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

Abstract

In this work we study the generalized Weierstrass semigroup $\hat{H}({\mathbf{P}}_m)$ at an $m$-tuple ${\mathbf{P}}_m=(P_1,\dots ,P_m)$ 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 $\hat{\Gamma }({\mathbf{P}}_m)$ of $\hat{H}({\mathbf{P}}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_1,\dots ,P_m\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.

17 pages