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