The Stacks project

Lemma 109.16.5. There is a decomposition into open and closed substacks

where each $\mathcal{M}_ g$ is characterized as follows:

1. given a family of curves $f : X \to S$ the following are equivalent

   1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{M}_ g$,

   2. $X \to S$ is smooth, $f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after any base change, and $R^1f_*\mathcal{O}_ X$ is a locally free $\mathcal{O}_ S$-module of rank $g$,

2. given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

   1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{M}_ g$,

   2. $X$ is smooth, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, and $X$ has genus $g$,

   3. $X$ is smooth, $\dim (X) = 1$, $X$ is geometrically connected, and $X$ has genus $g$,

   4. $X$ is smooth, $\dim (X) = 1$, $X$ is geometrically integral, and $X$ has genus $g$, and

   5. $X_{\overline{k}}$ is a smooth curve of genus $g$.