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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)].

14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group
14C20 Divisors, linear systems, invertible sheaves
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