The morphism
is smooth over a maximal open substack, see Morphisms of Stacks, Lemma 101.33.6. We want to give a criterion for when a curve is in this locus. We will do this using a bit of deformation theory.
Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Choose a Cohen ring $\Lambda $ for $k$, see Algebra, Lemma 10.160.6. Then we are in the situation described in Deformation Problems, Example 93.9.1 and Lemma 93.9.2. Thus we obtain a deformation category $\mathcal{D}\! \mathit{ef}_ X$ on the category $\mathcal{C}_\Lambda $ of Artinian local $\Lambda $-algebras with residue field $k$.
Lemma 109.15.1. In the situation above the following are equivalent
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through the open where $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,
the deformation category $\mathcal{D}\! \mathit{ef}_ X$ is unobstructed.
Proof. Since $\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation (Lemma 109.5.3) formation of the open substack where $\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth commutes with flat base change (Morphisms of Stacks, Lemma 101.33.6). Since the Cohen ring $\Lambda $ is flat over $\mathbf{Z}$, we may work over $\Lambda $. In other words, we are trying to prove that
is smooth in an open neighbourhood of the point $x_0 : \mathop{\mathrm{Spec}}(k) \to \Lambda \text{-}\mathcal{C}\! \mathit{urves}$ defined by $X/k$ if and only if $\mathcal{D}\! \mathit{ef}_ X$ is unobstructed.
The lemma now follows from Geometry of Stacks, Lemma 107.2.7 and the equality
This equality is not completely trivial to esthablish. Namely, on the left hand side we have the deformation category classifying all flat deformations $Y \to \mathop{\mathrm{Spec}}(A)$ of $X$ as a scheme over $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$. On the right hand side we have the deformation category classifying all flat morphisms $Y \to \mathop{\mathrm{Spec}}(A)$ with special fibre $X$ where $Y$ is an algebraic space and $Y \to \mathop{\mathrm{Spec}}(A)$ is proper, of finite presentation, and of relative dimension $\leq 1$. Since $A$ is Artinian, we find that $Y$ is a scheme for example by Spaces over Fields, Lemma 72.9.3. Thus it remains to show: a flat deformation $Y \to \mathop{\mathrm{Spec}}(A)$ of $X$ as a scheme over an Artinian local ring $A$ with residue field $k$ is proper, of finite presentation, and of relative dimension $\leq 1$. Relative dimension is defined in terms of fibres and hence holds automatically for $Y/A$ since it holds for $X/k$. The morphism $Y \to \mathop{\mathrm{Spec}}(A)$ is proper and locally of finite presentation as this is true for $X \to \mathop{\mathrm{Spec}}(k)$, see More on Morphisms, Lemma 37.10.3. $\square$
Here is a “large” open of the stack of curves which is contained in the smooth locus.
Lemma 109.15.2. The open substack has the following properties
$\mathcal{C}\! \mathit{urves}^{lci+} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,
given a family of curves $X \to S$ the following are equivalent
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,
$X \to S$ is a local complete intersection morphism and the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$,
given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,
$X$ is a local complete intersection over $k$ and $X \to \mathop{\mathrm{Spec}}(k)$ is smooth except at finitely many points.
\[ \mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \subset \mathcal{C}\! \mathit{urves} \]
Proof. If we can show that there is an open substack $\mathcal{C}\! \mathit{urves}^{lci+}$ whose points are characterized by (2), then we see that (1) holds by combining Lemma 109.15.1 with Deformation Problems, Lemma 93.16.4. Since
inside $\mathcal{C}\! \mathit{urves}$, we conclude by Lemmas 109.13.1 and 109.14.1. $\square$
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