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