## 107.23 Contraction morphisms

We urge the reader to familiarize themselves with Algebraic Curves, Sections 53.22, 53.23, and 53.24 before continuing here. The main result of this section is the existence of a “stabilization” morphism

$\mathcal{C}\! \mathit{urves}^{prestable}_ g \longrightarrow \overline{\mathcal{M}}_ g$

See Lemma 107.23.5. Loosely speaking, this morphism sends the moduli point of a nodal genus $g$ curve to the moduli point of the associated stable curve constructed in Algebraic Curves, Lemma 53.24.2.

Lemma 107.23.1. Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g : Y \to S$ be families of curves. Let $c : X \to Y$ be a morphism over $S$. If $c_{s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_ s}$ and $R^1c_{s, *}\mathcal{O}_{X_ s} = 0$, then after replacing $S$ by an open neighbourhood of $s$ we have $\mathcal{O}_ Y = c_*\mathcal{O}_ X$ and $R^1c_*\mathcal{O}_ X = 0$ and this remains true after base change by any morphism $S' \to S$.

Proof. Let $(U, u) \to (S, s)$ be an étale neighbourhood such that $\mathcal{O}_{Y_ U} = (X_ U \to Y_ U)_*\mathcal{O}_{X_ U}$ and $R^1(X_ U \to Y_ U)_*\mathcal{O}_{X_ U} = 0$ and the same is true after base change by $U' \to U$. Then we replace $S$ by the open image of $U \to S$. Given $S' \to S$ we set $U' = U \times _ S S'$ and we obtain étale coverings $\{ U' \to S'\}$ and $\{ Y_{U'} \to Y_{S'}\}$. Thus the truth of the statement for the base change of $c$ by $S' \to S$ follows from the truth of the statement for the base change of $X_ U \to Y_ U$ by $U' \to U$. In other words, the question is local in the étale topology on $S$. Thus by Lemma 107.4.3 we may assume $X$ and $Y$ are schemes. By More on Morphisms, Lemma 37.64.7 there exists an open subscheme $V \subset Y$ containing $Y_ s$ such that $c_*\mathcal{O}_ X|_ V = \mathcal{O}_ V$ and $R^1c_*\mathcal{O}_ X|_ V = 0$ and such that this remains true after any base change by $S' \to S$. Since $g : Y \to S$ is proper, we can find an open neighbourhood $U \subset S$ of $s$ such that $g^{-1}(U) \subset V$. Then $U$ works. $\square$

Lemma 107.23.2. Let $S$ be a scheme and $s \in S$ a point. Let $f : X \to S$ and $g_ i : Y_ i \to S$, $i = 1, 2$ be families of curves. Let $c_ i : X \to Y_ i$ be morphisms over $S$. Assume there is an isomorphism $Y_{1, s} \cong Y_{2, s}$ of fibres compatible with $c_{1, s}$ and $c_{2, s}$. If $c_{1, s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_ s} = 0$, then there exist an open neighbourhood $U$ of $s$ and an isomorphism $Y_{1, U} \cong Y_{2, U}$ of families of curves over $U$ compatible with the given isomorphism of fibres and with $c_1$ and $c_2$.

Proof. Recall that $\mathcal{O}_{S, s} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ S(U)$ where the colimit is over the system of affine neighbourhoods $U$ of $s$. Thus the category of algebraic spaces of finite presentation over the local ring is the colimit of the categories of algebraic spaces of finite presentation over the affine neighbourhoods of $s$. See Limits of Spaces, Lemma 68.7.1. In this way we reduce to the case where $S$ is the spectrum of a local ring and $s$ is the closed point.

Assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is a local ring and $s$ is the closed point. Write $A = \mathop{\mathrm{colim}}\nolimits A_ j$ with $A_ j$ local Noetherian (say essentially of finite type over $\mathbf{Z}$) and local transition homomorphisms. Set $S_ j = \mathop{\mathrm{Spec}}(A_ j)$ with closed point $s_ j$. We can find a $j$ and families of curves $X_ j \to S_ j$, $Y_{j, i} \to S_ j$, see Lemma 107.5.3 and Limits of Stacks, Lemma 100.3.5. After possibly increasing $j$ we can find morphisms $c_{j, i} : X_ j \to Y_{j, i}$ whose base change to $s$ is $c_ i$, see Limits of Spaces, Lemma 68.7.1. Since $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ j)$ we can similarly assume there is an isomorphism $Y_{j, 1, s_ j} \cong Y_{j, 2, s_ j}$ compatible with $c_{j, 1, s_ j}$ and $c_{j, 2, s_ j}$. Finally, the assumptions $c_{1, s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_ s} = 0$ are inherited by $c_{j, 1, s_ j}$ because $\{ s_ j \to s\}$ is an fpqc covering and $c_{1, s}$ is the base of $c_{j, 1, s_ j}$ by this covering (details omitted). In this way we reduce the lemma to the case discussed in the next paragraph.

