# Mathematics Colloquia and Seminars

A complex projective normal variety is said to be a Mori dream space if the Cox ring is finitely generated over the complex numbers. Mori dream paces were introduced and studied by Hu and Keel. From a Mori theory point of view they are the best possible varieties we can think of. A Mori dream space X is a GIT quotient of the affine variety Spec(Cox(X)) by the action of a torus and modifying the linearization one obtains all the small Q factorial modifications of X. We will describe all possible contractions from Mori dream spaces for blown-up projective space $P^n$ at $n+1$ points by investigating the structure of the movable cone, its decomposition into nef chambers and the action of the Weyl group on the set of nef chambers.