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On extensions of a double covering of plane curves and Weierstrass semigroups of the double covering type. (English) Zbl 1351.14022
In 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)].

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