Brill-Noether varieties of k-gonal curves

We consider a general curve of fixed gonality k and genus g. We propose an estimate (rho) over bar (g,k)(d, r) for the dimension of the variety W-d(r) (C) of special linear series on 0, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k >= 1/5g+2, and is an upper bound in all other cases. We also completely characterize the cases in which W-d(r)(C) has the same dimension as for a general curve of genus g. (C) 2017 Elsevier Inc. All rights reserved.