# Generalized Weierstrass semigroups and their Poincar\'e series

## TL;DR

This paper studies the structure of generalized Weierstrass semigroups at multiple points on algebraic curves over finite fields, revealing how their Poincaré series encode key semigroup properties and divisor information.

## Contribution

It provides a new description of these semigroups that links their structure to Poincaré series, advancing understanding of their arithmetical and geometric properties.

## Key findings

- Characterization of generalized Weierstrass semigroups
- Poincaré series fully describe the semigroups
- Insights into divisor and Riemann-Roch space properties

## Abstract

We 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.