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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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