Generalized Weierstrass semigroups and their Poincaré series

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.