Lemma 109.11.3 (0E1K)—The Stacks project

Lemma 109.11.3. There is a decomposition into open and closed substacks

\[ \mathcal{C}\! \mathit{urves}^{grc, 1} = \coprod \nolimits _{g \geq 0} \mathcal{C}\! \mathit{urves}^{grc, 1}_ g \]

where each $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$ is characterized as follows:

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 $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,
2. the geometric fibres of the morphism $f : X \to S$ are reduced, connected, of dimension $1$ and $R^1f_*\mathcal{O}_ X$ is a locally free $\mathcal{O}_ S$-module of rank $g$,
given a scheme $X$ 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 $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,
2. $X$ is geometrically reduced, geometrically connected, has dimension $1$, and has genus $g$.

Proof. First proof: set $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g = \mathcal{C}\! \mathit{urves}^{grc, 1} \cap \mathcal{C}\! \mathit{urves}_ g$ and combine Lemmas 109.11.2 and 109.9.4. Second proof: The existence of the decomposition into open and closed substacks follows immediately from the discussion in Section 109.6 and Lemma 109.11.2. This proves the characterization in (1). The characterization in (2) follows as well since the genus of a geometrically reduced and connected proper $1$-dimensional scheme $X/k$ is defined (Algebraic Curves, Definition 53.8.1 and Varieties, Lemma 33.9.3) and is equal to $\dim _ k H^1(X, \mathcal{O}_ X)$. $\square$

The Stacks project

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