Modular sheaves with many moduli

We exhibit moduli spaces of slope stable vector bundles on general polarized HK varieties $(X,h)$ of type K3[2] which have an irreducible component of dimension 2a²+2, with a an arbitrary integer greater than $1$. This is done by studying the case X=S[2] where S is an elliptic $K3$ surface. We show that in this case there is an irreducible component of the moduli space of stable vector bundles on S[2] which is birational to a moduli space of sheaves on S. We expect that if the moduli space of sheaves on S is a smooth HK variety (necessarily of type K3[a²+1]) then the following more precise version holds: the closure of the moduli space of slope stable vector bundles on $(X,h)$ in the moduli space of Gieseker-Maruyama semistable sheaves with its GIT polarization is a general polarized HK variety of type K3[a²+1].