## 107.24 Stable reduction theorem

In the chapter on semistable reduction we have proved the celebrated theorem on semistable reduction of curves. Let $K$ be the fraction field of a discrete valuation ring $R$. Let $C$ be a projective smooth curve over $K$ with $K = H^0(C, \mathcal{O}_ C)$. According to Semistable Reduction, Definition 55.14.6 we say $C$ has semistable reduction if either there is a prestable family of curves over $R$ with generic fibre $C$, or some (equivalently any) minimal regular model of $C$ over $R$ is prestable. In this section we show that for curves of genus $g \geq 2$ this is also equivalent to stable reduction.

Lemma 107.24.1. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $K = H^0(C, \mathcal{O}_ C)$ having genus $g \geq 2$. The following are equivalent

1. $C$ has semistable reduction (Semistable Reduction, Definition 55.14.6), or

2. there is a stable family of curves over $R$ with generic fibre $C$.

Proof. Since a stable family of curves is also prestable, it is immediate that (2) implies (1). Conversely, given a prestable family of curves over $R$ with generic fibre $C$, we can contract it to a stable family of curves by Lemma 107.23.4. Since the generic fibre already is stable, it does not get changed by this procedure and the proof is complete. $\square$

The following lemma tells us the stable family of curves over $R$ promised in Lemma 107.24.1 is unique up to unique isomorphism.

Lemma 107.24.2. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth proper curve over $K$ with $K = H^0(C, \mathcal{O}_ C)$ and genus $g$. If $X$ and $X'$ are models of $C$ (Semistable Reduction, Section 55.8) and $X$ and $X'$ are stable families of genus $g$ curves over $R$, then there exists an unique isomorphism $X \to X'$ of models.

Proof. Let $Y$ be the minimal model for $C$. Recall that $Y$ exists, is unique, and is at-worst-nodal of relative dimension $1$ over $R$, see Semistable Reduction, Proposition 55.8.6 and Lemmas 55.10.1 and 55.14.5 (applies because we have $X$). There is a contraction morphism

$Y \longrightarrow Z$

such that $Z$ is a stable family of curves of genus $g$ over $R$ (Lemma 107.23.4). We claim there is a unique isomorphism of models $X \to Z$. By symmetry the same is true for $X'$ and this will finish the proof.

By Semistable Reduction, Lemma 55.14.3 there exists a sequence

$X_ m \to \ldots \to X_1 \to X_0 = X$

such that $X_{i + 1} \to X_ i$ is the blowing up of a closed point $x_ i$ where $X_ i$ is singular, $X_ i \to \mathop{\mathrm{Spec}}(R)$ is at-worst-nodal of relative dimension $1$, and $X_ m$ is regular. By Semistable Reduction, Lemma 55.8.5 there is a sequence

$X_ m = Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to Y_0 = Y$

of proper regular models of $C$, such that each morphism is a contraction of an exceptional curve of the first kind1. By Semistable Reduction, Lemma 55.14.4 each $Y_ i$ is at-worst-nodal of relative dimension $1$ over $R$. To prove the claim it suffices to show that there is an isomorphism $X \to Z$ compatible with the morphisms $X_ m \to X$ and $X_ m = Y_ n \to Y \to Z$. Let $s \in \mathop{\mathrm{Spec}}(R)$ be the closed point. By either Lemma 107.23.2 or Lemma 107.23.4 we reduce to proving that the morphisms $X_{m, s} \to X_ s$ and $X_{m, s} \to Z_ s$ are both equal to the canonical morphism of Algebraic Curves, Lemma 53.24.2.

For a morphism $c : U \to V$ of schemes over $\kappa (s)$ we say $c$ has property (*) if $\dim (U_ v) \leq 1$ for $v \in V$, $\mathcal{O}_ V = c_*\mathcal{O}_ U$, and $R^1c_*\mathcal{O}_ U = 0$. This property is stable under composition. Since both $X_ s$ and $Z_ s$ are stable genus $g$ curves over $\kappa (s)$, it suffices to show that each of the morphisms $Y_ s \to Z_ s$, $X_{i + 1, s} \to X_{i, s}$, and $Y_{i + 1, s} \to Y_{i, s}$, satisfy property (*), see Algebraic Curves, Lemma 53.24.2.

Property (*) holds for $Y_ s \to Z_ s$ by construction.

