Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables

TL;DR

This paper characterizes generalized Weierstrass semigroups and explicitly constructs bases for Riemann-Roch spaces on certain algebraic curves with separated variables, extending previous results.

Contribution

It computes the generating set of the generalized Weierstrass semigroup and explicitly describes Riemann-Roch space bases for these curves, generalizing prior work.

Findings

Computed the generating set of the generalized Weierstrass semigroup.

Explicit basis for Riemann-Roch spaces on these curves.

Extended previous results to a broader class of curves.

Abstract

In this work we study the generalized Weierstrass semigroup $\widehat{H} (\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $\mathbb{F}_{q}$, where ${f(T),g(T)\in \mathbb{F}_q[T]}$. In particular, we compute the generating set $\widehat{\Gamma}(\mathbf{P}_m)$ of $\widehat{H} (\mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_{1}, \ldots , P_{m}\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.