reference query – Which texts (of differential geometry) should I read to understand Deligne-Mumford's article?

Until now, I have studied some fundamentals of algebraic geometry and algebraic number theory.
And then, in order to understand Fermat's last theorem, I want to study the stacks of curve modules.
So now I'm reading Deligne-Mumford's "The irreducibility of space curves of a given kind".

But his chapter 5 is very difficult.
I do not know why the authors used these theories (as what they called Dehn's theory and Teichmuller's theory).
It seems to me that these are the analogies of the fundamental techniques of differential geometry (in particular, I think, Riemann surface modules).
So I think I should study "classical" geometry, such as De Rham's cohomology and the fundamentals of differential geometry.

The question is therefore: suggest me some references to differential geometry, which suits me perfectly.

Thank you so much!