Exceptional linear systems on curves on Del Pezzo surfaces (Q757506)

We prove that, if C is a smooth irreducible curve on a Del Pezzo surface S such that \(K^ 2_ S\geq 2\), then - with one exception, involving curves of genus 3 - the gonality of the smooth curves in the linear system \(| C|\) is constant, and that, if the genus of C is at least 4, also the Clifford index of the smooth curves in \(| C|\) is constant. Both statements are not true for curves on Del Pezzo surfaces S such that \(K^ 2_ S=1\). Such results, especially the ones concerning the gonality, are predicted - and clarified - by Green and Lazarsfeld's conjectures on syzygies of curves embedded by line bundles of high degree.