Non-Cyclic Weierstrass Semigroups

Abstract

Morrison and Pinkham [5] gave a characterization of the semi-groups of Galois Weierstrass points, i.e., total ramification points of cyclic coverings of the projective line of degree n. They showed that such a semigroup must satisfy certain equalities, which we call the M-P equalities in this paper, and that the converse holds for any prime n = p ≤7. In this paper we consider the case when n = p is a prime number ≥ 11. In the case p = 11 we investigate whether a primitive numerical semigroup satisfying the M-P equalities is the semigroup of a Galois Weierstrass point. For each prime p ≥ 13, we give a Weierstrass semigroup which satisfies the M-P equalities but is not the semigroup of a Galois Weierstrass point. For these, we study the semigroups of Galois Weierstrass points using the equations defining curves which are cyclic covering of the projective line of degree p.

This work has been supported by the Japan Society for the Promotion of Science and the Korea Science and Engineering Foundation

The first author is partially supported by KOSEF-GARC and by the Basic Science Research Institute Program, Ministry of Education, Project No. BSRI-98–1406.