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@Jason: Also it seems from later on in Kawamata's paper, namely at the end of the proof of Theorem 6.5 on p. 22 (of the version on the Arxiv) where he says explicitly that the canonical covering stack restricted to one of the $U_x$ is the quotient stack $[\tilde{U}_x/\mu_{m_x}]$. So is his stack really the same as your $\mathcal X_m$? I admit that maybe your $\mathcal X_m$ (and not $\mathcal R$) is actually just Cadman's $X_{L,s,r}$. Is this the case?
@Jason: According to paper of Cadman you're referring to, he indicates in Example 2.4.1 that his root stack, which I'm guessing is your $\mathcal R$, is covered by the quotient stacks $[\tilde{U}_x/\mu_{m_x}]$, as the OP asked about. Is your $\mathcal{X}_m$ then more closely associated to the $\tilde{U}_x$? I'm just trying to see how your description fits in with Kawamata's. For example can you describe an atlas for each of these?
Why is the fiber product of finite type over $M$? Are you claiming that since $S$ is of finite-type over $\mathbb C$, it must then also be of finite-type over $\mathfrak M$, and then base-change?
Is $\phi$ the map defining the family or something in the radial direction? Is there an easy way to see that the variation map factors in this way, or a reference for it? Thank you both for your good answers. I'm starting to get the picture, and I think this will definitely help with what I'm working on. A follow-up question is if the vanishing spheres are independent in homology. More to the point, do you know what $Rank(S-I)$ is? It seems to depend on the number of nodes that appear, unless there's some relation between these spheres.
