On certain n-sheeted coverings of curves and numerical semigroups which cannot be realized as weierstrass semigroups

Abstract

A curve Xis said to be of type (iV, γ) if it is an iV-sheeted covering of a curve of genus γ with at least one totally ramified point. A numerical semigroup His said to be of type (iV, γ) if it has γ positive multiples of Nin [N, 2NJ] such that its γth element is 2Nγ and (2γ+1)NeH. If the genus of X is large enough and N is prime, X is of type (TV, γ) if and only if there is a point P6 X such that the Weierstrass semigroup at P is of type (N, γ) (this generalizes the case of double coverings of curves). Using the proof of this result and the Buchweitz's semigroup, we can construct numerical semigroups that cannot be realized as Weierstrass semigroups although they might satisfy Buchweitz's criterion