An example of the Weierstrass semigroup of a pointed curve on \(K3\) surfaces
type: Item
instance of: scholarly article
title: An example of the Weierstrass semigroup of a pointed curve on \(K3\) surfaces
zbMATH Open document ID: 1285.14036
author: Kenta Watanabe
published in: Semigroup Forum
publication date: 2013-06-18T00:00:00Z
review text: 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 \textit{S. J. Kim} and \textit{J. Komeda} and [J. Algebra 322, No. 1, 137--152 (2009; Zbl 1171.14020)], or those curves on non-singular toric surfaces computed by \textit{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 \(H(P)=2H(\pi(p))+ \sum_{i=0}^{d-4}((8d-9)+2(d-2)i){\mathbb N}_0\).
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6176651
zbMATH Keywords: Weierstrass semigroups of pointed curves
zbMATH Keywords: double coverings of curves
zbMATH Keywords: curves on \(K3\) surfaces
MaRDI profile type: Publication
full work available at URL: s00233-012-9464-5
OpenAlex ID: W1978175660
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DOI: 10.1007/S00233-012-9464-5
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description: scientific article; zbMATH DE number 6176651