$K3$-surfaces play the role of elliptic curves in the realm of algebraic surfaces. They are sophisticated enough to produce interesting and meaningful results that may hint possible generalizations, yet simple enough to make their study feasible. One remarkable feature of $K3$-surfaces is that, among all complete intersections of dimension at least two, they are the only ones whose group of projective automorphisms may (and typically is) much smaller than their group of birational automorphisms.

I will discuss a particular example of sextic $K3$-surfaces and a particular construction of non-projective automorphisms, related to lines. In particular, it will be shown that, whenever a sextic has at least two lines, its group of birational automorphisms is infinite.

This is a joint work with Igor Dolgachev, Shigeyuki Kondo, and Slawomir Rams.