Cones of divisors of blow-ups of projective spaces

We investigate Mori dream spaces obtained by blowing-up the n-dimensional complex projective space at n + 1, n + 2 or n + 3 points in very general position. Using toric techniques we study the movable cone of the blow-up of Pn at n + 1 points, its decomposition into nef chambers and the action of the Weyl group on the set of chambers. Moreover, using different methods, we explicitly write down the equations of the movable cone also for Pn blown-up at n + 2 points.