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By Section 8.3.1 of the book: A primer on mapping class groups by Farb and Margalit, a pair of pants is a compact surface of genus 0 with three boundary components. Let $S$ be a compact surface with $\chi(S)<0$. A pair of pants decomposition of $S$, or a pants decomposition of $S$, is a collection of disjoint simple closed curves in $S$ with the property that when we cut $S$ along these curves, we obtain a disjoint union of pairs of pants. Equivalently, a pants decomposition of $S$ is a maximal collection of disjoint, essential simple closed curves in $S$ with the property that no two of these curves are isotopic. In particular, a pants decomposition of $\Sigma_g$ (the orientable closed surface of genus $g$) for $g>1$ has $3g-3$ curves, cutting $\Sigma_g$ into $2g-2$ pairs of pants.

On the other hand, according to Riemann's computation, the dimension of the moduli space $\mathcal{M}_g$ of algebraic curves of genus $g>1$ is $3g-3$. See, for example, the book: Algebraic curves towards moduli spaces.

My question: why are the number of curves in the pants decomposition of $\Sigma_g$ for $g>1$ and the dimension of the moduli space of $\Sigma_g$ for $g>1$ the same number $3g-3$? Is there any deep relation?

Thank you!

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  • $\begingroup$ Are you looking for a deeper relation than the Teichmüller space? $\endgroup$
    – Aurelio
    Oct 11, 2020 at 9:47
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    $\begingroup$ Shaking hands. Let take real dimension, so the number of curves is $3g-3$, and the dimension is $6g-6$. The moduli essentially consist in choosing the size $\ell_c$ of these $3g-3$ curves $c$, and choose an angle $\alpha_c\in\mathbf{R}/\ell_c\mathbf{Z}$ of pasting along each of these curves, which gives $6g-6$ parameters of freedom (which can all be achieved since one can freely prescribe lengths of boundary curves of pants). A complete proof along these way of thoughts can be found in P. Buser, Geometry of Riemann surfaces. $\endgroup$
    – YCor
    Oct 11, 2020 at 10:07

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I am unsure if this is what you are looking for, but the moduli space $\mathcal{M}_g$ can be viewed as an orbifold quotient of the Teichmüller space by the mapping class group. I cannot offer more details, but this is explained in a paper by John H. Hubbard and Sarah Koch.

EDIT I notice that this has been addressed in another question as well.

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  • $\begingroup$ How does that explain the relation with the number of simple curves? $\endgroup$
    – abx
    Oct 11, 2020 at 10:28
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    $\begingroup$ @abx Indeed I did not give an explanation of the number of simple curves: this is described in the comment above by YCor, which I understand refers to the Fenchel–Nielsen coordinates. I was aiming at giving some insight into why the Teichmüller space and the Riemann moduli space have the same dimension. This is the "deep relation" that I assume the OP was looking for. $\endgroup$
    – Aurelio
    Oct 11, 2020 at 10:40
  • $\begingroup$ this is an essentially trivial relation. I don't think this is the sense of the question. $\endgroup$
    – abx
    Oct 11, 2020 at 11:50
  • $\begingroup$ @abx I agree that the relation is essentially trivial, but the Teichmüller space is not referred to in the original question, so I assumed it was worth mentioning. $\endgroup$
    – Aurelio
    Oct 11, 2020 at 11:54

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