On γ-hyperelliptic Numerical Semigroups

Abstract

We extend results on Weierstrass semigroups at rami ed 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 [23]. 0. Introduction Let H be a numerical semigroup, that is, a subsemigroup of (N;+) whose complement is nite. 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: De nition 0.1. Let 0 be an integer. H is called -hyperelliptic if the following conditions hold: (E1 ) H has even elements in [2; 4 ] . (E2 ) The ( + 1)th positive element of H is 4 + 2. A 0-hyperelliptic semigroup is usually called hyperelliptic. The motivation for study of such semigroups comes from the study of Weierstrass semigroups at rami ed points of double coverings of curves. Let : X ! ~ X be a double covering of projective, irreducible, non-singular algebraic curves over an algebraically closed eld k . Let g and be the genus of X and ~ X respectively. Assume that there exists P 2 X which is rami ed for , and denote by mi the ith non-gap at P . Then the Weierstrass semigroup H(P ) at P satis es the following properties (cf. [23], [24, Lemma 3.4]): (P1 ) H(P ) is -hyperelliptic, provided g 4 + 1 if char(k) 6= 2, and g 6 3 otherwise. (P2 ) m2 +1 = 6 + 2, provided g 5 + 1. (P3 ) m g 2 1 = g 2 or m g 1 2 = g 1, provided g 4 + 2. (P4 ) The weight w(P ) of H(P ) satis es g 2 2 ! w(P ) < g 2 + 2 2 ! : * This work was realized while the author was in ICTP (Trieste Italy) with a grant from the International Atomic Energy Agency and UNESCO.