Forgetful linear systems on the projective space and rational normal curves over ℳ0,2nGIT

Abstract

Let ℳ0,n be the moduli space of n-pointed rational curves. The aim of this note is to give a new geometric construction of ℳ 0 , 2 n GIT , the GIT compactification of ℳ0,2n, in terms of linear systems on ℙ2n−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 compactifications ℳ ― 0 , 2 n + 1 and ℳ ― 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 from ℳ ― 0 , 2 n to ℳ 0 , 2 n GIT and of the canonical forgetful maps. As a side result we also find a linear system on ℳ ― 0 , 2 n whose associated map is the contraction map c2n.