Let M_{0,n} be the moduli space of n-pointed rational curves. The aim of my talk is to give a new, geometric construction of M_{0,2n}^{GIT}, the GIT compactification of M_{0,2n}, in terms of linear systems on P^{2n-2} that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps \bar{M}_{0,2n+1} ----> \bar{M}_{0,2n} and have a deep relation with the Cremona inversion of the projective plane. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactication \bar{M}_{0,2n} to M_{0,2n}^{GIT} and of the natural forgetful maps.