Toric Varieties from Root Systems of Type A

Given a root system $R$ with Weyl chambers $\Sigma$, we can consider $\Sigma$ as a complete fan and build a toric variety $X$ from it. After briefly defining the objects I mentioned, I will walk us through this process with a few root systems of Type $A$. After that, we will see some of the consequences of this construction and an amazing connection with geometry: as it turns out, the toric variety of the root system $A_{n-1}$ coincides with the Losev--Manin moduli space $L_n$, which parametrizes chains of projective lines marked with $n$ points that may collide and 2 points that may not. This is meant as an introductory talk friendly to graduate students familiar with commutative algebra + varieties.