Brill-Noether theory for cyclic covers

We recall that the Brill-Noether Theorem gives necessary and sufficient conditions for the existence of a g(d)(r). Here we consider a general n-fold, etale, cyclic cover p : (C) over tilde -> C of a curve C of genus g and investigate for which numbers r, d a g(d)(r) exists on (C) over tilde. For r = 1 this means computing the gonality of (C) over tilde. 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 Plficker formula yields a necessary condition for the existence of a g(d)(r) which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all n, r, d. (C) 2016 Elsevier B.V. All rights reserved.