Gonality and Clifford index of projective curves on ruled surfaces

Abstract

Let X X be a smooth curve on a ruled surface π : S → C \pi : S\rightarrow C . In this paper, we deal with the questions on the gonality and the Clifford index of X X and on the composedness of line bundles on X X with the covering morphism π | X \pi |_X . The main theorem shows that if a smooth curve X ∼ a C o + b f X\sim aC_o +\textbf {b}f satisfies some conditions on the degree of b \bf b , then a line bundle L \mathcal {L} on X X with C l i f f ( L ) ≤ a g ( C ) − 1 \mathrm {Cliff}(\mathcal {L})\le ag(C)-1 is composed with π | X \pi |_X . This implies that a part of the gonality sequence of X X is computed by the gonality sequence of C C as follows: \[ d r ( X ) = a d r ( C )      for    r ≤ L , d_r (X)=ad_r (C) ~~\mbox { for }~r\le L, \] where L L is the length of the gonality sequence of C C .