A classification of trigonal Riemann surfaces
type: Item
instance of: scholarly article
title: A classification of trigonal Riemann surfaces
zbMATH Open document ID: 0966.14021
author: Robert D. M. Accola
published in: Kodai Mathematical Journal
publication date: 2001-05-09T00:00:00Z
review text: For a trigonal Riemann surface \(W_{p}\) of genus \(p>3\), there is a unique integer \(\ell\) so that the complete linear series \(|\ell g_{3}^{1}|\) and \(|(\ell+1) g_{3}^{1}|\) have dimensions \(\ell\) and \(\ell+\varepsilon\) respectively, where \(\varepsilon>1\). The author calls \(\ell\) the trigonal index of \(W_{p}\), and denotes it ti \((\text{ti} \leq p/2\)). If the canonical divisor \(K_{p}=(p-1-\text{ti})g_{3}^{1}+R\), then the residual divisor \(R\) is such that \(|R|=g_{\rho}^{\varepsilon}\), where \(\rho=3 \text{ ti}-(p-1)\), and \(\varepsilon=0\) if \(\text{ti}<p/2\) and \(\varepsilon=1\) if \(\text{ti}=p/2\). \(W_{p}\) admits a three sheeted covering of the Riemann sphere, and using Coppens terminology, ramification points of multiplicity two are called ordinary, and those of multiplicity three are called total. Ramification points occurring in the residual divisor are called to be of type II when \(\text{ti}<p/2\), and of type I otherwise. If \(\text{ti}=p/2\), then all ramification points are of type I. The author neatly formulates the theorem that necessary and sufficient conditions on the non-negative integers \(\ell,n,m,s,t,p\), in order that \(\text{Trig}_{p}(\ell,n,m,s,t)\), the set of trigonal Riemann surfaces, be non-empty are: (1) \(2p-2=n+m+2s+2t-6\), (2) there exist non-negative integers \(\nu,\mu,\sigma\), so that the equations \[ \begin{aligned} 2\ell+2 &= m+s+2t+\sigma,\\ 3\ell+3 &= 2m+s+2t+\mu,\\ 6\ell+6 &= 3m+2s+4t+n+2\nu\end{aligned} \] are satisfied, (3) \(\ell \leq p/2\), and he proves the sufficiency (the necessity is due to Kato-Horiuchi) using the Hurwitz existence theorem of coverings.
reviewed by: Ryutaro Horiuchi
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 1496084
zbMATH Keywords: trigonal Riemann surface
zbMATH Keywords: ramification point
zbMATH Keywords: trigonal index
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044156
OpenAlex ID: W2004165586
cites work: Q3720720
cites work: The Weierstrass gap sequence of the ordinary ramification points of trigonal coverings of \({\mathbb{P}}^ 1\); existence of a kind of Weierstrass gap sequence
cites work: Subspaces of trigonal Riemann surfaces
cites work: Weierstrass gap sequences at the ramification points of trigonal Riemann surfaces
cites work: Le serie lineari speciali sulle curve trigonali
DOI: 10.2996/KMJ/1138044156
description: scientific article; zbMATH DE number 1496084