Weierstrass semigroups whose minimum positive integers are even. (English) Zbl 1125.14015

Komeda, Jiryo

Arch. Math. 89, No. 1, 52-59 (2007).

Let \(C\) be a projective, non-singular, irreducible curve defined over an algebraically closed field \(k\) of characteristic zero and let \(P \in C\). Denote by \(\mathbb{N}_0\) the set of nonnegative integers and by \(\text{div}_\infty(f)\) the pole divisor of \(f \in k(C)\); the set \(H(P) := \{n \in \mathbb{N}_0 \mid \text{div}_\infty(f) = n P\}\) is the Weierstrass semigroup of \(P\), and it is a numerical semigroup i.e. \(\# (\mathbb{N}_0 \setminus H(P)) < \infty\).
The paper under review presents results on two classes of Weierstrass semigroups, namely: \(2n\)-cyclic semigroups, i.e. semigroups \(H(P)\) such that \(2n\) is the least positive element, \(C\) is a cyclic covering of \(\mathbb{P}^1(k)\) and \(P\) is a total ramification point; and \(2n\)-semigroups of double covering type, i.e. semigroups \(H(P)\) where \(2n\) is the least positive element, \(C\) is a double covering of a curve and \(P\) is a ramification point.
The main results in the paper prove that for any \(n > 2\) the set of \(2n\)-semigroups of double covering type contain properly the set of \(2n\)-cyclic semigroups; also the set of semigroups which may be realized as Weierstrass semigroups of some point \(P\) at some curve \(C\) and have \(2n\) as the least positive element contains properly the set of \(2n\)-semigroups of double covering type.

Reviewer: Cicero Carvalho (Uberlandia)

MSC:

14H55	Riemann surfaces; Weierstrass points; gap sequences
14H30	Coverings of curves, fundamental group
14M25	Toric varieties, Newton polyhedra, Okounkov bodies

Keywords:

Weierstrass semigroups; double coverings; cyclic coverings