96 research outputs found

    On Weierstra{\ss} semigroups at one and two points and their corresponding Poincar\'e series

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    The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.Comment: Beginning of Section 3 and Subsection 3.1 were modifie

    Generalized Weierstrass semigroups and their Poincaré series

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    Producción CientíficaWe investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincar´e series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.Ministerio de Economía, Industria y Competitividad ( grant MTM2015-65764-C3-2-P / MTM2016-81735-REDT / MTM2016-81932-REDT)Universitat Jaume I (grant P1-1B2015-02 / UJI-B2018-10)Consejo Nacional de Desarrollo Científico y Tecnológico (grants 201584/2015-8 / 159852/2014-5 / 310623/2017-0)IMAC-Institut de Matemàtiques i Aplicacions de Castell

    The Set of Pure Gaps at Several Rational Places in Function Fields

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    In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field FF over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters

    Irreducibility of Virasoro representations in Liouville CFT

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    In the context of Liouville conformal field theory, we construct the highest-weight representations of the Virasoro algebra at the degenerate values of the conformal weight (Kac table). We show that these modules are irreducible, giving a complete characterization of the algebraic structure of Liouville CFT. It also implies that all singular vectors vanish, which is one of the main assumptions usually made in theoretical physics. Our proof uses inputs from both probability theory and algebra, and gives new probabilistic content to the Kac table. Combining the information that singular vectors vanish with the main geometric properties of conformal blocks, we deduce that conformal blocks involving degenerate primary fields satisfy null-vector equations. These equations take the form of PDEs on the Teichm\"uller space of the underlying surface and generalize previous works in several directions.Comment: 44 pages,5 figure

    Quasi-ordinary singularities via toric geometry

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    Se estudian las singularidades casi-ordinarias de variedades analíticas complejas, por medio de técnicas de la geometría tórica, principalmente en el caso de gérmenes de hipersuperficie
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