I've just started studying algebraic stacks and I have a very basic question.

I've learned the notion of Deligne Mumford stack and I've seen as the stack of stable curves $\overline{\mathcal{M}_g}$ is one.

What I would like to understand is why "locally $\overline{\mathcal{M}_g}$ can be seen as the universal deformation of a stable curve". Can you suggest a good reference for that?

Thank you

  • $\begingroup$ Isn't this just the definition of the moduli stack? Look at the Deligne-Mumford paper, it contains everything you want (and more). $\endgroup$ – abx Jan 19 '15 at 17:07
  • $\begingroup$ Over an etale cover by a scheme (the existence of which is the content of it being a DM stack), the pullback of the "universal curve" over the complete local ring at any point is the (algebraization of) the universal deformation of the fiber curve based at that point. $\endgroup$ – user74230 Jan 19 '15 at 17:31
  • $\begingroup$ Deformation theory from the point of view of fibered categories Mattia Talpo, Angelo Vistoli arxiv.org/abs/1006.0497 $\endgroup$ – Niels Jan 19 '15 at 21:23

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