The Two Extremal Rays of Some Hyper–Kähler Fourfolds

We consider projective Hyper–Kähler manifolds of dimension 4 that are deformation equivalent to Hilbert squares of K3 surfaces. In case such a manifold admits a divisorial contraction, the exceptional divisor is a conic bundle over a K3 surface. A classification of lattice embeddings implies that there are five types of such conic bundles. In case the manifold has Picard rank 2 and has two (birational) divisorial contractions, we determine the types of these conic bundles. There are exactly seven cases. For the Fano varieties of lines of cubic fourfolds there are only four cases and we provide examples of these.