A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$

Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve of genus $g$. This picture starts from irreducible components of $W^r_d(C)$ restricted to an open subset of $Pic (C)$ satisfying Brill-Noether theory as in the case of a general curve of genus $g$. We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes $W^r_d(C)$ are translates of their closures in $Pic (C)$. We complete the proof that the schemes $W^r_d(C)$ are generically smooth in case $C$ is a general $k$-gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary $k$-gonal curve and we obtain some new smoothness results in case $C$ is a general $k$-gonal curve.