99.15 The stack of curves
In this section we prove the stack of curves is algebraic. For a further discussion of moduli of curves, we refer the reader to Moduli of Curves, Section 109.1.
A curve in the Stacks project is a variety of dimension $1$. However, when we speak of families of curves, we often allow the fibres to be reducible and/or nonreduced. In this section, the stack of curves will “parametrize proper schemes of dimension $\leq 1$”. However, it turns out that in order to get the correct notion of a family we need to allow the total space of our family to be an algebraic space. This leads to the following definition.
Situation 99.15.1. We define a category $\mathcal{C}\! \mathit{urves}$ as follows:
Objects are families of curves. More precisely, an object is a morphism $f : X \to S$ where the base $S$ is a scheme, the total space $X$ is an algebraic space, and $f$ is flat, proper, of finite presentation, and has relative dimension $\leq 1$ (Morphisms of Spaces, Definition 67.33.2).
A morphism $(X' \to S') \to (X \to S)$ between objects is given by a pair $(f, g)$ where $f : X' \to X$ is a morphism of algebraic spaces and $g : S' \to S$ is a morphism of schemes which fit into a commutative diagram
\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ S' \ar[r]^ g & S } \]
inducing an isomorphism $X' \to S' \times _ S X$, in other words, the diagram is cartesian.
The forgetful functor
\[ p : \mathcal{C}\! \mathit{urves}\longrightarrow \mathit{Sch}_{fppf},\quad (X \to S) \longmapsto S \]
is how we view $\mathcal{C}\! \mathit{urves}$ as a category over $\mathit{Sch}_{fppf}$ (see Section 99.2 for notation).
It follows from Spaces over Fields, Lemma 72.9.3 and more generally More on Morphisms of Spaces, Lemma 76.43.6 that if $S$ is the spectrum of a field, or an Artinian local ring, or a Noetherian complete local ring, then for any family of curves $X \to S$ the total space $X$ is a scheme. On the other hand, there are families of curves over $\mathbf{A}^1_ k$ where the total space is not a scheme, see Examples, Section 110.67.
It is clear that
99.15.1.1
\begin{equation} \label{quot-equation-curves-over-proper-spaces} \mathcal{C}\! \mathit{urves}\subset \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \end{equation}
and that an object $X \to S$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is in $\mathcal{C}\! \mathit{urves}$ if and only if $X \to S$ has relative dimension $\leq 1$. We will use this to verify Artin's axioms for $\mathcal{C}\! \mathit{urves}$.
Lemma 99.15.2. The category $\mathcal{C}\! \mathit{urves}$ is fibred in groupoids over $\mathit{Sch}_{fppf}$.
Proof.
Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.1) this reduces to the following statement: Given a morphism
\[ \xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]
in $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (recall that this implies in particular the diagram is cartesian) if $X \to S$ has relative dimension $\leq 1$, then $X' \to S'$ has relative dimension $\leq 1$. This follows from Morphisms of Spaces, Lemma 67.34.3.
$\square$
Lemma 99.15.3. The category $\mathcal{C}\! \mathit{urves}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$.
Proof.
Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.3) this reduces to the following statement: Given an object $X \to S$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ and an fppf covering $\{ S_ i \to S\} _{i \in I}$ the following are equivalent:
$X \to S$ has relative dimension $\leq 1$, and
for each $i$ the base change $X_ i \to S_ i$ has relative dimension $\leq 1$.
This follows from Morphisms of Spaces, Lemma 67.34.3.
$\square$
Lemma 99.15.4. The diagonal
\[ \Delta : \mathcal{C}\! \mathit{urves}\longrightarrow \mathcal{C}\! \mathit{urves}\times \mathcal{C}\! \mathit{urves} \]
is representable by algebraic spaces.
Proof.
This is immediate from the fully faithful embedding (99.15.1.1) and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.2).
$\square$
Lemma 99.15.6. The stack $\mathcal{C}\! \mathit{urves}\to \mathit{Sch}_{fppf}$ is limit preserving (Artin's Axioms, Definition 98.11.1).
Proof.
Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.6) this reduces to the following statement: Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be the limits of a directed inverse system of affine schemes. Let $i \in I$ and let $X_ i \to T_ i$ be an object of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ over $T_ i$. Assume that $T \times _{T_ i} X_ i \to T$ has relative dimension $\leq 1$. Then for some $i' \geq i$ the morphism $T_{i'} \times _{T_ i} X_ i \to T_ i$ has relative dimension $\leq 1$. This follows from Limits of Spaces, Lemma 70.6.14.
