The proportion of Weierstrass semigroups

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. A result of Eisenbud and Harris gives a sufficient condition for a semigroup to occur as a Weierstrass semigroup. We show that the family of semigroups satisfying this condition has density zero in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup. (C) 2012 Elsevier Inc. All rights reserved.