Inertia groups of a toric Deligne-Mumford stack, fake weighted projective stacks, and labeled sheared simplices

This paper determines the inertia groups (isotropy groups) of the points of a toric Deligne-Mumford stack $[Z/G]$ (considered over the category of smooth manifolds) that is realized from a quotient construction using a stacky fan or stacky polytope. The computation provides an explicit correspondence between certain geometric and combinatorial data. In particular, we obtain a computation of the connected component of the identity element $G_0 \subset G$ and the component group $G/G_0$ in terms of the underlying stacky fan, enabling us to characterize the toric DM stacks which are global quotients. As another application, we obtain a characterization of those stacky polytopes that yield stacks equivalent to weighted projective stacks and, more generally, to \textit {`fake' weighted projective stacks}. Finally, we illustrate our results in detail in the special case of \textit {labeled sheared simplices}, where explicit computations can be made in terms of the facet labels.