In this paper it is shown that the gonality of curves on an elliptic ruled surface is twice the degree of the restriction of the bundle map and the Clifford index of such curves is computed by pencils of minimal degree, under certain numerical conditions. It is also proved that any pencil computing the gonality and the Clifford index of curves is composed with the restriction of the bundle map under some stronger conditions. On the other hand, we found some counterexample to the constancy of gonality and Clifford index in a linear system.
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