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Recently I've coincidentally run into various relatives of the moduli stack $\mathcal{M}_g$ in several unrelated contexts. I tried reading Harris and Morrison's "Moduli of Curves," but it seems to require a substantial amount of foundation just to read the terminology. I can follow the definitions, of course, and I do want to learn all of the foundation in the long run, but I get the impression that, at least in some cases, these things are pretty straightforward, almost classical geometric objects, which leads me to wonder if I could learn about moduli of curves first with lighter machinery.

Is there a quick way to get a feel for these things -- possibly a text that constructs them using less machinery? Or should I just sit down and learn about schemes and stacks and representable functors and such? (I know the definitions of these things, but I imagine it takes some time to really get used to them; for instance, I can follow a proof using Yoneda's Lemma line-by-line, but I don't yet have a good intuition for what it's "really" for).

For the record, I also do not know any Teichmüller theory.

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You're probably looking for something that's longer and more detailed, but this is interesting: – Vitaly Lorman Dec 29 '11 at 1:41
It depends. If you really, really are going to need to work with these things, then I definitely recommend learning about functors of points, schemes, and stacks to the point of understanding why stacks are needed (which I don't think is that much actually). Historically speaking, it is exactly trying to think about these things purely geometrically that got people into trouble. – Matt Sep 18 '12 at 13:49

P.E.Newstead's "Lectures on Moduli Problems" at TIFR may be downloaded,I think.Please visit the publications of the ICTP esp the Workshop on Moduli 1999.

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Could you please provide a link? – draks ... Sep 24 '12 at 10:29
@draks I think he is refering to, and I don't think you can dowload it legally. – M Turgeon Sep 26 '12 at 16:05

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