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

For any smooth surface S, the Hilbert scheme S^[n] parameterizing 0-dimensional length-n subschemes of S is a smooth 2n-dimensional variety whose inner geometry is naturally related to that of S. For instance, if E ⊂ S^[n] is the exceptional divisor—that is, the exceptional locus of the Hilbert–Chow morphism μ: S^[n] -> Sym^n(S) — then irreducible (possibly singular) rational curves not contained in E roughly correspond to irreducible (possibly singular) curves on S with a linear series of degree k and dimension 1 on their normalizations, for some k ≤ n. One of the features of this paper is to show how ideas and techniques from one of the two sides of the correspondence make it possible to shed light on problems naturally arising on the other side. If S is moreover a K3 surface then S^[n] is a hyperkähler manifold, and rational curves play a fundamental role in the study of the (birational) geometry of S^[n].