Mori Dream Spaces which are extremal contractions of K3 surfaces

The talk is centered on the question: "is it possible to obtain a Mori Dream Space by an extremal contraction of a K3 surface which is not a Mori Dream Space?" The answer is yes. From one hand, we will describe an algorithm to find the admissible N\'eron-Severi groups of the K3 surfaces which provide a positive answer to this question. From the other hand we will give an explicit geometric description of some examples.