On the structure of numerical sparse semigroups and applications to Weierstrass points

Abstract

In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the realization of some sparse semigroups as Weierstrass semigroups. The smoothness property of monomial curves associated to (hyper)ordinary semigroups presented by Pinkham and Rim-Vitulli, and the results on double covering of curves by Torres are crucial in this.

On κ-sparse numerical semigroups

“…In [4], the authors work with the ordered pairs (ℓ i−1 , ℓ i ) such that ℓ i − ℓ i−1 = 1 and ℓ i − ℓ i−1 = 2, for sparse numerical semigroups of genus g ≥ 2 and 2 ≤ i ≤ g, not considering the element ℓ 0 = −1. These pairs were called single leap and double leap, respectively.…”