On extensions of a double covering of plane curves and Weierstrass semigroups of the double covering type.

*(English)*Zbl 1351.14022In this paper certain Weierstrass semigroups related to double covering of curves are investigated. For \(H\) a numerical semigroup of genus \(g=g(H)\) let \(d_2(H):=\{s\in\mathbb N: 2s\in H\}\), where \(\mathbb N\) stands for the set of nonnegative integers [J. C. Rosales et al., J. Number Theory 103, No. 2, 281–294 (2003; Zbl 1039.20036)]. A semigroup is the double covering type, if it is the Weierstrass semigroup \(H(\tilde P)\) at a totally ramified point \(\tilde P\) of a double covering \(\pi:\tilde C\to C\) of (projective, irreducible, nonsingular, algebraic) curves over the complex numbers; in this case, \(d_2(H(\tilde P))=H(P)\) being \(P=\pi(\tilde P)\) [T. Kato, Kodai Math. J. 2, 275–285 (1979; Zbl 0425.30038)].

If \(g(d_2(H))\in\{0,1,2,3\}\) and \(g(H)\) is large enough, \(H\) is the double covering type; see e.g. [J. Komeda, J. Reine Angew. Math. 341, 68–86 (1983; Zbl 0498.30053)], [Res. Rep. Kamagawa Inst. Technol. B-33, 37–42 (2009)], [G. Oliveira and F. L. R. Pimentel, Semigroup Forum 77, No. 2, 152–162 (2008; Zbl 1161.14023)] , [J. Gilvan de Oliveira et al., J. Pure Appl. Algebra 214, No. 11, 1955–1961 (2010; Zbl 1194.14048)], [J. Komeda, Semigroup Forum 83, No. 3, 479–488 (2011; Zbl 1244.14025)].

Let \(\pi:\tilde C\to C\), \(\tilde P\), \(P\) be as above. In the paper under review, \(C\) is a plane curve of degree \(d\geq 4\), \(T_P\) stands for the tangent line to \(C\) at \(P\). Let \(M_d\) be the proposition: \(\pi\) extends to a double covering \(\tilde \pi: X\to{\mathbb P}^2\) branched along a reduced divisor of degree six containing \(P\). The main result here is a characterization of certain semigroups of the double covering type: (a) If \(I_P(T_P\cap C)=d\), then \(M_d\) holds if and only if \(H(\tilde P)=2H(P)+(6d-1)\mathbb N\). (b) Let \(I_P(T_P\cap C)=d-1\) and \(T_P\cdot C=(d-1)P+Q\). If \(I_Q(T_Q\cap C)=d\), then \(M_d\) holds if and only if \(H(\tilde P)=2H(P)+ \sum_{i-0}^{d-4}(8d-9+2i(d-2))\mathbb N\). Particular cases of this result were already computed in [K. Watanabe, Semigroup Forum 86, No. 2, 395–403 (2013; Zbl 1285.14036)].

If \(g(d_2(H))\in\{0,1,2,3\}\) and \(g(H)\) is large enough, \(H\) is the double covering type; see e.g. [J. Komeda, J. Reine Angew. Math. 341, 68–86 (1983; Zbl 0498.30053)], [Res. Rep. Kamagawa Inst. Technol. B-33, 37–42 (2009)], [G. Oliveira and F. L. R. Pimentel, Semigroup Forum 77, No. 2, 152–162 (2008; Zbl 1161.14023)] , [J. Gilvan de Oliveira et al., J. Pure Appl. Algebra 214, No. 11, 1955–1961 (2010; Zbl 1194.14048)], [J. Komeda, Semigroup Forum 83, No. 3, 479–488 (2011; Zbl 1244.14025)].

Let \(\pi:\tilde C\to C\), \(\tilde P\), \(P\) be as above. In the paper under review, \(C\) is a plane curve of degree \(d\geq 4\), \(T_P\) stands for the tangent line to \(C\) at \(P\). Let \(M_d\) be the proposition: \(\pi\) extends to a double covering \(\tilde \pi: X\to{\mathbb P}^2\) branched along a reduced divisor of degree six containing \(P\). The main result here is a characterization of certain semigroups of the double covering type: (a) If \(I_P(T_P\cap C)=d\), then \(M_d\) holds if and only if \(H(\tilde P)=2H(P)+(6d-1)\mathbb N\). (b) Let \(I_P(T_P\cap C)=d-1\) and \(T_P\cdot C=(d-1)P+Q\). If \(I_Q(T_Q\cap C)=d\), then \(M_d\) holds if and only if \(H(\tilde P)=2H(P)+ \sum_{i-0}^{d-4}(8d-9+2i(d-2))\mathbb N\). Particular cases of this result were already computed in [K. Watanabe, Semigroup Forum 86, No. 2, 395–403 (2013; Zbl 1285.14036)].

Reviewer: Fernando Torres (Campinas)

##### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

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\textit{J. Komeda} and \textit{K. Watanabe}, Semigroup Forum 91, No. 2, 517--523 (2015; Zbl 1351.14022)

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##### References:

[1] | Kang, E; Kim, SJ, A Weierstrass semigroup at a pair of inflection points on a smooth plane curve, Bull Korean Math. Soc., 44, 369-378, (2007) · Zbl 1143.14026 |

[2] | Komeda, J, On Weierstrass semigroups of double coverings of genus three curves, Semigroup Forum, 83, 479-488, (2011) · Zbl 1244.14025 |

[3] | Oliveira, G; Pimentel, FLR, On Weierstrass semigroups of double covering of genus two curves, Semigroup Forum, 77, 77-152, (2008) · Zbl 1161.14023 |

[4] | 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 |

[5] | Watanabe, K, An example of the Weierstrass semigroup of a pointed curve on K3 surfaces, Semigroup Forum, 86, 395-403, (2013) · Zbl 1285.14036 |

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