Assume $S$ is the spectrum of a Noetherian local ring $\Lambda$ and $s$ is the closed point. Consider the scheme theoretic image $Z$ of

$(c_1, c_2) : X \longrightarrow Y_1 \times _ S Y_2$

The statement of the lemma is equivalent to the assertion that $Z$ maps isomorphically to $Y_1$ and $Y_2$ via the projection morphisms. Since taking the scheme theoretic image of this morphism commutes with flat base change (Morphisms of Spaces, Lemma 65.30.12, we may replace $\Lambda$ by its completion (More on Algebra, Section 15.42).

Assume $S$ is the spectrum of a complete Noetherian local ring $\Lambda$. Observe that $X$, $Y_1$, $Y_2$ are schemes in this case (More on Morphisms of Spaces, Lemma 74.43.5). Denote $X_ n$, $Y_{1, n}$, $Y_{2, n}$ the base changes of $X$, $Y_1$, $Y_2$ to $\mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^{n + 1})$. Recall that the arrow

$\mathcal{D}\! \mathit{ef}_{X_ s \to Y_{2, s}} \cong \mathcal{D}\! \mathit{ef}_{X_ s \to Y_{1, s}} \longrightarrow \mathcal{D}\! \mathit{ef}_{X_ s}$

is an equivalence, see Deformation Problems, Lemma 91.10.6. Thus there is an isomorphism of formal objects $(X_ n \to Y_{1, n}) \cong (X_ n \to Y_{2, n})$ of $\mathcal{D}\! \mathit{ef}_{X_ s \to Y_{1, s}}$. Finally, by Grothendieck's algebraization theorem (Cohomology of Schemes, Lemma 30.28.3) this produces an isomorphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$. $\square$

Lemma 107.23.3. Let $f : X \to S$ be a family of curves. Let $s \in S$ be a point. Let $h_0 : X_ s \to Y_0$ be a morphism to a proper scheme $Y_0$ over $\kappa (s)$ such that $h_{0, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_0}$ and $R^1h_{0, *}\mathcal{O}_{X_ s} = 0$. Then there exist an elementary étale neighbourhood $(U, u) \to (S, s)$, a family of curves $Y \to U$, and a morphism $h : X_ U \to Y$ over $U$ whose fibre in $u$ is isomorphic to $h_0$.

Proof. We first do some reductions; we urge the reader to skip ahead. The question is local on $S$, hence we may assume $S$ is affine. Write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a cofiltered limit of affine schemes $S_ i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a family of curves $X_ i \to S_ i$ whose base change is $X \to S$. This follows from Lemma 107.5.3 and Limits of Stacks, Lemma 100.3.5. Let $s_ i \in S_ i$ be the image of $s$. Observe that $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ i)$ and that $X_ s$ is a scheme (Spaces over Fields, Lemma 70.9.3). After increasing $i$ we may assume there exists a morphism $h_{i, 0} : X_{i, s_ i} \to Y_ i$ of finite type schemes over $\kappa (s_ i)$ whose base change to $\kappa (s)$ is $h_0$, see Limits, Lemma 32.10.1. After increasing $i$ we may assume $Y_ i$ is proper over $\kappa (s_ i)$, see Limits, Lemma 32.13.1. Let $g_{i, 0} : Y_0 \to Y_{i, 0}$ be the projection. Observe that this is a faithfully flat morphism as the base change of $\mathop{\mathrm{Spec}}(\kappa (s)) \to \mathop{\mathrm{Spec}}(\kappa (s_ i))$. By flat base change we have

$h_{0, *}\mathcal{O}_{X_ s} = g_{i, 0}^*h_{i, 0, *}\mathcal{O}_{X_{i, s_ i}} \quad \text{and}\quad R^1h_{0, *}\mathcal{O}_{X_ s} = g_{i, 0}^*Rh_{i, 0, *}\mathcal{O}_{X_{i, s_ i}}$

see Cohomology of Schemes, Lemma 30.5.2. By faithful flatness we see that $X_ i \to S_ i$, $s_ i \in S_ i$, and $X_{i, s_ i} \to Y_ i$ satisfies all the assumptions of the lemma. This reduces us to the case discussed in the next paragraph.

