Punctured Riemann surface moduli dimension (${\dim }_{\mathbb{C}}{\mathcal{M}}_{g,n}=3g-3+n$)

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 $g\ge 2$, 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.