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

TL;DR

This paper investigates the structure and classification of sparse semigroups, establishes bounds on their genus, and explores their realization as Weierstrass semigroups, with applications to algebraic curves and Weierstrass points.

Contribution

It provides a comprehensive description, classification, and genus bounds for sparse semigroups, and studies their realization as Weierstrass semigroups, connecting algebraic geometry and semigroup theory.

Findings

Upper bound for the genus of sparse semigroups

Classification of sparse semigroups based on structure

Conditions for realizing sparse semigroups as Weierstrass semigroups

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.