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What are the different coordinates systems that have been considered on the moduli spaces of curves $\mathcal M_{g,n}$ (or $\overline{\mathcal M}_{g,n}$)? What are their main properties? How are they related?

References are welcome.

Thanks.

Ps: my main interest concerns the case when $n=0$.

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  • $\begingroup$ Depending on your definition of "coordinate system" you may be out of luck in most cases, since the moduli spaces transition to general type for large $g$. $\endgroup$
    – S. Carnahan
    May 31, 2013 at 10:05
  • $\begingroup$ If by "coordinate system" you mean "local isomorphism with an analytic open subset of affine space", then I suggest you look up "Teichmueller theory". The universal cover of $\overline{M}_g$ (in the orbifold sense) is naturally an open ball in an affine space. $\endgroup$ May 31, 2013 at 13:57
  • $\begingroup$ Correction: $\overline{M}_g$ --> $M_g$ (force of habit). $\endgroup$ May 31, 2013 at 13:57

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