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Number of weak Galois-Weierstrass points with Weierstrass semigroups generated by two elements. (English) Zbl 1428.14055
Let $$C$$ 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.
##### 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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##### References:
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