On the existence of Weierstrass gap sequences on curves of genus \(\leq 8\)
type: Item
instance of: scholarly article
title: On the existence of Weierstrass gap sequences on curves of genus \(\leq 8\)
zbMATH Open document ID: 0849.14011
DOI: 10.1016/0022-4049(94)90039-6
author: Jiryo Komeda
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-11T00:00:00Z
review text: Let \(C\) be a complete nonsingular irreducible 1-dimensional algebraic variety of genus \(g\) over the field \(\mathbb{C}\) of complex numbers. Let \(\mathbb{N}\) be the additive semigroup of non-negative integers. Let \(K(C)\) denote the field of rational functions on \(C\). An subsemigroup \(H\) of \(\mathbb{N}\) is Weierstrass if there exists a pointed curve \((C,P)\) such that \(H(P) = \{h \in \mathbb{N} \mid\) there exists \(f\in K(C)\) with \((f)_\infty = hP\} = H\). In this paper the author proves that any numerical semigroup \(H\) (a subsemigroup of \(\mathbb{N}\) whose complement \(\mathbb{N} \backslash H\) in \(\mathbb{N}\) is finite) of genus \(g \leq 7\) is Weierstrass. Moreover, in the cases \(g = 8\) he proves that all primitive numerical semigroups are Weierstrass, i.e., twice the smallest positive integer in \(H > \) the largest integer in \(\mathbb{N} \backslash H\).
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 719442
zbMATH Keywords: Weierstrass gap sequences
zbMATH Keywords: genus
zbMATH Keywords: Weierstrass group
reviewed by: Emilio Bujalance
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90039-6
OpenAlex ID: W2084359694
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description: scientific article; zbMATH DE number 719442