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arXiv · 0705.0302

On two conjectures for curves on $K3$ surfaces

Andreas Leopold Knutsen
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Abstract

We prove that the gonality among the smooth curves in a complete linear system on a $K3$ surface is constant except for the Donagi-Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. As a consequence we prove that exceptional curves on $K3$ surfaces satisfy the Eisenbud-Lange-Martens-Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud-Lange-Martens-Schreyer examples of exceptional curves on $K3$ surfaces.

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Andreas Leopold Knutsen. 2007-05-02. On two conjectures for curves on $K3$ surfaces. https://arxiv.org/abs/0705.0302

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