Inertia groups of a toric DM stack, fake weighted projective spaces, and labelled 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 `fake' weighted projective stacks. Finally, we illustrate our results in detail in the special case of labelled sheared simplices, where explicit computations can be made in terms of the facet labels.