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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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