109.11 Geometrically reduced and connected curves
There is an open substack of $\mathcal{C}\! \mathit{urves}$ parametrizing the geometrically reduced and connected “curves”. We will get rid of $0$-dimensional objects right away.
Lemma 109.11.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{grc, 1} \subset \mathcal{C}\! \mathit{urves}$ such that
given a family of curves $X \to S$ the following are equivalent
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{grc, 1}$,
the geometric fibres of the morphism $X \to S$ are reduced, connected, and 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 $\mathcal{C}\! \mathit{urves}^{grc, 1}$,
$X$ is geometrically reduced, geometrically connected, and has dimension $1$.
Proof.
By Lemmas 109.10.1, 109.10.2, 109.8.1, and 109.8.2 it is clear that we have
\[ \mathcal{C}\! \mathit{urves}^{grc, 1} \subset \mathcal{C}\! \mathit{urves}^{geomred} \cap \mathcal{C}\! \mathit{urves}^{CM, 1} \]
if it exists. Let $f : X \to S$ be a family of curves such that $f$ is Cohen-Macaulay, has geometrically reduced fibres, and has relative dimension $1$. By More on Morphisms of Spaces, Lemma 76.36.9 in the Stein factorization
\[ X \to T \to S \]
the morphism $T \to S$ is étale. This implies that there is an open and closed subscheme $S' \subset S$ such that $X \times _ S S' \to S'$ has geometrically connected fibres (in the decomposition of Morphisms, Lemma 29.48.5 for the finite locally free morphism $T \to S$ this corresponds to $S_1$). Formation of this open commutes with arbitrary base change because the number of connected components of geometric fibres is invariant under base change (it is also true that the Stein factorization commutes with base change in our particular case but we don't need this to conclude). Thus we get the open substack with the desired properties by the method discussed in Section 109.6.
$\square$
Lemma 109.11.2. We have $\mathcal{C}\! \mathit{urves}^{grc, 1} \subset \mathcal{C}\! \mathit{urves}^{h0, 1}$ as open substacks of $\mathcal{C}\! \mathit{urves}$. In particular, given a family of curves $f : X \to S$ whose geometric fibres are reduced, connected and of dimension $1$, then $R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module whose formation commutes with arbitrary base change.
Proof.
This follows from Varieties, Lemma 33.9.3 and Lemmas 109.9.1 and 109.11.1. The final statement follows from Lemma 109.9.3.
$\square$
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
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,
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
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{grc, 1}_ g$,
$X$ 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 $\dim _ k H^1(X, \mathcal{O}_ X)$.
$\square$
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