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

TL;DR

This paper describes the functor of toric varieties linked to Weyl chambers of root systems and demonstrates that for type A root systems, the associated toric variety is the moduli space of stable pointed chains of projective lines, providing a new proof.

Contribution

It offers a simple description of the functor of toric varieties from root systems and connects these to Losev-Manin moduli spaces for type A root systems.

Findings

The functor of $X(R)$ can be described in terms of the root system $R$.

For type $A$, $X(A_n)$ is the moduli space $ar{L}_{n+1}$ of stable pointed chains.

Provides a new proof of the isomorphism between $X(A_n)$ and $ar{L}_{n+1}$.

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 $\bar{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.