Gorenstein Braid Cones and Crepant Resolutions

To any poset $P$, we associate a convex cone called a braid cone. We also associate a fan and study the toric varieties the cone and fan define. The fan always defines a smooth toric variety $X_P$, while the toric variety $U_P$ of the cone may be singular. We show that $X_{P} \dashrightarrow U_{P}$ is a crepant resolution of singularities if and only if $P$ is bounded. Next, we aim to determine when $U_P$ is Gorenstein or $\mathbb{Q}$-Gorenstein. We prove that whether or not $U_P$ is ($\mathbb{Q}$)-Gorenstein depends only on the biconnected components of the Hasse diagram of $P$. In the case that $P$ has a minimum or maximum element, we show that the Gorenstein property of $U_P$ is completely determined by the M\"obius function of $P$. We also provide a recursive method that determines if $U_P$ is ($\mathbb{Q}$)-Gorenstein in this case. We conjecture that $U_P$ is Gorenstein if and only if it is $\mathbb{Q}$-Gorenstein. We verify this conjecture for posets of length $1$ and also for posets with a minimum or maximum element.