## 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:

1. 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).

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 110.66.

It is clear that

99.15.1.1
$$\label{quot-equation-curves-over-proper-spaces} \mathcal{C}\! \mathit{urves}\subset \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$$

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:

1. $X \to S$ has relative dimension $\leq 1$, and

2. 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). $\square$

Remark 99.15.5. Let $B$ be an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $B\text{-}\mathcal{C}\! \mathit{urves}$ be the category consisting of pairs $(X \to S, h : S \to B)$ where $X \to S$ is an object of $\mathcal{C}\! \mathit{urves}$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', h') \to (X \to S, h)$ in $B\text{-}\mathcal{C}\! \mathit{urves}$ is a morphism $(f, g)$ in $\mathcal{C}\! \mathit{urves}$ such that $h \circ g = h'$. In this situation the diagram

$\xymatrix{ B\text{-}\mathcal{C}\! \mathit{urves}\ar[r] \ar[d] & \mathcal{C}\! \mathit{urves}\ar[d] \\ (\mathit{Sch}/B)_{fppf} \ar[r] & \mathit{Sch}_{fppf} }$

is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\mathcal{C}\! \mathit{urves}$ to the case of families of curves over a given base algebraic space.

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. $\square$

$\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }$

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 $\mathop{\mathrm{Spec}}(k)$.

1. 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

2. 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). $\square$

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. $\square$

Remark 99.15.13. Consider the $2$-fibre product

$\xymatrix{ \mathcal{C}\! \mathit{urves}\times _{\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}} \mathcal{P}\! \mathit{olarized}\ar[r] \ar[d] & \mathcal{P}\! \mathit{olarized}\ar[d] \\ \mathcal{C}\! \mathit{urves}\ar[r] & \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} }$

This fibre product parametrized polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. It turns out that the left vertical arrow

$\textit{PolarizedCurves} \longrightarrow \mathcal{C}\! \mathit{urves}$

is algebraic, smooth, and surjective. Namely, this $1$-morphism is algebraic (as base change of the arrow in Lemma 99.14.6), every point is in the image, and there are no obstructions to deforming invertible sheaves on curves (see proof of Lemma 99.15.9). This gives another approach to the algebraicity of $\mathcal{C}\! \mathit{urves}$. Namely, by Lemma 99.15.12 we see that $\textit{PolarizedCurves}$ is an open and closed substack of the algebraic stack $\mathcal{P}\! \mathit{olarized}$ and any stack in groupoids which is the target of a smooth algebraic morphism from an algebraic stack is an algebraic stack.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).