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Forgetful linear systems on the projective space and rational normal curves over $\cM_{0,2n}^{GIT}$

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 $c_{2n}$.