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An example of the Weierstrass semigroup of a pointed curve on $$K3$$ surfaces. (English) Zbl 1285.14036
Let $$C$$ be a non-singular, projective, irreducible, algebraic curve defined over an algebraically closed field of characteristic zero, and let $$P\in C$$. The Weierstrass semigroup $$H(P)$$ of $$C$$ at $$P$$ is the set of poles of regular function on $$C\setminus\{P\}$$. Thus $$H(P)$$ is indeed a subsemigroup of the additive semigroup $$\mathbb N_0$$ such that $$\#(\mathbb N_0\setminus H(P))$$ equals the genus of $$C$$ (The Weierstrass gap theorem). In general it is a difficult problem to compute Weierstrass semigroups.
In the paper under review, the author considers certain curves contained in $$K3$$ surfaces over the complex numbers with Picard number one. In this way he continues the computation of Weierstrass semigroups for non-singular curves on non-singular surfaces such as those of plane curves of degree at most seven computed by S. J. Kim and J. Komeda and [J. Algebra 322, No. 1, 137–152 (2009; Zbl 1171.14020)], or those curves on non-singular toric surfaces computed by R. Kawaguchi [Kodai Math. J. 33, No. 1, 63–86 (2010; Zbl 1221.14042)].
The main result in the paper under review is the following. Let $$X$$ be a non-singular $$K3$$ surface over the complex number with Picard number one and let $$X$$ be defined by a double morphism $$\pi: X\to \mathbb P^2$$. Let $$C\subseteq X$$ be a non-singular, projective, irreducible algebraic curve of degree $$d\geq 4$$ which is not the ramification divisor of $$\pi$$ and such that $$\pi^{-1}(\pi(C))=C$$. For $$R\in C$$ let $$I(R)$$ denote the intersection divisor of $$\pi(C)$$ and the tangent line of $$\pi(C)$$ at $$\pi(R)$$. Let $$P$$ be a ramification point of $$\pi|C: C\to\pi(C)$$. If $$I(P)=d\pi(P)$$, then $$H(P)=2H(\pi(P))+(6d-1){\mathbb N}_0$$. If $$I(P)=(d-1)\pi(P)+Q$$ with $$I(Q)=d\pi(Q)$$, then $$H(P)=2H(\pi(p))+ \sum_{i=0}^{d-4}((8d-9)+2(d-2)i){\mathbb N}_0$$.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H30 Coverings of curves, fundamental group 14J28 $$K3$$ surfaces and Enriques surfaces
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##### References:
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