The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

A root system R of rank n defines an n-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system R and apply this result in the case of root systems of type A to give a new proof of the fact that the toric variety X(Aₙ) is the fine moduli space L̄ₙ₊₁ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.