Brill-Noether varieties of k-gonal curves

We consider a general curve of fixed gonality k and genus g. We propose an estimate ρ‾g,k(d,r) for the dimension of the variety Wdr(C) of special linear series on C, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k≥15g+2, and is an upper bound in all other cases. We also completely characterize the cases in which Wdr(C) has the same dimension as for a general curve of genus g.