In this talk we study the birational geometry of X_sⁿ the blow-up of the n-dimensional complex projective space at s points in (very) general position, encoded in the Mori chamber decomposition (MCD) of its effective cone.  It is known following work of Mukai that when X_sⁿ is a Mori dream space, the Mori chamber decomposition coincides with the stable base locus de- composition and they are given by a hyperplane arrangement. We show that the set of curve classes that are dual to such hyperplanes are governed by the action of the Weyl group, yield- ing the equivalent notion of Weyl chamber decomposition.  This viewpoint can be naturally extended to the cases where X_sⁿ is not a Mori dream space. In this case we obtain an infinitie wall-and-chamber decomposition that partially describes the MCD.
This is joint work with Maria Chiara Brambilla, Olivia Dumitrescu and Elisa Postinghel.