On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi)

Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S)

0 via the study of the associated families of rational curves in Hilb^2(S). We use this result to prove the existence of nodal curves of geometric genus 3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining specific 2-dimensional families of rational curves in its Hilbert square. We describe two infinite series of examples of general, primitively polarized K3’s such that their Hilbert squares contain a IP^2 or a threefold birational to a IP^1-bundle over a K3. We discuss some consequences on the Mori cone of the Hilbert square of a general K3.