Gonality and Clifford index of projective curves on ruled surfaces

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 $X\sim aC_o +\textbf {b}f$ satisfies some conditions on the degree of $\bf b$, then a line bundle $\mathcal {L}$ on $X$ with $\mathrm {Cliff}(\mathcal {L})\le ag(C)-1$ 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: \[ d_r (X)=ad_r (C) ~~\mbox { for }~r\le L,\] where $L$ is the length of the gonality sequence of $C$.