  Let $\cM_{0,n}$ the moduli space of $n$-pointed rational curves. The aim of
this note is to give a new, geometric construction of $\cM_{0,2n}^{GIT}$, the
GIT compacification of $\cM_{0,2n}$, in terms of linear systems on $\PP^{2n-2}$
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 between $\bar{\cM}_{0,2n+1}$ and $\bar{\cM}_{0,2n}$. The
construction is performed via a study of the so-called $\textit{contraction}$
maps from the Knudsen-Mumford compactification $\bar{\cM}_{0,2n}$ to
$\cM_{0,2n}^{GIT}$ and of the canonical forgetful maps. As a side result we
also find a linear system on $\bar{\cM}_{0,2n}$ whose associated map is the
contraction map $c_{2n}$.

Algebraic Geometry