# Smoothness of the Hilbert Scheme of genus $g$ $n$-canonically embedded stable curves

(My notation and my question are from Deligne-Mumford's famous 1969 paper.)

A genus $g$ $n$-canonically embedded stable curve $C$ over a field $k$ with $n\geq 3$ has two related "local" moduli spaces. The first is the universal deformation space $\mathcal M$, where $\mathcal M\cong \text{Spec } \mathfrak o_k[[t_1,...,t_N]]$, where $\mathfrak o_k$ is $k$ if the char. $k=0$ and the complete regular local ring with maximal ideal $p.\mathfrak o_k$ and residue field $k$, if char. $k=p>0$, and $N=\dim_k \text{Ext}^1(\Omega_C,\mathcal O_C)$. This has a universal family of deformations $\mathcal C$.

The second is $T=\text{Spec }(\hat{\mathcal O}_x)$, where $x=[C]\in H_g$ is the point representing $C$ in the Hilbert scheme of genus $g$ $n$-canonically embedded stable curves, and we've taken the completion of the local ring of the Hilbert scheme at the point $x$. Denote by $Z_g\subset H_g\times \mathbb P$ the universal family over $H_g$. The natural morphism $T\rightarrow H_g$ induces the pull-back family $Z'\subset T\times \mathbb P$ whose closed fiber is of course $C$.

Now from the definition of the deformation functor and universality of the deformation space we get a unique morphism $f:T\rightarrow \mathcal M$ such that $Z'\cong \mathcal C\times_{\mathcal M} T$. Since $\pi_*(\omega_{\mathcal C/\mathcal M}^n)$ is locally free over a local scheme, it is in fact free, and thus we get an $n$-canonical embeddeding $\mathcal C\subset \mathcal M\times \mathbb P$, and thus a unique morphism $s: \mathcal M\rightarrow H_g$ such that $\mathcal C$ is the pull-back of $Z_g$. But then $s$ factors through $T$ and gives us that $s$ is a section of $f$.

From the above and the fact that the stabilizer of $x=[C]\in H_g$ under the natural $PGL$-action is finite and reduced, supposedly it follows that the action of $PGL$ on $T$ is "formally free" and thus that $T$ is formally a principle fibre bundle over $\mathcal M$ with structure group $PGL$.

My questions are: 1) What does "formally free" mean, and 2) why does it follow that $T$ is formally a principle fiber bundle over $\mathcal M$ and thus that $T$ is formally smooth over $\mathcal M$?

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