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Maximal variation of curves on K3 surfaces

Yajnaseni Dutta and Daniel Huybrechts
Vol. 4 (2022), No. 3, 443–464
DOI: 10.2140/tunis.2022.4.443
Abstract

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
Mathematical Subject Classification
Primary: 14J28
Milestones
Received: 17 June 2021
Revised: 3 February 2022
Accepted: 19 February 2022
Published: 9 November 2022
Authors
Yajnaseni Dutta
Hausdorff Center for Mathematics
Universität Bonn
Bonn
Germany
Daniel Huybrechts
Mathematisches Institut
Universität Bonn
Bonn
Germany