Let \(C\) be a general curve of gonality \(k\) and genus \(g\). For any integer \(r\geq 2\) let \(W^r_d(C)\) be the scheme of special linear series on \(C\). In this paper the author (using tropical geometry) gives a very good upper bound \(\overline{\rho}_{g,k}(d,r)\) for it. [M. Coppens and G. Martens, Indag. Math., New Ser. 13, No. 1, 29–45 (2002; Zbl 1059.14036)] proved strong existence theorems and dimension theorems for \(W^r_d(C)\) and a useful feature of the paper under review is the translation of their results as \(\dim W^r_d(C) \geq \underline{\rho}_{g,k}(d,r)\) for a certain function \(\underline{\rho}_{g,k}(d,r)\). He proves here the exact value \(\overline{\rho}_{g,k}(d,r)\) of \(W^r_d(C)\) if either \(k\leq 5\) or \(k\geq \frac{1}{5}g+2\) and conjecture that it is always the case if \(g-d+r>0\) and \(2\leq k\leq (g+3)/2\). \(W^r_d(C)\) is usually reducible and he asks if all irreducible components of \(W^r_d(C)\) have dimension \(\rho _g(d,r-\ell)-\ell k\) for some \(\ell\), where \(\rho _g(d,r)\) is the usual Brill-Noether number.