Remarks on Numerical Semigroups

We extend results on Weierstrass semigroups at ramified points of double covering of curves to any numerical semigroup whose genus is large enough. As an application we strengthen the properties concerning Weierstrass weights stated in [To]. 0. Introduction Let H be a numerical semigroup, that is, a subsemigroup of (N,+) whose complement is finite. Examples of such semigroups are the Weierstrass semigroups at non-singular points of algebraic curves. In this paper we deal with the following type of semigroups: Definition 0.1. Let γ ≥ 0 an integer. H is called γ-hyperelliptic if the following conditions hold: (E1) H has γ even elements in [2, 4γ]. (E2) The (γ + 1)th positive element of H is 4γ + 2. A 0-hyperelliptic semigroup is usually called hyperelliptic. The motivation for study of such semigroups comes from the study of Weierstrass semigroups at ramified points of double coverings of curves. Let π : X → X̃ be a double covering of projective, irreducible, non-singular algebraic curves over an algebraically closed field k. Let g and γ be the genus of X and X̃ respectively. Assume that there exists P ∈ X which is ramified for π, and denote by mi the ith non-gap at P . Then the Weierstrass semigroup H(P ) at P satisfies the following properties (cf. [To], [To1, Lemma 3.4]): (P1) H(P ) is γ-hyperelliptic, provided g ≥ 4γ+1 if char(k) 6= 2, and g ≥ 6γ−3 otherwise. (P2) m2γ+1 = 6γ + 2, provided g ≥ 5γ + 1. (P3) m g 2 −γ−1 = g − 2 or m g−1 2 −γ = g − 1, provided g ≥ 4γ + 2. (P4) The weight w(P ) of H(P ) satisfies