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Abstract
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We prove that curves in a nonprimitive, base point free, ample linear system on a K3
surface have maximal variation. The result is deduced from general restriction theorems
applied to the tangent bundle. We also show how to use specialisation to spectral
curves to deduce information about the variation of curves contained in a K3
surface more directly. The situation for primitive linear systems is not clear at
the moment. However, the maximal variation holds in genus two and can,
in many cases, be deduced from a recent result of van Geemen and Voisin
(Int. Math. Res. Not. 2016:10 (2016), 3111–3123) confirming a conjecture due to
Matsushita.
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Keywords
K3 surfaces, variation of curves, Hitchin system, Mukai
system, tangent bundle, stability
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Mathematical Subject Classification
Primary: 14J28
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Milestones
Received: 17 June 2021
Revised: 3 February 2022
Accepted: 19 February 2022
Published: 9 November 2022
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