Curves and line bundles on Del Pezzo surfaces

Abstract

We extend the results of Pareschi on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree 1, where we explicitly classify the cases where the gonality and Clifford index are not constant. We also classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension 3. Moreover, the property of being exceptional holds for all curves in the complete linear system. Furthermore, we relate the Clifford index and gonality of smooth curves in $|L|$ to the higher order embedding properties of $|L+K_S|$. More precisely, we show that for a nef line bundle $L$ on a Del Pezzo surface, $L+K_S$ is birationally $k$-very ample if and only if all the smooth curves in $|L|$ have gonality $\geq k+2$, and we also find numerical criteria for birational $k$-very ampleness.