GONALITY AND CLIFFORD INDEX OF PROJECTIVE CURVES ON RULED SURFACES.

Let X be a smooth curve on a ruled surface π : S → C. In this paper, we deal with the questions on the gonality and the Clifford index of X and on the composedness of line bundles on X with the covering morphism π∣X. The main theorem shows that if a smooth curve X ∼ aCo + bf satisfies some conditions on the degree of b, then a line bundle L on X with Cliff(L) ⩽ ag(C) - 1 is composed with π∣X. This implies that a part of the gonality sequence of X is computed by the gonality sequence of C as follows: dr(X) = adr(C) for r ⩽ L, where L is the length of the gonality sequence of C. [ABSTRACT FROM AUTHOR]