| Let (S,H) be a general
primitively polarized K3 surface. I will talk about a joint work with
Concettina Galati, where we prove the existence of curves in the
complete linear system |nH| with A_k-singularities and living in
families of the "expected dimension". This generalizes results in the
nodal case of Mumford for n=1 and Chen for n>1. The proof is by degenerating the K3 surface to a union of rational normal scrolls and deform a higher order tacnode on a curve on the degerated surface. |