Coverings over $d$-gonal curves

Abstract

Let M be a compact Riemann surface and / be a meromorphic function on M. Let (/) be the principal divisor associated to / and (/)≪,be the polar divisor of /. We call/ a meromorphic function ofdegree d if rf=degree (/)≪. If d is the minimal integer in which a meromorphic function of degree d existson M, then we callM a d-gonal curve. Now we assume that M is d-gonal,and considera covering map it':M'^M that M' stillremains rf-gonal. The purpose of thispaper is to show how such %' can be characterized. The case that it'is a normal covering and d=2 (i.e.,M is hyperelliptic) has been already studied([2], [3], [4] and [7]). In this case the existence of the hyperellipticinvolution v' on M' plays an important role. More precisely, as v' commutes with each element of the Galois group G―Gal{M''/M), v'induces the hyperellipticinvolution v on M and we can reduce rc'to a normal covering x: P'^Pi with Galois group G, where P[ and Px are Riemann spheres isomorphic to quotient Riemann surfaces M'/ and M/ respectively. On the other hand itisknown thatfinitesubgroups of thelinear transformation group are cyclic, dihedral, tetrahedral,octahedral and icosahedral. Horiuchi [3] decided all the different normal coverings it':M'-*M over a hyperellipticcurve M that M' stillremains a hyperellipticcurve by investigating each of above fivetypes. Let M be a rf-gonal curve. In this paper we willshow at firstthat a covering map %': M'―>M (not necessarily normal) with rf-gona!M' canonically induces some covering map n: P[-^PX (Theorem 2.1§2). Moreover if both M and M' have unique linearsystem gldand it'is normal, then we can see that itis also normal (Cor. 2.3). In §3,§4 and §5 we assume that M is a cyclic/>-gonalcurve fora prime number p. We will determine allramification types of normal coverings it':M'^M with 6-gonal M' by thesame way as Horiuchi did in case ￡=2(§4),