The Stacks project

107.15 The smooth locus of the stack of curves

The morphism

\[ \mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z}) \]

is smooth over a maximal open substack, see Morphisms of Stacks, Lemma 99.32.6. 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 10.158.6. Then we are in the situation described in Deformation Problems, Example 91.9.1 and Lemma 91.9.2. 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 107.15.1. In the situation above the following are equivalent

  1. 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,

  2. 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 107.5.3) 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 99.32.6). Since the Cohen ring $\Lambda $ is flat over $\mathbf{Z}$, we may work over $\Lambda $. In other words, we are trying to prove that

\[ \Lambda \text{-}\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\Lambda ) \]

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 105.2.7 and the equality

\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{F}_{\Lambda \text{-}\mathcal{C}\! \mathit{urves}, k, x_0} \]

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 70.9.3. 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 107.15.2. The open substack

\[ \mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \subset \mathcal{C}\! \mathit{urves} \]

has the following properties

  1. $\mathcal{C}\! \mathit{urves}^{lci+} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,

  2. given a family of curves $X \to S$ the following are equivalent

    1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

    2. $X \to S$ is a local complete intersection morphism and the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$,

  3. given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent

    1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

    2. $X$ is a local complete intersection over $k$ and $X \to \mathop{\mathrm{Spec}}(k)$ is smooth except at finitely many points.

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 107.15.1 with Deformation Problems, Lemma 91.16.4. Since

\[ \mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \]

inside $\mathcal{C}\! \mathit{urves}$, we conclude by Lemmas 107.13.1 and 107.14.1. $\square$


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