Minimal Non-Weierstrass Semigroups

Abstract

Let p in X be a point on a compact Riemann surface. The Weierstrass semigroup of p is the semigroup of pole orders of meromorphic functions on X that are regular at all but p. Hurwitz asked in 1892 whether all numerical semigroups occur as Weierstrass semigroups. In this paper we give a new method for showing that certain numerical semigroups are not Weierstrass, including some of every genus g in which non-Weierstrass examples could possibly exist, except g=18. Our example for g=13 has at once the smallest possible genus, multiplicity and number of generators of any possible non-Weierstrass semigroup.

This version contains a proof that all semigroups of genus < 13 are Weierstrass, as well as new examples including non-Weierstrass semigroups of every genus >= 13 except 18; hence the change in the title