Bounding the number of points on a curve using a generalization of Weierstrass semigroups

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup [P. Beelen, N. Tuta\c{s}: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in [O. Geil, R. Matsumoto: Bounding the number of $\mathbb{F}_q$-rational places in algebraic function fields using Weierstrass semigroups. Pure Appl. Algebra, 213(6), 2009]

An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes

On linear series with negative Brill-Noether number

Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle having at least $(r+1)$ independent global sections. The Brill-Noether theorem asserts that the map $\mathcal{W}^r_{d,g} \rightarrow \mathcal{M}_g$ is surjective with general fiber dimension given by the number $\rho = g - (r+1)(g-d+r)$, under the hypothesis that $0 \leq \rho \leq g$. One may naturally conjecture that for $\rho < 0$, this map is generically finite onto a subvariety of codimension $-\rho$ in $\mathcal{M}_g$. This conjecture fails in general, but seemingly only when $-\rho$ is large compared to $g$. This paper proves that this conjecture does hold for at least one irreducible component of $\mathcal{W}^r_{d,g}$, under the hypothesis that $0 < -\rho \leq \frac{r}{r+2} g - 3r+3$. We conjecture that this result should hold for all $0 < -\rho \leq g + C$ for some constant $C$, and we give a purely combinatorial conjecture that would imply this stronger result.Comment: 16 page

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