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ON TWO CONJECTURES FOR CURVES ON K3 SURFACES

ANDREAS LEOPOLD KNUTSEN

Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, Norway

https://doi.org/10.1142/S0129167X09005881Cited by:15 (Source: Crossref)

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. The constancy was originally conjectured by Harris and Mumford.

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.

Keywords:

AMSC: 14J28, 14H10, 14H51

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