On certain N--sheeted coverings and numerical semigroups which cannot be realized as Weierstrass semigroups

Abstract

A curve $X$ is said to be of type $(N,\gamma)$ if it is an $N$--sheeted covering of a curve of genus $\gamma$ with at least one totally ramified point. A numerical semigroup $H$ is said to be of type $(N,\gamma)$ if it has $\gamma$ positive multiples of $N$ in $[N,2N\gamma]$ such that its $\gamma^{th}$ element is $2N\gamma $ and $(2\gamma+1)N \in H$. If the genus of $X$ is large enough and $N$ is prime, $X$ is of type $(N,\gamma)$ if and only if there is a point $P \in X$ such that the Weierstrass semigroup at $P$ is of type $(N,\gamma)$ (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.