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On Weierstrass semigroups of double coverings of genus three curves. (English) Zbl 1244.14025
For a numerical semigroup \(\tilde H\) of genus \(\tilde g\), let \(H:=\{\tilde h/2: \tilde h\in \tilde H, \tilde h \text{even}\}\) and let \(g\) be its genus. The subject matter of this paper is concerning the existence of a double covering \(\pi: \tilde C\to C\) of curves of genus \(\tilde g\) and \(g\), respectively, and a totally ramified point \(\tilde P\in \tilde C\) in such a way that the Weierstrass semigroup at \(\tilde P\) equals \(\tilde H\) (so that the Weierstrass semigroup at \(\pi(\tilde P)\) equals \(H\); (cf. [F. Torres, Manuscr. Math. 83, 39–59 (1994; Zbl 0838.14025)]).
If \(g=0\), \(\tilde H\) is a hyperelliptic semigroup and the existence of \(\pi\) is well-known. If \(g=1,2\), the possibilities for \(\tilde H\) were also determined and for each case the corresponding map \(\pi\) were constructed (see [J. Reine Angew. Math. 341, 68–86 (1983; Zbl 0498.30053); Res. Rep. Kamagawa Inst. Technol. B-33, 37–42 (2009); Semigroup Forum 77, 152–162 (2008; Zbl 1161.14023)]).
In this paper the case \(g=3\) is considered where some results were already obtained in [J. Pure Appl. Algebra 214, 1955–1961 (2010; Zbl 1194.14048)]. As a matter of fact, all the possibilities for \(\tilde H\) (with \(g=3\)) are realized as a Weierstrass semigroups at totally ramified points of certain double covering of curves.

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
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