Remarks on \(d\)-gonal curves (Q1916077)

We assume that \(M\) is a \(p\)-gonal curve of genus \(g\) with a prime number \(p\). Then Namba has shown that \(M\) has a unique linear system \(g^1_p\) of projective dimension one and degree \(p\) provided \(g>(p-1)^2\) [\textit{M. Namba}, ``Families of meromorphic functions on compact Riemann surfaces'', Lect. Notes Math. 767 (1979; Zbl 0417.32008)]. In this paper we treat a compact Riemann surface \(M\) defined by an equation (*) \(y^d-(x-a_1)^{r_1} \cdots (x-a_s)^{r_s} =0\) with \(\sum r_i \equiv 0\bmod d\) and \(1\leq r_i<d\), where \(d\) is not necessarily a prime number. In \S 2, we will show that \(M\) is \(d\)-gonal with the function \(x\) of degree \(d\) if there are enough \(r_i\)'s relatively prime to \(p\) for each prime number \(p\) dividing \(d\). In this case we call \(M\) a cyclic \(d\)-gonal curve. We will also show that \(M\) has a unique \(g^1_d\) if there are more sufficient such \(r_i\)'s as above. -- In \S 3, let \(M\) be a cyclic \(d\)-gonal curve defined by (*) having a unique \(g^1_d\) and \(M'\) be a compact Riemann surface defined by \(y^d-(x-b_1)^{t_1} \cdots (x-b_s)^{t_s}=0\). We will study the relations among \(a_i\), \(b_i\), \(r_i\) and \(t_i\) \((1\leq i\leq s)\) in the case \(M\) and \(M'\) are conformally equivalent. M. Namba and T. Kato have already studied this problem in the case \(d\) is a prime number. We will give similar results for an arbitrary \(d\) (\S 3). -- In \S 4, we consider a covering map \(\pi': M'\to M\), where \(M\) is a cyclic \(d\)-gonal curve with a unique \(g^1_d\) and \(M'\) is a \(d'\)-gonal curve. In the case \(d=d'\), we can apply the same methods as in a previous paper [\textit{N. Ishii}, Tsukuba J. Math. 16, No. 1, 173-189 (1992; Zbl 0771.30042)], and we will see that \(M'\) is also cyclic \(d\)-gonal. Moreover if \(\pi'\) is normal and \(d=d'\), then the covering group of \(\pi'\) is isomorphic to cyclic, dihedral, tetrahedral, octahedral or isohedral. For a general case \(d\leq d'\), we will show some relations between \(d\) and \(d'\). -- In \S 5, we will give some remarks about coverings \(M\to N\) with a cyclic \(d\)-gonal curve \(M\) having a unique \(g^1_d\). -- Finally we determine the equation (*), which defines the curve \(M\) (with a unique \(g^1_d)\) having an automorphism \(V\) \((\notin \langle T\rangle)\) of order \(N\), where \(T\) is the automorphism defined by \(T^*x=x\) and \(T^*y= e^{2\pi i/d}y\) (\S 6).