## Tohoku Mathematical Journal 2011 December SECOND SERIES VOL. 63, NO. 4

 Tohoku Math. J. 63 (2011), 581-604

Title THE FUNCTOR OF TORIC VARIETIES ASSOCIATED WITH WEYL CHAMBERS AND LOSEV-MANIN MODULI SPACES

Author Victor Batyrev and Mark Blume

Abstract. 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 $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.