= Punctured Riemann surface moduli dimension
{title2=$\dim_{\mathbb C}\mathcal M_{g,n}=3g-3+n$}
For stable compact <Riemann surfaces> with $n$ distinct labelled marked points, where $2g-2+n>0$, the displayed complex dimension counts independent conformal shapes and positions after conformal automorphisms are removed. On the sphere it is $n-3$, since three points fix a <Möbius transformation>. On a torus it is $1+n-1=n$: one shape modulus and $n$ positions, less one translation. For <genus> $g\ge2$, the unmarked shape dimension is $3g-3$ and continuous conformal automorphisms vanish. The shape dimension also follows from the <Riemann-Roch theorem> applied to <holomorphic quadratic differentials>.
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