2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129383Under 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.Submitted preprint. Paper 1: On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi). Authors: Flaminio Flamini, Andreas Leopold Knutsen and Gianluca Pacienza. Pages: 1 -- 34. Figures: 1. Paper 2: Partial desingularizations of families of nodal curves. Author: Edoardo Sernesi. Pages: 35--37Algebraic GeometryPrimary: 14H10, 14H51, 14J28. Secondary 14C05, 14C25, 14D15, 14E30text