The study of configurations of rational curves on various surfaces has a long history. In the case of K3 surfaces one can mention Bogomolov and Mumford's proof of the fact that every complex projective K3 surface contains a (possibly singular) rational curve, and fairly recent result by Chen, Gounelas and Liedtke on the existence of infinitely many rational curves on every K3 surface over an algebraically closed field of characteristic zero.
The problem of rational curves assumes a different flavour when we consider quasi-polarized K3 surfaces of a fixed degree h (i.e. pairs (X,H), such that the Picard divisor H is big, nef, base-point free and non-hyperelliptic with self-intersection h) and take the degrees of the rational curves relative to the polarization H into account. In my talk I will present some upper bounds on the number of low degree rational curves on quasi-polarized K3 surfaces. In particular, if time allows me, I will sketch the proof of the fact that a complex projective K3 quartic with non-empty singular locus contains at most 52 lines (a sharp bound, joint work with A. Degtyarev (Ankara)) .
