Non-Weierstrass numerical semigroups
type: Item
instance of: scholarly article
title: Non-Weierstrass numerical semigroups
zbMATH Open document ID: 0922.14022
DOI: 10.1007/PL00005972
published in: Semigroup Forum
publication date: 1999-10-17T00:00:00Z
review text: A subsemigroup \(H\) of \(\mathbb{N}\) is called numerical if \(\mathbb{N} \setminus H\) is finite; \(\text{card}(\mathbb{N}\setminus H)\) is called the genus of \(H\). Let \(C\) be a smooth complex projective curve of genus \(g\geq 2\) and \(P\in C\). Set \(H(P):=\{t \in\mathbb{N}: h^0(C,{\mathcal O}_C (tP))> h^0(C, {\mathcal O}_C((t-1)P)) \}\); \(H(P)\) is a numerical semigroup of genus \(g\); such semigroups are called Weierstrass. In 1980 Buchweitz solved a long-standing problem proving the existence of a genus 16 numerical semigroup which is not Weierstrass. His easy proof used properties of a semigroup \(H(P)\) arising from the Riemann-Roch formula for \(h^0(C,K_C^{\otimes 2})\). In the paper under review the author makes an abstract study of semigroups, \(H\), such that for some integer \(t\geq 2\) Buchweitz's method and the corresponding value for \(h^0(C,K_C^{\otimes t})\) show that \(H\) is not Weierstrass.
reviewed by: E. Ballico
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 20M99
zbMATH DE Number: 1209919
zbMATH Keywords: gap sequence
zbMATH Keywords: numerical semigroups of a Weierstrass point
zbMATH Keywords: Weierstrass semigroup
author: Jiryo Komeda
MaRDI profile type: Publication
full work available at URL: pl00005972
OpenAlex ID: W2015281608
description: scientific article; zbMATH DE number 1209919