TITLE: Intuition behind moduli space of curves
QUESTION [34 upvotes]: For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the moduli space of conformal structures on $M$. A conformal structure is an equivalence class of Riemmanian metrics that give the same angle measurements.
If I embed $M$ in $\mathbb{R}^3$, I get a metric on $M$. This gives me a conformal structure, hence a point in the moduli space. The moduli space is known to have complex dimension 3g-3 (except for g=0, where the moduli space is a point, and g=1, where the moduli space is 1-dimensional).
My question is: can we visualize the 6g-6 real dimensions as deformations of the embedding of $M$ in $\mathbb{R}^3$. In particular, how can we see that small deformations of a 2-sphere are conformally equivalent to the original 2-sphere.

REPLY [5 votes]: This is related to Kevin's answer but goes back to the 19th century.  A Riemann surface is planar if a simple closed curve separates it.  A Riemann surface of genus g becomes planar after cutting g handles along simple closed curves.  Then the classical uniformaization theorem states that it is equivalent to a sphere with 2g slits parallel to the "x axis".  Then the location of the slits is determine by the 2g centers (4g real coordinates) and the 2g lengths (2g real coordinates).  This give 6g real coordinates, which are then reduced by the 6 real dimensions of the conformal automorphisms of the sphere.  Thus Riemann surfaces of genus g have 6g-6 real parameters, plus the number of parameters of the conformal automorphisms of the generic surface of genus g, which adds 6 for g=0 and 2 for g=1.  (Alan Mayer explained it this way in a colloquium at Brandeis about 1967.)
As usual, induction is the easiest way to see anything, and here one degenerates a Riemann surface to one of lower genus by acquiring a node (shrinking off a loop).  Then the lower genus Riemann surface has 3(g-1)-3 complex parameters. the two identified points have 2 parameters, and we add one more for the degeneration direction.  (Such degeneration methods occur in Wirtinger's book on theta functions in 1895.)