A family of Non-Weierstrass Semigroups

TL;DR

This paper introduces a new syzygy-based method to identify non-Weierstrass numerical semigroups, providing the first example with multiplicity 6 and genus 13, and expanding understanding of their properties.

Contribution

The paper presents a novel approach using syzygies to determine non-Weierstrass semigroups, including the first known example with multiplicity 6 and genus 13.

Findings

Identified the first non-Weierstrass semigroup with multiplicity 6 and genus 13.

Developed a new method using syzygies for classifying semigroups.

Provided numerous examples demonstrating the method's applicability.

Abstract

A numerical semigroup is said to be Weierstrass if it is the semigroup of pole orders of rational functions that are regular at all but one point of some compact Riemann surface or smooth algebraic curve. Hurwitz asked in 1892 whether all numerical semigroups can occur. In this paper we give a new method, using syzygies,to show that certain semigroups are not Weierstrass, including the first one of multiplicity 6 (the lowest possible) and genus 13 (the lowest known). We give many other examples to which the method applies.