A double covering of curves on a Hirzebruch surface of degree one and Weierstrass semigroups. (English) Zbl 1409.14058
Let \(C\) be a smooth projective curve over \(\mathbb{C}\), and \(P\) a point on \(C\). The Weierstrass semigroup \(H(P)\) at \(P\) is the set of non-negative integers \(n\) such that there exists a meromorphic function \(f\) on \(C\) with \(nP\) the pole of \(f\) at \(P\). The author investigates in this paper when a semigroup \(H\) is the Weierstrass semigroup \(H(\tilde {P})\) at a ramification point \(\tilde {P}\) of a double covering \(\pi : \tilde{C} \to C\) of a smooth curve \(C\). The main result is Theorem 1.2, which has a very technical statement, allowing to relate \(H(\tilde {P})\) and \(H(P)\). A couple of standard examples show how to deal with the two cases arising in Theorem 1.2.
Reviewer: José Javier Etayo (Madrid)
MSC:
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 20M14 | Commutative semigroups |
| 14H30 | Coverings of curves, fundamental group |
References:
| [1] | Kang, E., Kim, S.J.: A Weierstrass semigroup at a pair of inflection points on a smooth plane curve. Bull. Korean Math. Soc. 44(2), 369-378 (2007) · Zbl 1143.14026 · doi:10.4134/BKMS.2007.44.2.369 |
| [2] | Komeda, J.: On Weierstrass semigroups of double coverings of genus three curves. Semigroup Forum 83, 479-488 (2011) · Zbl 1244.14025 · doi:10.1007/s00233-011-9331-9 |
| [3] | 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 |
| [4] | Watanabe, K.: An example of the Weierstrass semigroup of a pointed curve on K3 surfaces. Semigroup Forum 86, 395-403 (2013) · Zbl 1285.14036 · doi:10.1007/s00233-012-9464-5 |
| [5] | Watanabe, K.: On extensions of a double covering of plane curves and Weierstrass semigroups of the double covering type. Semigroup Forum 91, 517-523 (2015) · Zbl 1351.14022 · doi:10.1007/s00233-015-9718-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
