Self-Correspondences of K3 Surfaces via Moduli of Sheaves

Abstract

Let X be an algebraic K3 surface with Picard lattice N(X), and M X (v) the moduli space of sheaves on X with given primitive isotropic Mukai vector v = (r, H, s). In [14] and [3], we described all the divisors in moduli of polarized K3 surfaces (X, H) (that is, all pairs H ∈ N(X) with rank N(X) = 2) for which M X (v)≅X. These provide certain Mukai self-correspondences of X. Applying these results, we show that there exists a Mukai vector v and a codimension-2 subspace in moduli of (X, H) (that is, a pair H ∈ N(X) with rank N(X) = 3) for which M X (v)≅X, but such that this subspace does not extend to a divisor in moduli having the same property. There are many similar examples. Aiming to generalize the results of [14] and [3], we discuss the general problem of describing all subspaces of moduli of K3 surfaces with this property, and the Mukai self-correspondences defined by these and their composites, in an attempt to outline a possible general theory.