an:06540857
Zbl 1351.14022
Komeda, Jiryo; Watanabe, Kenta
On extensions of a double covering of plane curves and Weierstrass semigroups of the double covering type
EN
Semigroup Forum 91, No. 2, 517-523 (2015).
00349052
2015
j
14H55
Weierstrass semigroups; double coverings of curves; Weierstrass gap
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 [\textit{J. C. Rosales} et al., J. Number Theory 103, No. 2, 281--294 (2003; Zbl 1039.20036)]. A semigroup is the \textit{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 \(P=\pi(\tilde P)\) [\textit{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. [\textit{J. Komeda}, J. Reine Angew. Math. 341, 68--86 (1983; Zbl 0498.30053)], [Res. Rep. Kamagawa Inst. Technol. B-33, 37--42 (2009)], [\textit{G. Oliveira} and \textit{F. L. R. Pimentel}, Semigroup Forum 77, No. 2, 152--162 (2008; Zbl 1161.14023)] , [\textit{J. Gilvan de Oliveira} et al., J. Pure Appl. Algebra 214, No. 11, 1955--1961 (2010; Zbl 1194.14048)], [\textit{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 [\textit{K. Watanabe}, Semigroup Forum 86, No. 2, 395--403 (2013; Zbl 1285.14036)].
Fernando Torres (Campinas)
Zbl 1039.20036; Zbl 0425.30038; Zbl 0498.30053; Zbl 1161.14023; Zbl 1194.14048; Zbl 1244.14025; Zbl 1285.14036