Forum of Mathematics Sigma最新文献

{"title":"A remark on Gibbs measures with log-correlated Gaussian fields","authors":"Tadahiro Oh, Kihoon Seong, Leonardo Tolomeo","doi":"10.1017/fms.2024.28","DOIUrl":"https://doi.org/10.1017/fms.2024.28","url":null,"abstract":"We study Gibbs measures with log-correlated base Gaussian fields on the <jats:italic>d</jats:italic>-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson’s argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove nonnormalizability of the Gibbs measure. When <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000288_inline1.png\" /> <jats:tex-math> $d = 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, our argument provides an alternative proof of the nonnormalizability result for the focusing <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000288_inline2.png\" /> <jats:tex-math> $Phi ^4_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein’s inequality on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000288_inline3.png\" /> <jats:tex-math> $mathbb R^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) nonnormalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579263","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":"Cohomological Descent for Faltings Ringed Topos","authors":"Tongmu He","doi":"10.1017/fms.2024.26","DOIUrl":"https://doi.org/10.1017/fms.2024.26","url":null,"abstract":"<p>Faltings ringed topos, the keystone of Faltings’ approach to <span>p</span>-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings’ approach to any integral model, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579078","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":"Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity","authors":"P. Habegger, H. Schmidt","doi":"10.1017/fms.2023.112","DOIUrl":"https://doi.org/10.1017/fms.2023.112","url":null,"abstract":"<p>In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$T^p+c$</span></span></img></span></span>, where <span>p</span> is a prime number and where the orbit of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span> is finite. For example, if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p=2$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span> is periodic under <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$T^2+c$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$cin mathbb {R}$</span></span></img></span></span>, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these <span>f</span>, our method has application to the irreducibility of polynomials. Indeed, say <span>y</span> is preperiodic under <span>f</span> but not periodic. Then any iteration of <span>f</span> minus <span>y</span> is irreducible in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Q}(y)[T]$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579080","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":"Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs","authors":"Chih-Whi Chen, Shun-Jen Cheng","doi":"10.1017/fms.2024.17","DOIUrl":"https://doi.org/10.1017/fms.2024.17","url":null,"abstract":"<p>We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb {C}}$</span></span></img></span></span> with a triangular decomposition and a character <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$zeta $</span></span></img></span></span> of the nilpotent radical, the associated Backelin functor <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma _zeta $</span></span></img></span></span> sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {O}}$</span></span></img></span></span>. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma _zeta $</span></span></img></span></span> and its target category, respectively, categorify a <span>q</span>-symmetrizing map and the corresponding <span>q</span>-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$AIII$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578951","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":"Extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes","authors":"Minki Kim, Alan Lew","doi":"10.1017/fms.2024.23","DOIUrl":"https://doi.org/10.1017/fms.2024.23","url":null,"abstract":"<p>We present extensions of the colorful Helly theorem for <span>d</span>-collapsible and <span>d</span>-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev.</p><p>As an application, we obtain the following extension of Tverberg’s theorem: Let <span>A</span> be a finite set of points in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb R}^d$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$|A|&gt;(r-1)(d+1)$</span></span></img></span></span>. Then, there exist a partition <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$A_1,ldots ,A_r$</span></span></img></span></span> of <span>A</span> and a subset <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Bsubset A$</span></span></img></span></span> of size <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(r-1)(d+1)$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$cap _{i=1}^r operatorname {mathrm {text {conv}}}( (Bcup {p})cap A_i)neq emptyset $</span></span></img></span></span> for all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$pin Asetminus B$</span></span></img></span></span>. That is, we obtain a partition of <span>A</span> into <span>r</span> parts that remains a Tverberg partition even after removing all but one arbitrary point from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridg","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578950","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":"Stability of isometric immersions of hypersurfaces","authors":"Itai Alpern, Raz Kupferman, Cy Maor","doi":"10.1017/fms.2024.30","DOIUrl":"https://doi.org/10.1017/fms.2024.30","url":null,"abstract":"<p>We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$L^p$</span></span></img></span></span>-perturbations of their fundamental forms: For a manifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal M}^d$</span></span></img></span></span> endowed with a reference metric and a reference shape operator, we show that a sequence of immersions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_n:{mathcal M}^dto {mathcal N}^{d+1}$</span></span></img></span></span>, whose pullback metrics and shape operators are arbitrary close in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$L^p$</span></span></img></span></span> to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal N}$</span></span></img></span></span>, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579196","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":"Evaluation of and period polynomial relations","authors":"Steven Charlton, Adam Keilthy","doi":"10.1017/fms.2024.16","DOIUrl":"https://doi.org/10.1017/fms.2024.16","url":null,"abstract":"<p>In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {SL}}_2({mathbb {Z}})$</span></span></img></span></span>. In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$zeta (2,ldots ,2,4,2,ldots ,2)$</span></span></img></span></span> as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple <span>t</span> values [22] <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$t(2ell ,2k)$</span></span></img></span></span> in terms of classical double zeta values.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578943","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":"Cohomological","authors":"Woonam Lim, Miguel Moreira, Weite Pi","doi":"10.1017/fms.2024.31","DOIUrl":"https://doi.org/10.1017/fms.2024.31","url":null,"abstract":"<p>We prove that the cohomology rings of the moduli space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$M_{d,chi }$</span></span></img></span></span> of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$chi $</span></span></img></span></span>-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$M_{d,chi }$</span></span></img></span></span> are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579189","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":"K-stable smooth Fano threefolds of Picard rank two","authors":"Ivan Cheltsov, Elena Denisova, Kento Fujita","doi":"10.1017/fms.2024.5","DOIUrl":"https://doi.org/10.1017/fms.2024.5","url":null,"abstract":"<p>We prove that all smooth Fano threefolds in the families <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline1.png?pub-status=live\" type=\"\"> and <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline2.png?pub-status=live\" type=\"\"> are K-stable, and we also prove that smooth Fano threefolds in the family <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline3.png?pub-status=live\" type=\"\"> that satisfy one very explicit generality condition are K-stable.</img></img></img></p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166669","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":"Finite skew braces of square-free order and supersolubility","authors":"A. Ballester-Bolinches, R. Esteban-Romero, M. Ferrara, V. Pérez-Calabuig, M. Trombetti","doi":"10.1017/fms.2024.29","DOIUrl":"https://doi.org/10.1017/fms.2024.29","url":null,"abstract":"<p>The aim of this paper is to study <span>supersoluble</span> skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace <span>B</span> many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of <span>B</span> is <span>B</span>-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, <span>B</span> has finite multipermutational level if and only if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315044131610-0210:S205050942400029X:S205050942400029X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(B,+)$</span></span></img></span></span> is nilpotent.</p><p>Given a finite presentation of the structure skew brace <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315044131610-0210:S205050942400029X:S205050942400029X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$G(X,r)$</span></span></img></span></span> associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315044131610-0210:S205050942400029X:S205050942400029X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G(X,r)$</span></span></img></span></span> is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149743","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}