$K3$-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.