Number of weak Galois-Weierstrass points with Weierstrass semigroups generated by two elements.

*(English)*Zbl 1428.14055Let \(C\) be a non-singular projective curve of genus \(g \geq 2\) over an algebraically closed field of characteristic \(0\). Take a point \(P\) on \(C\). Its Weierstrass semigroup \(H(P)\) is the set of non-negative integers \(n\) for which there exists a rational function \(f\) on \(C\) such that \(f\) has a pole of order \(n\) at \(P\), and is regular away from \(P\). The point \(P\) is a Galois Weierstrass point (GW point, in short), if \(\Phi_{|aP|} : C \rightarrow \mathbb{P}^1\) is a Galois covering where \(a\) is the smallest positive integer of \(H(P)\). Besides, \(P\) is said to be a weak Galois-Weierstrass point (weak GW point), if it is a Weierstrass point and there exists a Galois morphism \(\varphi : C \rightarrow \mathbb{P}^1\) such that \(P\) is a total ramification point of \(\varphi\).

The paper under review is devoted to study the number of weak GW points which satisfy that their Weierstrass semigroup \(H(P)\) is generated by two positive integers, \(a\) and \(b\), such that gcd\((a,b) = 1\) and \(2 < a < b-1\). The main result of the article is Theorem 1.3. In its first part it is proved that the number of GW points \(P\) with \(H(P) = \langle a,b \rangle\) is \(0\) or \(b+1\) if \(b \equiv -1\) (mod \(a\)), and it is \(0\) or \(1\) if \(b \not\equiv -1\) (mod \(a\)). For second and third parts of Theorem 1.3, let \(P\) be a weak GW point, and call \(\mathrm{degGW}(P)\) the set of degrees of the Galois coverings of \(C\) totally ramified at \(P\). Then, the number of weak GW points \(P\) with \(H(P) = \langle a,b \rangle\) and \(b \in\mathrm{degGW}(P)\) is \(0\) or \(1\), and there exists a weak GW point \(P\) with \(H(P) = \langle a,b \rangle\), and \(a, b \in\mathrm{degGW}(P)\) if and only if \(C\) is birationally equivalent to the curve \(X^b = Y^a Z^{b-a} + Z^b\).

It is important to note that M. Coppens has obtained in [Abh. Math. Semin. Univ. Hamb. 89, No. 1, 1–16 (2019; Zbl. 07100734)] results which overlap partially the mentioned Theorem 1.3.

The paper under review is devoted to study the number of weak GW points which satisfy that their Weierstrass semigroup \(H(P)\) is generated by two positive integers, \(a\) and \(b\), such that gcd\((a,b) = 1\) and \(2 < a < b-1\). The main result of the article is Theorem 1.3. In its first part it is proved that the number of GW points \(P\) with \(H(P) = \langle a,b \rangle\) is \(0\) or \(b+1\) if \(b \equiv -1\) (mod \(a\)), and it is \(0\) or \(1\) if \(b \not\equiv -1\) (mod \(a\)). For second and third parts of Theorem 1.3, let \(P\) be a weak GW point, and call \(\mathrm{degGW}(P)\) the set of degrees of the Galois coverings of \(C\) totally ramified at \(P\). Then, the number of weak GW points \(P\) with \(H(P) = \langle a,b \rangle\) and \(b \in\mathrm{degGW}(P)\) is \(0\) or \(1\), and there exists a weak GW point \(P\) with \(H(P) = \langle a,b \rangle\), and \(a, b \in\mathrm{degGW}(P)\) if and only if \(C\) is birationally equivalent to the curve \(X^b = Y^a Z^{b-a} + Z^b\).

It is important to note that M. Coppens has obtained in [Abh. Math. Semin. Univ. Hamb. 89, No. 1, 1–16 (2019; Zbl. 07100734)] results which overlap partially the mentioned Theorem 1.3.

Reviewer: JosĂ© Javier Etayo (Madrid)

##### MSC:

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

14H50 | Plane and space curves |

14H30 | Coverings of curves, fundamental group |

20M14 | Commutative semigroups |

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\textit{J. Komeda} and \textit{T. Takahashi}, J. Korean Math. Soc. 56, No. 6, 1463--1474 (2019; Zbl 1428.14055)

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##### References:

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