Let \(|L_g|\) , be the genus g du Val linear system on a Halphen surface Y of index k. We prove that the Clifford index \({\text {Cliff}}(C)\) is constant on smooth curves \(C\in |L_g|\) . Let \(\gamma (C)\) be the gonality of C. When \({\text {Cliff}}(C)<\lfloor {\frac{g-1}{2}}\rfloor \) (the relevant case), we show that \(\gamma (C)={\text {Cliff}}(C)+2=k\) , and that the gonality is realized by the Weierstrass linear series \(|-{kK_Y}_{|C}|\) , which is totally ramified at one point. The proof of the first statement follows closely the path indicated by Green and Lazarsfeld for a similar statement regarding K3 surfaces.
