On the gonality sequence of an algebraic curve

For any smooth irreducible projective curve X, the gonality sequence ${\{d_r | r \in \mathbb N\}}$ is a strictly increasing sequence of positive integer invariants of X. In most known cases d _r+1 is not much bigger than d _r . In our terminology this means the numbers d _r satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in ${{\mathbb P}^r}$ , for curves on a general K3-surface in ${{\mathbb P}^r}$ and for complete intersections in ${{\mathbb P}^3}$ .