Assume $S$ is affine of finite type over $\mathbf{Z}$. Let $\mathcal{O}_{S, s}^ h$ be the henselization of the local ring of $S$ at $s$. Observe that $\mathcal{O}_{S, s}^ h$ is a G-ring by More on Algebra, Lemma 15.49.8 and Proposition 15.49.12. Suppose we can construct a family of curves $Y' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ and a morphism

$h' : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h) \longrightarrow Y'$

over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ whose base change to the closed point is $h_0$. This will be enough. Namely, first we use that

$\mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(U, u)} \mathcal{O}_ U(U)$

where the colimit is over the filtered category of elementary étale neighbourhoods (More on Morphisms, Lemma 37.31.5). Next, we use again that given $Y'$ we can descend it to $Y \to U$ for some $U$ (see references given above). Then we use Limits, Lemma 32.10.1 to descend $h'$ to some $h$. This reduces us to the case discussed in the next paragraph.

Assume $S = \mathop{\mathrm{Spec}}(\Lambda )$ where $(\Lambda , \mathfrak m, \kappa )$ is a henselian Noetherian local G-ring and $s$ is the closed point of $S$. Recall that the map

$\mathcal{D}\! \mathit{ef}_{X_ s \to Y_0} \to \mathcal{D}\! \mathit{ef}_{X_ s}$

is an equivalence, see Deformation Problems, Lemma 91.10.6. (This is the only important step in the proof; everything else is technique.) Denote $\Lambda ^\wedge$ the $\mathfrak m$-adic completion. The pullbacks $X_ n$ of $X$ to $\Lambda /\mathfrak m^{n + 1}$ define a formal object $\xi$ of $\mathcal{D}\! \mathit{ef}_{X_ s}$ over $\Lambda ^\wedge$. From the equivalence we obtain a formal object $\xi '$ of $\mathcal{D}\! \mathit{ef}_{X_ s \to Y_0}$ over $\Lambda ^\wedge$. Thus we obtain a huge commutative diagram

$\xymatrix{ \ldots \ar[r] & X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_ s \ar[d] \\ \ldots \ar[r] & Y_ n \ar[r] \ar[d] & Y_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & Y_0 \ar[d] \\ \ldots \ar[r] & \mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^{n + 1}) \ar[r] & \mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^ n) \ar[r] & \ldots \ar[r] & \mathop{\mathrm{Spec}}(\kappa ) }$

The formal object $(Y_ n)$ comes from a family of curves $Y' \to \mathop{\mathrm{Spec}}(\Lambda ^\wedge )$ by Quot, Lemma 97.15.9. By More on Morphisms of Spaces, Lemma 74.43.3 we get a morphism $h' : X_{\Lambda ^\wedge } \to Y'$ inducing the given morphisms $X_ n \to Y_ n$ for all $n$ and in particular the given morphism $X_ s \to Y_0$.

To finish we do a standard algebraization/approximation argument. First, we observe that we can find a finitely generated $\Lambda$-subalgebra $\Lambda \subset A \subset \Lambda ^\wedge$, a family of curves $Y'' \to \mathop{\mathrm{Spec}}(A)$ and a morphism $h'' : X_ A \to Y''$ over $A$ whose base change to $\Lambda ^\wedge$ is $h'$. This is true because $\Lambda ^\wedge$ is the filtered colimit of these rings $A$ and we can argue as before using that $\mathcal{C}\! \mathit{urves}$ is locally of finite presentation (which gives us $Y''$ over $A$ by Limits of Stacks, Lemma 100.3.5) and using Limits of Spaces, Lemma 68.7.1 to descend $h'$ to some $h''$. Then we can apply the approximation property for G-rings (in the form of Smoothing Ring Maps, Theorem 16.13.1) to find a map $A \to \Lambda$ which induces the same map $A \to \kappa$ as we obtain from $A \to \Lambda ^\wedge$. Base changing $h''$ to $\Lambda$ the proof is complete. $\square$

