The gonality of Riemann surfaces under projections by normal coverings

A compact Riemann surface X of genus g≥2 which can be realized as a q-fold, normal covering of a compact Riemann surface of genus p is said to be (q,p)-gonal. In particular the notion of (2,p)-gonality coincides with p-hyperellipticity and (q,0)-gonality coincides with ordinary q-gonality. Here we completely determine the relationship between the gonalities of X and Y for an N-fold normal covering X→Y between compact Riemann surfaces X and Y. As a consequence we obtain classical results due to Maclachlan (1971) [5], and Martens (1977) [6].