$\square$
Lemma 99.15.7. Let
\[ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } \]
be a pushout in the category of schemes where $T \to T'$ is a thickening and $T \to S$ is affine, see More on Morphisms, Lemma 37.14.3. Then the functor on fibre categories
\[ \mathcal{C}\! \mathit{urves}_{S'} \longrightarrow \mathcal{C}\! \mathit{urves}_ S \times _{\mathcal{C}\! \mathit{urves}_ T} \mathcal{C}\! \mathit{urves}_{T'} \]
is an equivalence.
Proof.
Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.7) this reduces to the following statement: given a morphism $X' \to S'$ of an algebraic space to $S'$ which is of finite presentation, flat, proper then $X' \to S'$ has relative dimension $\leq 1$ if and only if $S \times _{S'} X' \to S$ and $T' \times _{S'} X' \to T'$ have relative dimension $\leq 1$. One implication follows from the fact that having relative dimension $\leq 1$ is preserved under base change (Morphisms of Spaces, Lemma 67.34.3). The other follows from the fact that having relative dimension $\leq 1$ is checked on the fibres and that the fibres of $X' \to S'$ (over points of the scheme $S'$) are the same as the fibres of $S \times _{S'} X' \to S$ since $S \to S'$ is a thickening by More on Morphisms, Lemma 37.14.3.
$\square$
Lemma 99.15.8. Let $k$ be a field and let $x = (X \to \mathop{\mathrm{Spec}}(k))$ be an object of $\mathcal{X} = \mathcal{C}\! \mathit{urves}$ over $\mathop{\mathrm{Spec}}(k)$.
If $k$ is of finite type over $\mathbf{Z}$, then the vector spaces $T\mathcal{F}_{\mathcal{X}, k, x}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x})$ (see Artin's Axioms, Section 98.8) are finite dimensional, and
in general the vector spaces $T_ x(k)$ and $\text{Inf}_ x(k)$ (see Artin's Axioms, Section 98.21) are finite dimensional.
Proof.
This is immediate from the fully faithful embedding (99.15.1.1) and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.8).
$\square$
Lemma 99.15.9. Consider the stack $\mathcal{C}\! \mathit{urves}$ over the base scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Then every formal object is effective.
Proof.
For definitions of the notions in the lemma, please see Artin's Axioms, Section 98.9. Let $(A, \mathfrak m, \kappa )$ be a Noetherian complete local ring. Let $(X_ n \to \mathop{\mathrm{Spec}}(A/\mathfrak m^ n))$ be a formal object of $\mathcal{C}\! \mathit{urves}$ over $A$. By More on Morphisms of Spaces, Lemma 76.43.5 there exists a projective morphism $X \to \mathop{\mathrm{Spec}}(A)$ and a compatible system of ismomorphisms $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^ n) \cong X_ n$. By More on Morphisms, Lemma 37.12.4 we see that $X \to \mathop{\mathrm{Spec}}(A)$ is flat. By More on Morphisms, Lemma 37.30.6 we see that $X \to \mathop{\mathrm{Spec}}(A)$ has relative dimension $\leq 1$. This proves the lemma.
$\square$
Lemma 99.15.10. The stack in groupoids $\mathcal{X} = \mathcal{C}\! \mathit{urves}$ satisfies openness of versality over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Similarly, after base change (Remark 99.15.5) openness of versality holds over any Noetherian base scheme $S$.
Proof.
This is immediate from the fully faithful embedding (99.15.1.1) and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.9).
$\square$
reference
Theorem 99.15.11 (Algebraicity of the stack of curves). The stack $\mathcal{C}\! \mathit{urves}$ (Situation 99.15.1) is algebraic. In fact, for any algebraic space $B$ the stack $B\text{-}\mathcal{C}\! \mathit{urves}$ (Remark 99.15.5) is algebraic.
Proof.
The absolute case follows from Artin's Axioms, Lemma 98.17.1 and Lemmas 99.15.4, 99.15.7, 99.15.6, 99.15.9, and 99.15.10. The case over $B$ follows from this, the description of $B\text{-}\mathcal{C}\! \mathit{urves}$ as a $2$-fibre product in Remark 99.15.5, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma 94.14.3.
$\square$
Lemma 99.15.12. The $1$-morphism (99.15.1.1)
\[ \mathcal{C}\! \mathit{urves}\longrightarrow \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \]
is representable by open and closed immersions.
Proof.
Since (99.15.1.1) is a fully faithful embedding of categories it suffices to show the following: given an object $X \to S$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ there exists an open and closed subscheme $U \subset S$ such that a morphism $S' \to S$ factors through $U$ if and only if the base change $X' \to S'$ of $X \to S$ has relative dimension $\leq 1$. This follows immediately from More on Morphisms of Spaces, Lemma 76.31.5.
$\square$
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