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 Poincaré series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.

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

“…Notice that, since Γ(Q) ∩ C m is finite and Θ m is finitely generated, Γ(Q) is determined by a finite number of elements in H(Q). We also observe that the set C m and the elements η i 's are slightly modified from that defined in [20,Section 3]. This modification does not affect the conclusions and its purpose is to make our subsequent computations simpler.…”

“…Moyano-Fernández, Tenório, and Torres [20] studied the generalized Weierstrass semigroups at several points on a curve over a finite field. Motivated by the description of the classical Weierstrass semigroups at several points given by Matthews [16], they characterized the semigroups H(Q) in terms of the absolute maximal elements of H(Q) (see Definition 2.2) introduced by Delgado [7], providing a generating set for H(Q) in the sense of [16].…”

On Weierstrass Gaps at Several Points

“…Beelen and Tutas [3] studied these objects for curves defined over finite fields-in [7] was assumed curves over algebraically closed fields-and present many interesting properties, especially for a pair of points. These structures were recently explored in [16], where there is established connections between the concepts of maximal elements in [7] and generating sets for H(Q) in the sense of [15]. Explicit computations relying on the approach of [16] and its outcomes are presented in [17] for certain special curves with separable variables, denoted by X f,g (see Sec.…”

“…These structures were recently explored in [16], where there is established connections between the concepts of maximal elements in [7] and generating sets for H(Q) in the sense of [15]. Explicit computations relying on the approach of [16] and its outcomes are presented in [17] for certain special curves with separable variables, denoted by X f,g (see Sec. 4).…”

“…The concept of absolute maximality was used in [16,Theorem 3.4] to provide a description of H(Q), whose motivation comes from [15] for classical Weierstrass semigroups.…”