The automorphism group of a cyclic \(p\)-gonal curve (Q2472516)

Let \(p\geq 2\) be a prime number and suppose that \(X\) is a compact Riemann surface of genus \(g\) which admits a group of conformal automorphisms \(C_{p}\) of prime order \(p\) such that the quotient space \(X/C_{p}\) has genus \(0\). We call the surface \(X\) a cyclic \(p\)-gonal surface. Suppose that the quotient map \(\pi \colon X\rightarrow X/C_{p}\) is branched over the points \(\{a_{1} ,\dots ,a_{t}\}\). If \(a_{i} \neq \infty\) for any \(i\), then \(X\) admits a defining equation of the form \[ y^p=\prod_{i=1}^{t} (x-a_{i})^{r_{i}} \] where \(\sum_{i=1}^{t} r_{i} = 0\mod{(p)}\). Else if \(a_{i}=\infty\) for some \(i\) (without loss of generality, assume \(a_{t}=\infty\)), then \(X\) admits a defining equation of the form \[ y^p=\prod_{i=1}^{t-1} (x-a_{i})^{r_{i}}, \] where \(\sum_{i=1}^{t-1} r_{i} \neq 0\mod{(p)}\). In the paper under review, the authors consider the problem of determining explicit defining equations for \(X\) dependent upon the quotient group \(H=G/C_{p}\) where \(G\) is a group of automorphisms of \(X\) with \(C_{p} \vartriangleleft G\). That is, the authors consider the problem of determining the \(r_{i}\) and the \(a_{i}\). It should be noted that the results developed by the authors hold only under the assumption that \(C_{p} \leq C(G)\) (where \(C(G)\) denotes the center of \(G\)). Lemma 2.1 claims to show that \(C_{p} \leq C(G)\) provided \(g>(p-1)^2\) (and in fact more generally), but unfortunately the argument presented is invalid. An example of a family of cyclic \(p\)-gonal surfaces for which \(C_{p}\nleq C(G)\) with genus \(g> (p-1)^2\) is given by Theorem \(1\) of \textit{J. Wolfart} and \textit{M. Streit} [ Rev. Mat. Complut. 13, No. 1, 49--81 (2000; Zbl 1053.14021)] and other families appear in the literature. The general approach taken by the authors is as follows. If \(G\) is a group of automorphisms of \(X\) with \(C_{p}\vartriangleleft G\), then the group \(H=G/C_{p}\) acts on \({\mathbb P}^{1}\) and the \(a_{i}\) lie in \(H\)-orbits of this action. Moreover, if \(C_{p}\leq C(G)\), then if \(a_{i}\) and \(a_{j}\) lie in the same \(H\)-orbit, then \(r_{i}=r_{j}\). This provides a general formula for \(X\) which can be made more explicit by choosing a specific representation of \(H \leq \text{Aut} ({\mathbb P}^{1})\). The authors illustrate their results by producing explicit equations for hyperelliptic curves (when \(p=2\)) for genus \(2\) and trigonal curves (when \(p=3\)) for genera \(5\), \(7\) and \(9\) for all the different possible \(H\) which arise. Note that for hyperelliptic curves, \(C_{2} \leq C(G)\) for all \(G\) and \(g\), and for trigonal curves of genus \(5\), \(7\) and \(9\), \(C_{3} \leq C(G)\) for all possible \(G\), and thus complete results are presented for all these cases. For similar results without the assumption that \(C_{p}\leq C(G)\), see \textit{A. Wootton} [Isr. J. Math. 157, 103--122 (2007; Zbl 1109.30036)] where Fuchsian groups are used to determine defining equations. Also, more generally, for related results for general \(n\)-gonal surfaces (\(n\) not necessarily prime), see \textit{A. Kontogeorgis} [J. Algebra 216, No. 2, 665--706 (1999; Zbl 0938.11056)]. Editorial remark: A correction to Lemma 2.1 is given in [Tsukuba J. Math. 32, No. 2, 407--408 (2008; Zbl 1158.14307)] by assuming that \(V\) is in the center of \(G\).