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On double coverings of a pointed non-singular curve with any Weierstrass semigroup. (English) Zbl 1154.14023
Let $$H$$ be the Weierstrass semigroup of a point $$P$$ of a genus $$g$$ complete non-singular curve $$C.$$ In the paper under review, given an odd positive integer $$n$$ satisfying certain conditions, the authors construct a double covering $$\pi :\widetilde{C}\rightarrow C$$ with a ramification point $$\widetilde{P}$$ lying over $$P$$ such that the genus of the curve $$\widetilde {C}$$ is $$2g+(n-1)/2$$ and the Weierstrass semigroup of $$\widetilde{P}$$ is $$H_n=2H+n\mathbb N,$$ where $$\mathbb N$$ is the set of non-negative integers. By giving bounds for the weight of the Weierstrass semigroup of some ramification point F. Torres [Manuscr. Math. 83, No. 1, 39–58 (1994; Zbl 0838.14025)] obtained a characterization of double coverings of the curve $$C$$.
In the case that $$3$$ and $$6$$ are respectively the smallest positive integers of $$H$$ and $$H_n$$ the authors describe necessary and sufficient conditions for $$H_n$$ be the Weierstrass semigroup of a total ramification point of a cyclic covering of the projective line. In particular they conclude that the semigroup generated by $$6, 8, 10$$ and $$2g-7$$ is cyclic if and only if $$g\equiv2 \pmod 3.$$ Explicit examples of non-singular curve realizing this semigroup was given by A. Garcia [Manuscr. Math. 55, 419–432 (1986; Zbl 0603.14014)] and in the opposite case by G. Oliveira and F. L. R. Pimentel [Semigroup Forum 77, No. 2, 152–162 (2008; Zbl 1161.14023)].

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H30 Coverings of curves, fundamental group 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
Weierstrass semigroup; double covering; cyclic covering
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