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Let ℳ 0, n   be the moduli space of  n ‐pointed rational curves. The aim of this note is to give a new geometric construction ofℳ   0  ,  2  n    GIT, the GIT compactification of ℳ 0,2 n  , in terms of linear systems on ℙ 2 n −2  that contract all the rational normal curves passing through the points of a projective base. These linear systems are a projective analogue of the forgetful maps between the Mumford–Knudsen compactificationsℳ  ―    0  ,  2  n  +  1andℳ  ―    0  ,  2  n    , but on the other hand they contract some components of the boundary, yielding a rational map onto    ℳ   0  ,  2  n    GIT    . The construction is performed via a study of the so‐called  contraction  maps fromℳ  ―0  ,  2  nto    ℳ   0  ,  2  n    GITand of the canonical forgetful maps. As a side result we also find a linear system onℳ  ―0  ,  2  nwhose associated map is the contraction map  c  2 n  .

Forgetful linear systems on the projective space and rational normal curves over ℳ 0 , 2 n GIT