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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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##### References:
 [1] Komeda, J.: On Weierstrass points whose first non-gaps are four. J. Reine Angew. Math. 341, 68–86 (1983) · Zbl 0498.30053 [2] Komeda, J.: A numerical semigroup from which the semigroup gained by dividing by two is either $$\mathbb{N}$$0 or a 2-semigroup or ,4,5 Res. Rep. Kanagawa Inst. Technol. B-33, 37–42 (2009) [3] Komeda, J., Ohbuchi, A.: Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve. Serdica Math. J. 30, 43–54 (2004) · Zbl 1075.14029 [4] Komeda, J., Ohbuchi, A.: On double coverings of a pointed non-singular curve with any Weierstrass semigroup. Tsukuba J. Math. 31, 205–215 (2007) · Zbl 1154.14023 [5] Komeda, J., Ohbuchi, A.: Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve II. Serdica Math. J. 34, 771–782 (2008) · Zbl 1224.14008 [6] Oliveira, G., Pimentel, F.L.R.: On Weierstrass semigroups of double covering of genus two curves. Semigroup Forum 77, 152–162 (2008) · Zbl 1161.14023 · doi:10.1007/s00233-007-9038-0 [7] Oliveira, G., Torres, F., Villanueva, J.: On the weight of numerical semigroups. J. Pure Appl. Algebra 214, 1955–1961 (2010) · Zbl 1194.14048 · doi:10.1016/j.jpaa.2009.12.032 [8] Torres, F.: Weierstrass points and double coverings of curves with application: Symmetric numerical semigroups which cannot be realized as Weierstrass semigroups. Manuscr. Math. 83, 39–58 (1994) · Zbl 0838.14025 · doi:10.1007/BF02567599
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