A classification of trigonal Riemann surfaces
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.
scientific article; zbMATH DE number 1496084