Abstract
Green’s Conjecture is proved for smooth curves $$C$$ lying on a rational surface $$S$$ with an anticanonical pencil, under some mild hypotheses on the line bundle $$L=\mathcal{O }_S(C)$$ . Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system $$\vert L\vert $$ is also obtained.
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