Maximal variation of curves on K3 surfaces

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.

19 pages. Comments welcome. Proper credit to the work of Ciliberto, Dedieu and Sernesi added

References in corpus (3)

• Fano threefolds and K3 surfaces
• $K3$ curves with index $k>1$
• La conjecture de Green generique [d'apres C. Voisin]

Cited by in corpus (2)

• Infinitesimal variation functions for families of smooth varieties
• General infinitesimal variations of Hodge structure of ample curves in surfaces