The Proportion of Weierstrass Semigroups

TL;DR

This paper investigates the density of Weierstrass semigroups among numerical semigroups, showing that those satisfying certain criteria are rare and establishing new structural results about typical semigroups.

Contribution

It resolves a problem by Komeda regarding the proportion of Weierstrass semigroups and demonstrates that known families of such semigroups have zero density among all numerical semigroups.

Findings

Semigroups satisfying Buchweitz's criterion are rare.

Known Weierstrass semigroup families have zero density.

New structural properties of typical numerical semigroups are established.

Abstract

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.