Explicit correspondences of a K3 surface with itself

Abstract

Let be a K3-surface with a polarization of degree , . We consider the moduli space of sheaves over with a primitive isotropic Mukai vector . This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice ) for and to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between and which is a composite of certain universal geometric isomorphisms between moduli of sheaves over and Tyurin's geometric isomorphism between moduli of sheaves over and itself. It follows that a general K3-surface with is isomorphic to if and only if there is an isomorphism which is a composite of universal isomorphisms and Tyurin's isomorphism.