The moduli stacks of smooth and stable curves

$\mathcal M_g$ is smooth of dimension $3g-3$ but not proper; adding stable curves gives the smooth proper compactification $\overline{\mathcal M}_g$.Suggest a slogan

(Deligne--Mumford)

Let $g \geq 2$.

The moduli stack $\mathcal{M}_g$ of smooth projective curves of genus $g$ is a smooth Deligne–Mumford stack over $\Spec \ZZ$ of relative dimension $3g - 3$, with irreducible geometric fibres.
$\mathcal{M}_g$ is not proper.
It is an open dense substack of the moduli stack $\overline{\mathcal{M}}_g$ of stable curves, which is smooth and proper over $\Spec \ZZ$.

The coarse moduli space $M_g$ is irreducible of dimension $3g-3$ but is singular for $g \geq 4$, at curves with extra automorphisms; smoothness is a property of the stack.

Why is the moduli space of curves $\mathcal{M}_g$ smooth? Is it proper?

Dependencies

T-DEFEXTInfinitesimal automorphisms and first-order deformationsuses

Backlinks

D-ALGSTACKAlgebraic stacks, substacks, smooth stacks, and orbifoldsrelated-to
D-STABLEMAPStable mapsrelated-to

What links to this

Moduli of curves