109.9 Curves of a given genus
The convention in the Stacks project is that the genus $g$ of a proper $1$-dimensional scheme $X$ over a field $k$ is defined only if $H^0(X, \mathcal{O}_ X) = k$. In this case $g = \dim _ k H^1(X, \mathcal{O}_ X)$. See Algebraic Curves, Section 53.8. The conditions needed to define the genus define an open substack which is then a disjoint union of open substacks, one for each genus.
Lemma 109.9.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{h0, 1} \subset \mathcal{C}\! \mathit{urves}$ such that
given a family of curves $f : X \to S$ the following are equivalent
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{h0, 1}$,
$f_*\mathcal{O}_ X = \mathcal{O}_ S$, this holds after arbitrary base change, and the fibres of $f$ have dimension $1$,
given a scheme $X$ proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{h0, 1}$,
$H^0(X, \mathcal{O}_ X) = k$ and $\dim (X) = 1$.
Proof.
Given a family of curves $X \to S$ the set of $s \in S$ where $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ is open in $S$ by Derived Categories of Spaces, Lemma 75.26.2. Also, the set of points in $S$ where the fibre has dimension $1$ is open by More on Morphisms of Spaces, Lemma 76.31.5. Moreover, if $f : X \to S$ is a family of curves all of whose fibres have dimension $1$ (and in particular $f$ is surjective), then condition (1)(b) is equivalent to $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ for every $s \in S$, see Derived Categories of Spaces, Lemma 75.26.7. Thus we see that the lemma follows from the general discussion in Section 109.6.
$\square$
Lemma 109.9.2. We have $\mathcal{C}\! \mathit{urves}^{h0, 1} \subset \mathcal{C}\! \mathit{urves}^{CM, 1}$ as open substacks of $\mathcal{C}\! \mathit{urves}$.
Proof.
See Algebraic Curves, Lemma 53.6.1 and Lemmas 109.9.1 and 109.8.2.
$\square$
Lemma 109.9.3. Let $f : X \to S$ be a family of curves such that $\kappa (s) = H^0(X_ s, \mathcal{O}_{X_ s})$ for all $s \in S$, i.e., the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{h0, 1}$ (Lemma 109.9.1). Then
$f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds universally,
$R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module,
for any morphism $h : S' \to S$ if $f' : X' \to S'$ is the base change, then $h^*(R^1f_*\mathcal{O}_ X) = R^1f'_*\mathcal{O}_{X'}$.
Proof.
We apply Derived Categories of Spaces, Lemma 75.26.7. This proves part (1). It also implies that locally on $S$ we can write $Rf_*\mathcal{O}_ X = \mathcal{O}_ S \oplus P$ where $P$ is perfect of tor amplitude in $[1, \infty )$. Recall that formation of $Rf_*\mathcal{O}_ X$ commutes with arbitrary base change (Derived Categories of Spaces, Lemma 75.25.4). Thus for $s \in S$ we have
\[ H^ i(P \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s)) = H^ i(X_ s, \mathcal{O}_{X_ s}) \text{ for }i \geq 1 \]
This is zero unless $i = 1$ since $X_ s$ is a $1$-dimensional Noetherian scheme, see Cohomology, Proposition 20.20.7. Then $P = H^1(P)[-1]$ and $H^1(P)$ is finite locally free for example by More on Algebra, Lemma 15.75.7. Since everything is compatible with base change we also see that (3) holds.
$\square$
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
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}_ g$,
$f_*\mathcal{O}_ X = \mathcal{O}_ S$, 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
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}_ g$,
$\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, 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.
$\square$
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