A family of Non-Weierstrass Semigroups

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.