Lemma 107.23.4. Let $f : X \to S$ be a prestable family of curves of genus $g \geq 1$. There is a factorization $X \to Y \to S$ of $f$ where $g : Y \to S$ is a stable family of curves and $c : X \to Y$ has the following properties

1. $\mathcal{O}_ Y = c_*\mathcal{O}_ X$ and $R^1c_*\mathcal{O}_ X = 0$ and this remains true after base change by any morphism $S' \to S$, and

2. for any $s \in S$ the morphism $c_ s : X_ s \to Y_ s$ is the contraction of rational tails and bridges discussed in Algebraic Curves, Section 53.24.

Moreover $c : X \to Y$ is unique up to unique isomorphism.

Proof. Let $s \in S$. Let $c_0 : X_ s \to Y_0$ be the contraction of Algebraic Curves, Section 53.24 (more precisely Algebraic Curves, Lemma 53.24.2). By Lemma 107.23.3 there exists an elementary étale neighbourhood $(U, u)$ and a morphism $c : X_ U \to Y$ of families of curves over $U$ which recovers $c_0$ as the fibre at $u$. Since $\omega _{Y_0}$ is ample, after possibly shrinking $U$, we see that $Y \to U$ is a stable family of genus $g$ by the openness inherent in Lemmas 107.22.3 and 107.22.5. After possibly shrinking $U$ once more, assertion (1) of the lemma for $c : X_ U \to Y$ follows from Lemma 107.23.1. Moreover, part (2) holds by the uniqueness in Algebraic Curves, Lemma 53.24.2. We conclude that a morphism $c$ as in the lemma exists étale locally on $S$. More precisely, there exists an étale covering $\{ U_ i \to S\}$ and morphisms $c_ i : X_{U_ i} \to Y_ i$ over $U_ i$ where $Y_ i \to U_ i$ is a stable family of curves having properties (1) and (2) stated in the lemma.

To finish the proof it suffices to prove uniqueness of $c : X \to Y$ (up to unique isomorphism). Namely, once this is done, then we obtain isomorphisms

$\varphi _{ij} : Y_ i \times _{U_ i} (U_ i \times _ S U_ j) \longrightarrow Y_ i \times _{U_ j} (U_ i \times _ S U_ j)$

satisfying the cocycle condition (by uniqueness) over $U_ i \times U_ j \times U_ k$. Since $\overline{\mathcal{M}_ g}$ is an algebraic stack, we have effectiveness of descent data and we obtain $Y \to S$. The morphisms $c_ i$ descend to a morphism $c : X \to Y$ over $S$. Finally, properties (1) and (2) for $c$ are immediate from properties (1) and (2) for $c_ i$.

Finally, if $c_1 : X \to Y_ i$, $i = 1, 2$ are two morphisms towards stably families of curves over $S$ satisfying (1) and (2), then we obtain a morphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$ at least locally on $S$ by Lemma 107.23.3. We omit the verification that these morphisms are unique (hint: this follows from the fact that the scheme theoretic image of $c_1$ is $Y_1$). Hence these locally given morphisms glue and the proof is complete. $\square$

Lemma 107.23.5. Let $g \geq 2$. There is a morphism of algebraic stacks over $\mathbf{Z}$

$stabilization : \mathcal{C}\! \mathit{urves}^{prestable}_ g \longrightarrow \overline{\mathcal{M}}_ g$

which sends a prestable family of curves $X \to S$ of genus $g$ to the stable family $Y \to S$ asssociated to it in Lemma 107.23.4.

Proof. To see this is true, it suffices to check that the construction of Lemma 107.23.4 is compatible with base change (and isomorphisms but that's immediate), see the (abuse of) language for algebraic stacks introduced in Properties of Stacks, Section 98.2. To see this it suffices to check properties (1) and (2) of Lemma 107.23.4 are stable under base change. This is immediately clear for (1). For (2) this follows either from the fact that the contractions of Algebraic Curves, Lemmas 53.22.6 and 53.23.6 are stable under ground field extensions, or because the conditions characterizing the morphisms on fibres in Algebraic Curves, Lemma 53.24.2 are preserved under ground field extensions. $\square$

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