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The purpose of this paper is to show that for a complete intersection curve C in projective space (other than a few exceptions stated below), any morphism f : C → P r satisfying deg f * O P r (1) < deg C is obtained by projection from a linear space. In particular, we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves. We also prove a special case of one of the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris.Picard group generated by the hyperplane class. However, Theorem 1.2 also holds in the more general setting of arbitrary curves lying on surfaces with Picard group Z: Theorem 1.5. Let C ⊂ P n be a smooth non-degenerate curve lying on a smooth surface S with Pic(S) = Z • [O S (1)], and C ∈ |O S (α)|, where α ≥ 4. Then any morphism f : C → P r with r < n satisfying deg f * O Pr (1) < deg C is obtained by projecting from an (n − r − 1)-plane.Remark 1.6. In [Ras15], Rasmussen independently showed that, under stronger hypotheses, the fibers of a morphism C → P 1 of degree less than the degree of C must lie in hyperplanes, which follows from our Lemma 4.4.Remark 1.7. A special case of one of the Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris (Conjecture CB12 in [EGH96]) follows easily from the proof of Theorem 1.5. We discuss this in Section 5.We also obtain a bound on the gonality in this more general setting:

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