On numerical semigroups related to covering of curves
A projective, irreducible, non-singular algebraic curve \(C\) of genus \(g\) is of type \((N,\gamma)\) if it is an \(N\)-sheeted covering of a curve of genus \(\gamma\) having at least one totally ramification point. Corresponding to the Weierstrass semigroup at a totally ramification point of such coverings, a numerical semigroup \(H\) is of type \((N,\gamma)\) if the first \(\gamma\) positive non-gaps are multiples of \(N\) with the \(\gamma\)-th nongap equal to \(2\gamma N\) and the number \((2\gamma +1)N\in H\). The authors continue to generalize characterizations of the Weierstrass semigroup of hyperelliptic curves, a work which was started by the second author [Manuscr. Math. 83, 39--58 (1994; Zbl 0838.14025), Commun. Algebra 23, 4211--4228 (1995; Zbl 0842.14023), Semigroup Forum 55, 364--379 (1997; Zbl 0931.14017), Arch. Math. 68, 139--144 (1997; Zbl 0872.14026)]. Using a generalization of a theorem in \textit{G. A. Freiman}'s book [Foundations of a structural theory of set addition, Transl. Math. Monogr. 37 (1973; Zbl 0271.10044)] the authors first prove that for a numerical semigroup \(H\) of genus large enough \(H\) is of type \((N,\gamma)\) if and only if \(2\gamma N, (N +1)N\gamma \in H\) where \(N\) is an odd prime, and also another similar result. Secondly they obtained a sharp lower bound for the weight of semigroups of type \((N,\gamma)\), and upper bounds for the weight in special cases.
scientific article; zbMATH DE number 2078225
scientific article