Lemma 109.9.4 (0E6K)—The Stacks project

Lemma 109.9.4. There is a decomposition into open and closed substacks

\[ \mathcal{C}\! \mathit{urves}^{h0, 1} = \coprod \nolimits _{g \geq 0} \mathcal{C}\! \mathit{urves}_ g \]

where each $\mathcal{C}\! \mathit{urves}_ g$ 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 $\mathcal{C}\! \mathit{urves}_ g$,
2. $f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after arbitrary base change, the fibres of $f$ have 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 $\mathcal{C}\! \mathit{urves}_ g$,
2. $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, and the genus of $X$ is $g$.

Proof. We already have the existence of $\mathcal{C}\! \mathit{urves}^{h0, 1}$ as an open substack of $\mathcal{C}\! \mathit{urves}$ characterized by the conditions of the lemma not involving $R^1f_*$ or $H^1$, see Lemma 109.9.1. The existence of the decomposition into open and closed substacks follows immediately from the discussion in Section 109.6 and Lemma 109.9.3. This proves the characterization in (1). The characterization in (2) follows from the definition of the genus in Algebraic Curves, Definition 53.8.1. $\square$

The Stacks project

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