Maximal variation of curves on K3 surfaces

TL;DR

This paper demonstrates that curves in certain linear systems on K3 surfaces exhibit maximal variation, using restriction theorems and specialization techniques, with specific results for genus two cases.

Contribution

It establishes maximal variation for curves in non-primitive, base point free, ample linear systems on K3 surfaces, extending understanding of their geometric properties.

Findings

Maximal variation proven for non-primitive linear systems on K3 surfaces

Specialization to spectral curves provides alternative insights

Maximal variation confirmed in genus two cases

Abstract

We prove that curves in a non-primitive, 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 confirming a conjecture due to Matsushita.