> associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with <span>Schmidt rank</span> (also called <span>strength</span>). We also introduce the notion of <span>relative bias</span>. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is <span>polynomial</span> in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal P=(P_i)_{i=1}^c$</span></span></img></span></span> of degrees <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$le d$</span></span></img></span></span> in a polynomial ring over an algebraically closed field of characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$&gt;d$</span></span></img></span></span> is contained in an ideal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal I({mathcal Q})$</span></span></img></span></span>, generated by a collection <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal Q}$</span></span></img></span></span> of polynomials of degrees <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305102827475-0522:S205050942400015X:S205050942400015X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$le d$</spa","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045999","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":"Nef cones of fiber products and an application to the cone conjecture","authors":"Cécile Gachet, Hsueh-Yung Lin, Long Wang","doi":"10.1017/fms.2024.22","DOIUrl":"https://doi.org/10.1017/fms.2024.22","url":null,"abstract":"<p>We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron–Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher-dimensional analogues of the Calabi–Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi–Yau pairs, and in each dimension at least three, there exist Schoen varieties with nonpolyhedral nef cone. We prove the Kawamata–Morrison–Totaro cone conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034360","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":"Dirac geometry II: coherent cohomology","authors":"Lars Hesselholt, Piotr Pstrągowski","doi":"10.1017/fms.2024.2","DOIUrl":"https://doi.org/10.1017/fms.2024.2","url":null,"abstract":"Dirac rings are commutative algebras in the symmetric monoidal category of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000021_inline1.png\" /> <jats:tex-math> $mathbb {Z}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000021_inline2.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000021_inline3.png\" /> <jats:tex-math> $operatorname {MU}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000021_inline4.png\" /> <jats:tex-math> $mathbb {F}_p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140005183","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":"On the lack of compactness in the axisymmetric neo-Hookean model","authors":"Marco Barchiesi, Duvan Henao, Carlos Mora-Corral, Rémy Rodiac","doi":"10.1017/fms.2024.9","DOIUrl":"https://doi.org/10.1017/fms.2024.9","url":null,"abstract":"<p>We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223115422166-0971:S2050509424000094:S2050509424000094_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {S}^2$</span></span></img></span></span>-valued harmonic maps.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139968035","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":"On determining and breaking the gauge class in inverse problems for reaction-diffusion equations","authors":"Yavar Kian, Tony Liimatainen, Yi-Hsuan Lin","doi":"10.1017/fms.2024.18","DOIUrl":"https://doi.org/10.1017/fms.2024.18","url":null,"abstract":"<p>We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223055244077-0176:S2050509424000185:S2050509424000185_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb R}^n$</span></span></img></span></span>.</p><p>Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139968241","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":"Two-Point Concentration of the Independence Number of the Random Graph","authors":"Tom Bohman, Jakob Hofstad","doi":"10.1017/fms.2024.6","DOIUrl":"https://doi.org/10.1017/fms.2024.6","url":null,"abstract":"We show that the independence number of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000069_inline1.png\" /> <jats:tex-math> $ G_{n,p}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is concentrated on two values if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000069_inline2.png\" /> <jats:tex-math> $ n^{-2/3+ epsilon } &lt; p le 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on two values for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000069_inline3.png\" /> <jats:tex-math> $ p = o ( (log (n)/n)^{2/3} )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The extent of concentration of the independence number of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000069_inline4.png\" /> <jats:tex-math> $ G_{n,p}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000069_inline5.png\" /> <jats:tex-math> $ omega (1/n) &lt; p le n^{-2/3}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> remains an interesting open question.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946424","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":"Minimal log discrepancies of hypersurface mirrors","authors":"Louis Esser","doi":"10.1017/fms.2024.10","DOIUrl":"https://doi.org/10.1017/fms.2024.10","url":null,"abstract":"For certain quasismooth Calabi–Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi–Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139910435","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":"Combinatorial formulas for shifted dual stable Grothendieck polynomials","authors":"Joel Lewis, Eric Marberg","doi":"10.1017/fms.2024.8","DOIUrl":"https://doi.org/10.1017/fms.2024.8","url":null,"abstract":"The <jats:italic>K</jats:italic>-theoretic Schur <jats:italic>P</jats:italic>- and <jats:italic>Q</jats:italic>-functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline1.png\" /> <jats:tex-math> $Ghspace {-0.2mm}P_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline2.png\" /> <jats:tex-math> $Ghspace {-0.2mm}Q_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual <jats:italic>K</jats:italic>-theoretic Schur <jats:italic>P</jats:italic>- and <jats:italic>Q</jats:italic>-functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline3.png\" /> <jats:tex-math> $ghspace {-0.1mm}p_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline4.png\" /> <jats:tex-math> $ghspace {-0.1mm}q_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> via a Cauchy identity involving <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline5.png\" /> <jats:tex-math> $Ghspace {-0.2mm}P_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline6.png\" /> <jats:tex-math> $Ghspace {-0.2mm}Q_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline7.png\" /> <jats:tex-math> $ghspace {-0.1mm}p_lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000082_inline8.png\" /> <jats:tex-math> $ghspace {-0.1mm}q_lambda $ </jats:tex-math> </jats:alternatives> </jats:i","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772079","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":"Local-global compatibility for regular algebraic cuspidal automorphic representations when","authors":"Ila Varma","doi":"10.1017/fms.2024.7","DOIUrl":"https://doi.org/10.1017/fms.2024.7","url":null,"abstract":"We prove the compatibility of local and global Langlands correspondences for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline2.png\" /> <jats:tex-math> $operatorname {GL}_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline3.png\" /> <jats:tex-math> $r_p(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote an <jats:italic>n</jats:italic>-dimensional <jats:italic>p</jats:italic>-adic representation of the Galois group of a CM field <jats:italic>F</jats:italic> attached to a regular algebraic cuspidal automorphic representation <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline4.png\" /> <jats:tex-math> $pi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline5.png\" /> <jats:tex-math> $operatorname {GL}_n(mathbb {A}_F)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that the restriction of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline6.png\" /> <jats:tex-math> $r_p(pi )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to the decomposition group of a place <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline7.png\" /> <jats:tex-math> $vnmid p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>F</jats:italic> corresponds up to semisimplification to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline8.png\" /> <jats:tex-math> $operatorname {rec}(pi _v)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the image of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000070_inline9.png\" /> <jats:tex-math> $pi _v$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772259","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":"Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts","authors":"Jarod Alper, Jack Hall, Daniel Halpern-Leistner, David Rydh","doi":"10.1017/fms.2023.60","DOIUrl":"https://doi.org/10.1017/fms.2023.60","url":null,"abstract":"<p>We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin–Tyomkin.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656952","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}