Explicit correspondences of a K3-surface with itself

摘要

Let X be a K3-surface with a polarization H of degree H 2rs, r, s >= 1. We consider the moduli space Y of sheaves over X with a primitive isotropic Mukai vector (r, H, s). This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice N(X)) for Y and X to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between Y and X which is a composite of certain universal geometric isomorphisms between moduli of sheaves over X and Tyurin's geometric isomorphism between moduli of sheaves over X and X itself. It follows that 14 a general K3-surface X with rho(X) = rk N(X) <= 2 is isomorphic to Y if and only if there is an isomorphism Y congruent to X which is a composite of universal isomorphisms and Tyurin's isomorphism.