Brill-Noether theory for cyclic covers

We recall that the Brill-Noether Theorem gives necessary and sufficient conditions for the existence of a gdr. Here we consider a general n-fold, étale, cyclic cover p:C˜→C of a curve C of genus g and investigate for which numbers r,d a gdr exists on C˜. For r=1 this means computing the gonality of C˜. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a gdr which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all n,r,d.