Local structure of hyperkähler singularities in O'Grady's examples

We study the local structure of the singularity in the moduli space of sheaves on a K3 surface which has been resolved by K. O'Grady in his construction of new examples of hyperkaehler manifolds. In particular, we identify the singularity with the closure of a certain nilpotent orbit in the coadjoint representation of the group ${\rm Sp}(4)$. We also prove that the moduli spaces for some other sets of numerical parameters do not admit a smooth symplectic resolution of singularities.