The proportion of Weierstrass semigroups

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. 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.

A Graph-Theoretic Approach to Wilf's Conjecture

“…Specifically, among all 23 022 228 615 numerical semigroups of genus g 45, the proportion of those satisfying |P | m/3 exceeds 99.999%. It is likely, though it remains to be proved, that this proportion tends to 1 as g goes to infinity, in complete analogy to Zhai's result regarding the condition c m/3, where c is the conductor [23,16]. In addition, Delgado discovered that the condition of Theorem 1 is well suited to efficiently trim the tree of numerical semigroups while probing certain open problems concerning them [6].…”

Weierstrass Sets on Finite Graphs

“…In 1893 Hurwitz [13] asks if all the numerical semigroups arise in this manner. Several years later, in 1980, Buchweitz [5] showed that the follwoing numerical semigroup: S = 13, 14, 15,16,17,18,20,22,23 is not Weierstrass (see also [9, page 499]). The proof essentially gives the following necessary condition for a semigroup to be Weierstrass: the m-sumset of the set of gaps must satisfy |mG(P )| (2m−1)(g −1) for any integer m 2.…”

“…Despite these results, little is known more generally about the family of Weierstrass semigroups. For instance, the problem of determining its density in the set of all numerical semigroups is still open [15].…”

Weierstrass sets on finite graphs

“…In 1893 Hurwitz [13] asks if all the numerical semigroups arise in this manner. Several years later, in 1980, Buchweitz [5] showed that the numerical semigroup S = 13,14,15,16,17,18,20,22,23 is not Weierstrass (see also [9, page 499]). The proof essentially gives the following necessary condition for a semigroup to be Weierstrass: the m-sumset of the set of gaps must satisfy |mG(P )| ≤ (2m − 1)(g − 1) for any integer m ≥ 2.…”