Coverings over \(d\)-gonal curves (Q1202798)

Let \(M'\) and \(M\) be two \(d\)-gonal compact Riemann surfaces with functions of degree \(d\), \(\pi':M'\to P_ 1'\) and \(\pi:M\to P_ 1\). Let \(\varphi: M'\to M\) be a covering map. The author showed at first that \(\varphi'\) canonically induces a covering map \(\varphi: P_ 1'\to P_ 1\), and that if \(M'\) has unique linear system \(g_ d^ 1\) and \(\varphi'\) is normal, then \(\varphi\) is also normal. Then restricting \(d\) to be a prime number and a defining equation of \(M\) to have the form: \(w^ d=f(z)\), he got the equation of \(M'\) and \(\varphi'\) explicitly from that of \(M\) and \(\pi\). Using these results be determined all the ramification types of normal coverings \(\varphi'\) on condition that \(d\) is a prime number, \(M'\), \(M\) have a unique linear system \(g_ d^ 1\) respectively and a defining equation of \(M\) have the form: \(w^ d=f(z)\). The case that \(d=2\), i.e., that \(M'\), \(M\) are both hyperelliptic had been studied by Machlachlan, Farkas, Kato and Horiuchi.