On birational transformations of Hilbert schemes of points on K3 surfaces

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk"ahler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.