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