A study of general Martens-special chains of cycles

For a general Martens-special chain of cycles $\Gamma$ of type $k$ we prove that the gonality is equal to $k+2$. Although $\dim (W^1_{k+2} (\Gamma))=k$ we prove that $w^1_{k+2}(\Gamma)=0$. We also compute the gonality sequence of $\Gamma$ and we prove it is divisorial complete. We prove that a general Martens-special discrete chain of cycles $G$ of type $k$ has the same gonality sequence.