## 109.22 Stable curves

The following lemma will help us understand families of stable curves.

Lemma 109.22.1. Let $f : X \to S$ be a prestable family of curves of genus $g \geq 2$. Let $s \in S$ be a point of the base scheme. The following are equivalent

1. $X_ s$ does not have a rational tail and does not have a rational bridge (Algebraic Curves, Examples 53.22.1 and 53.23.1), and

2. $\omega _{X/S}$ is ample on $f^{-1}(U)$ for some $s \in U \subset S$ open.

Proof. Assume (2). Then $\omega _{X_ s}$ is ample on $X_ s$. By Algebraic Curves, Lemmas 53.22.2 and 53.23.2 we conclude that (1) holds (we also use the characterization of ample invertible sheaves in Varieties, Lemma 33.44.15).

Assume (1). Then $\omega _{X_ s}$ is ample on $X_ s$ by Algebraic Curves, Lemmas 53.23.6. We conclude by Descent on Spaces, Lemma 74.13.2. $\square$

Motivated by Lemma 109.22.1 we make the following definition.

Definition 109.22.2. Let $f : X \to S$ be a family of curves. We say $f$ is a stable family of curves if

1. $X \to S$ is a prestable family of curves, and

2. $X_ s$ has genus $\geq 2$ and does not have a rational tails or bridges for all $s \in S$.

In particular, a prestable family of curves of genus $0$ or $1$ is never stable. Let $X$ be a proper scheme over a field $k$ with $\dim (X) \leq 1$. Then $X \to \mathop{\mathrm{Spec}}(k)$ is a family of curves and hence we can ask whether or not it is stable. Unwinding the definitions we see the following are equivalent

1. $X$ is stable,

2. $X$ is prestable, has genus $\geq 2$, does not have a rational tail, and does not have a rational bridge,

3. $X$ is geometrically connected, is smooth over $k$ apart from a finite number of nodes, and $\omega _ X$ is ample.

To see the equivalence of (2) and (3) use Lemma 109.22.1 above. This shows that our definition agrees with most definitions one finds in the literature.

Lemma 109.22.3. There exist an open substack $\mathcal{C}\! \mathit{urves}^{stable} \subset \mathcal{C}\! \mathit{urves}$ such that

1. given a family of curves $f : X \to S$ the following are equivalent

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

2. $X \to S$ is a stable family of curves,

2. given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

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

2. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $\geq 2$, and $X$ has no rational tails or bridges,

3. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, and $\omega _{X_ s}$ is ample.

Proof. By the discussion in Section 109.6 it suffices to look at families $f : X \to S$ of prestable curves. By Lemma 109.22.1 we obtain the desired openness of the locus in question. Formation of this open commutes with arbitrary base change, either because the (non)existence of rational tails or bridges is insensitive to ground field extensions by Algebraic Curves, Lemmas 53.22.6 and 53.23.6 or because ampleness is insenstive to base field extensions by Descent, Lemma 35.25.6. $\square$

Definition 109.22.4. We denote $\overline{\mathcal{M}}$ and we name the moduli stack of stable curves the algebraic stack $\mathcal{C}\! \mathit{urves}^{stable}$ parametrizing stable families of curves introduced in Lemma 109.22.3. For $g \geq 2$ we denote $\overline{\mathcal{M}}_ g$ and we name the moduli stack of stable curves of genus $g$ the algebraic stack introduced in Lemma 109.22.5.

Here is the obligatory lemma.

Lemma 109.22.5. There is a decomposition into open and closed substacks

$\overline{\mathcal{M}} = \coprod \nolimits _{g \geq 2} \overline{\mathcal{M}}_ g$

where each $\overline{\mathcal{M}}_ g$ is characterized as follows:

1. given a family of curves $f : X \to S$ the following are equivalent

1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\overline{\mathcal{M}}_ g$,

2. $X \to S$ is a stable family of curves and $R^1f_*\mathcal{O}_ X$ is a locally free $\mathcal{O}_ S$-module of rank $g$,

2. given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\overline{\mathcal{M}}_ g$,

2. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $g$, and $X$ has no rational tails or bridges.

3. the singularities of $X$ are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $g$, and $\omega _{X_ s}$ is ample.

Lemma 109.22.6. The morphisms $\overline{\mathcal{M}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ and $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ are smooth.

Proof. Since $\overline{\mathcal{M}}$ is an open substack of $\mathcal{C}\! \mathit{urves}^{nodal}$ this follows from Lemma 109.18.2. $\square$

Lemma 109.22.7. The stacks $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_ g$ are open substacks of $\mathcal{C}\! \mathit{urves}^{DM}$. In particular, $\overline{\mathcal{M}}$ and $\overline{\mathcal{M}}_ g$ are DM (Morphisms of Stacks, Definition 101.4.2) as well as Deligne-Mumford stacks (Algebraic Stacks, Definition 94.12.2).

Proof. Proof of the first assertion. Let $X$ be a scheme proper over a field $k$ whose singularities are at-worst-nodal, $\dim (X) = 1$, $k = H^0(X, \mathcal{O}_ X)$, the genus of $X$ is $\geq 2$, and $X$ has no rational tails or bridges. We have to show that the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \overline{\mathcal{M}} \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{DM}$. We may first replace $k$ by the algebraic closure (since we already know the relevant stacks are open substacks of the algebraic stack $\mathcal{C}\! \mathit{urves}$). By Lemmas 109.22.3, 109.7.3, and 109.7.4 it suffices to show that $\text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0$. This is proven in Algebraic Curves, Lemma 53.25.3.

Since $\mathcal{C}\! \mathit{urves}^{DM}$ is the maximal open substack of $\mathcal{C}\! \mathit{urves}$ which is DM, we see this is true also for the open substack $\overline{\mathcal{M}}$ of $\mathcal{C}\! \mathit{urves}^{DM}$. Finally, a DM algebraic stack is Deligne-Mumford by Morphisms of Stacks, Theorem 101.21.6. $\square$

Lemma 109.22.8. Let $g \geq 2$. The inclusion

$|\mathcal{M}_ g| \subset |\overline{\mathcal{M}}_ g|$

is that of an open dense subset.

Proof. Since $\overline{\mathcal{M}}_ g \subset \mathcal{C}\! \mathit{urves}^{lci+}$ is open and since $\mathcal{C}\! \mathit{urves}^{smooth} \cap \overline{\mathcal{M}}_ g = \mathcal{M}_ g$ this follows immediately from Lemma 109.17.1. $\square$

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