Smoothness of Modulus space

Question

Let $M_g$ be the modulus space of genus $g$ Riemann surfaces. I know that $M_g$ is of dimension $3g-3$, which is also equal to the dimension of $H^1(T_X)$ for any Riemann surface $X$ (which is supposed to be the tangent space of $M_g$ at $X$). This suggests that $M_g$ is smooth. Yet, the comments in this question suggest that $M_g$ is singular. What is going on here?

Answer 1

The moduli stack $\mathcal{M}_g$ of smooth proper geometrically connected curves of genus $g$ is smooth. Its coarse space $M_g$ (which is sometimes also referred to as the moduli space of genus $g$ curves) is a variety which "approximates" the moduli stack the best way an algebraic variety could; its dimension equals the dimension of $\mathcal{M}_g$ (which is $3g-3$). However, this coarse space is not really what we want the moduli of genus $g$ curves to be. Certainly, it suffices to consider the space $M_g$ to study certain problems, but the more natural object to consider is the stack $\mathcal{M}_g$. It is the variety $M_g$ which is singular when $g>1$.

The reason the stack $\mathcal{M}_g$ is smooth is precisely by what you say (essentially). Its tangent space at an object corresponding to a curve $X$ is given by $\mathrm{H}^1(X,T_X)$ and this has the right dimension, as you note, so the stack is smooth (for this to make sense, you'd have to familiarize yourself with the language of algebraic stacks). On the other hand, the tangent space to the point $[X]$ of the coarse space corresponding to $X$ is not $\mathrm{H}^1(X,T_X)$ in general.