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Uniformization of Deligne-Mumford curves

K. Behrend·B. Noohi
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Abstract

We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks $\mathcal{P}(n_1,n_2,...,n_r)$. We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups.

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K. Behrend, B. Noohi. 2005-04-14. Uniformization of Deligne-Mumford curves. https://arxiv.org/abs/math/0504309

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