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For any smooth irreducible projective curve X, the gonality sequence {d r | r ∈ 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 P r , for curves on a general K3-surface in P r and for complete intersections in P 3 .

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