The morphisms $c : X_{i + 1, s} \to X_{i, s}$ are constructed and studied in the proof of Semistable Reduction, Lemma 55.14.3. It suffices to check (*) étale locally on $X_{i, s}$. Hence it suffices to check (*) for the base change of the morphism “$X_1 \to X_0$” in Semistable Reduction, Example 55.14.1 to $R/\pi R$. We leave the explicit calculation to the reader.

The morphism $c : Y_{i + 1, s} \to Y_{i, s}$ is the restriction of the blow down of an exceptional curve $E \subset Y_{i + 1}$ of the first kind, i.e., $b : Y_{i + 1} \to Y_ i$ is a contraction of $E$, i.e., $b$ is a blowing up of a regular point on the surface $Y_ i$ (Resolution of Surfaces, Section 54.16). Then $\mathcal{O}_{Y_ i} = b_*\mathcal{O}_{Y_{i + 1}}$ and $R^1b_*\mathcal{O}_{Y_{i + 1}} = 0$, see for example Resolution of Surfaces, Lemma 54.3.4. We conclude that $\mathcal{O}_{Y_{i, s}} = c_*\mathcal{O}_{Y_{i + 1, s}}$ and $R^1c_*\mathcal{O}_{Y_{i + 1, s}} = 0$ by More on Morphisms, Lemmas 37.64.1, 37.64.2, and 37.64.4 (only gives surjectivity of $\mathcal{O}_{Y_{i, s}} \to c_*\mathcal{O}_{Y_{i + 1, s}}$ but injectivity follows easily from the fact that $Y_{i, s}$ is reduced and $c$ changes things only over one closed point). This finishes the proof. $\square$

From Lemma 107.24.1 and Semistable Reduction, Theorem 55.18.1 we immediately deduce the stable reduction theorem.

Theorem 107.24.3. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$ and genus $g \geq 2$. Then

1. there exists an extension of discrete valuation rings $R \subset R'$ inducing a finite separable extension of fraction fields $K'/K$ and a stable family of curves $Y \to \mathop{\mathrm{Spec}}(R')$ of genus $g$ with $Y_{K'} \cong C_{K'}$ over $K'$, and

2. there exists a finite separable extension $L/K$ and a stable family of curves $Y \to \mathop{\mathrm{Spec}}(A)$ of genus $g$ where $A \subset L$ is the integral closure of $R$ in $L$ such that $Y_ L \cong C_ L$ over $L$.

Proof. Part (1) is an immediate consequence of Lemma 107.24.1 and Semistable Reduction, Theorem 55.18.1.

Proof of (2). Let $L/K$ be the finite separable extension found in part (3) of Semistable Reduction, Theorem 55.18.1. Let $A \subset L$ be the integral closure of $R$. Recall that $A$ is a Dedekind domain finite over $R$ with finitely many maximal ideals $\mathfrak m_1, \ldots , \mathfrak m_ n$, see More on Algebra, Remark 15.102.6. Set $S = \mathop{\mathrm{Spec}}(A)$, $S_ i = \mathop{\mathrm{Spec}}(A_{\mathfrak m_ i})$, $U = \mathop{\mathrm{Spec}}(L)$, and $U_ i = S_ i \setminus \{ \mathfrak m_ i\}$. Observe that $U \cong U_ i$ for $i = 1, \ldots , n$. Set $X = C_ L$ viewed as a scheme over the open subscheme $U$ of $S$. By our choice of $L$ and $A$ and Lemma 107.24.1 we have a stable families of curves $X_ i \to S_ i$ and isomorphisms $X \times _ U U_ i \cong X_ i \times _{S_ i} U_ i$. By Limits of Spaces, Lemma 68.18.4 we can find a finitely presented morphism $Y \to S$ whose base change to $S_ i$ is isomorphic to $X_ i$ for $i = 1, \ldots , n$. Alternatively, you can use that $S = \bigcup _{i = 1, \ldots , n} S_ i$ is an open covering of $S$ and $S_ i \cap S_ j = U$ for $i \not= j$ and use $n - 1$ applications of Limits of Spaces, Lemma 68.18.1 to get $Y \to S$ whose base change to $S_ i$ is isomorphic to $X_ i$ for $i = 1, \ldots , n$. Clearly $Y \to S$ is the stable family of curves we were looking for. $\square$

[1] In fact we have $X_ m = Y$, i.e., $X_ m$ does not contain any exceptional curves of the first kind. We encourage the reader to think this through as it simplifies the proof somewhat.

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).