On Weierstrass semigroups of double coverings of genus three curves.

*(English)*Zbl 1244.14025For 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.

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.

Reviewer: Fernando Torres (Campinas)

##### 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 |

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[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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