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Gonality and Clifford index of projective curves on ruled surfaces. (English) Zbl 1234.14026
Authors’ abstract: Let $$X$$ be a smooth curve on a ruled surface $$\pi : S\rightarrow C$$. 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 $$\pi | _X$$. The main theorem shows that if a smooth curve $$X\sim aC_o +bf$$ satisfies some conditions on the degree of $$b$$, then a line bundle $$\mathcal{L}$$ on $$X$$ with $$\mathrm{Cliff}(\mathcal L)\leq ag(C)-1$$ is composed with $$\pi | _X$$. This implies that a part of the gonality sequence of $$X$$ is computed by the gonality sequence of $$C$$ as follows: $d_r (X)=ad_r (C) \text{ for } r\leq L,$ where $$L$$ is the length of the gonality sequence of $$C$$.

##### MSC:
 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14J26 Rational and ruled surfaces 14H45 Special algebraic curves and curves of low genus
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##### References:
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