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55.16 Semistable reduction in genus one

In this section we prove the semistable reduction theorem (Theorem 55.18.1) for curves of genus one. We suggest the reader first read the proof in the case of genus $\geq 2$ (Section 55.17). We are going to use as much as possible the classification of minimal numerical types of genus $1$ given in Lemma 55.6.2.

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. Assume the genus of $C$ is $1$. Choose a prime $\ell \geq 7$ different from the characteristic of $k$. Choose a finite separable extension $K'/K$ of such that $C(K') \not= \emptyset $ and such that $\mathop{\mathrm{Pic}}\nolimits (C_{K'})[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2}$. See Algebraic Curves, Lemma 53.17.2. Let $R' \subset K'$ be the integral closure of $R$, see discussion in More on Algebra, Remark 15.111.6. We may replace $R$ by $R'_{\mathfrak m}$ for some maximal ideal $\mathfrak m$ in $R'$ and $C$ by $C_{K'}$. This reduces us to the case discussed in the next paragraph.

In the rest of this section $R$ is a discrete valuation ring with fraction field $K$, $C$ is a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$, with genus $1$, having a $K$-rational point, and with $\mathop{\mathrm{Pic}}\nolimits (C)[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2}$ for some prime $\ell \geq 7$ different from the characteristic of $k$. We will prove that $C$ has semistable reduction.

Let $X$ be a minimal model for $C$, see Proposition 55.8.6. Let $T = (n, m_ i, (a_{ij}), w_ i, g_ i)$ be the numerical type associated to $X$ (Definition 55.11.4). Then $T$ is a minimal numerical type (Lemma 55.11.5). As $C$ has a rational point, there exists an $i$ such that $m_ i = w_ i = 1$ by Lemma 55.11.7. Looking at the classification of minimal numerical types of genus $1$ in Lemma 55.6.2 we see that $m = w = 1$ and that cases (3), (6), (7), (9), (11), (13), (15), (18), (19), (21), (24), (26), (28), (30) are disallowed (because there is no index where both $w_ i$ and $m_ i$ is equal to $1$). Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. For the remaining cases we have

  1. $e = n - 1$ for cases (1), (2), (5), (8), (12), (14), (17), (20), (22), (23), (27), (29), (31), (32), (33), and (34), and

  2. $e = n$ for cases (4), (10), (16), and (25).

We will argue these cases separately.

Case (A). In this case $\mathop{\mathrm{Pic}}\nolimits (T)[\ell ]$ is trivial (the Picard group of a numerical type is defined in Section 55.4). The vanishing follows as $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Coker}}(A)$ (Lemma 55.4.3) and $\mathop{\mathrm{Coker}}(A)[\ell ] = 0$ by Lemma 55.2.6 and the fact that $\ell $ was chosen relatively prime to $a_{ij}$ and $m_ i$. By Lemmas 55.13.3 and 55.13.4 we conclude that there is an embedding

\[ (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2} \subset \mathop{\mathrm{Pic}}\nolimits ((X_ k)_{red})[\ell ]. \]

By Algebraic Curves, Lemma 53.18.6 we obtain

\[ 2 \leq \dim _ k H^1((X_ k)_{red}, \mathcal{O}_{(X_ k)_{red}}) + g_{geom}((X_ k)_{red}/k) \]

By Algebraic Curves, Lemmas 53.18.1 and 53.18.5 we see that $g_{geom}((X_ k)_{red}/k) \leq \sum w_ ig_ i$. The assumptions of Lemma 55.11.8 hold by Lemma 55.11.7 and we conclude that we have $\dim _ k H^1((X_ k)_{red}, \mathcal{O}_{(X_ k)_{red}}) \leq g = 1$. Combining these we see

\[ 2 \leq 1 + \sum w_ i g_ i \]

Looking at the list we conclude that the numerical type is given by $n = 1$, $w_1 = m_1 = g_1 = 1$. Because we have equality everywhere we see that $g_{geom}(C_1/k) = 1$. On the other hand, we know that $C_1$ has a $k$-rational point $x$ such that $C_1 \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$. It follows that $C_1$ is geometrically integral (Varieties, Lemma 33.25.10). Thus $g_{geom}(C_1/k) = 1$ is both equal to the genus of the normalization of $C_{1, \overline{k}}$ and the genus of $C_{1, \overline{k}}$. It follows that the normalization morphism $C_{1, \overline{k}}^\nu \to C_{1, \overline{k}}$ is an isomorphism (Algebraic Curves, Lemma 53.18.4). We conclude that $C_1$ is smooth over $k$ as desired.

Case (B). Here we only conclude that there is an embedding

\[ \mathbf{Z}/\ell \mathbf{Z} \subset \mathop{\mathrm{Pic}}\nolimits (X_ k)[\ell ] \]

From the classification of types we see that $m_ i = w_ i = 1$ and $g_ i = 0$ for each $i$. Thus each $C_ i$ is a genus zero curve over $k$. Moreover, for each $i$ there is a $j$ such that $C_ i \cap C_ j$ is a $k$-rational point. Then it follows that $C_ i \cong \mathbf{P}^1_ k$ by Algebraic Curves, Proposition 53.10.4. In particular, since $X_ k$ is the scheme theoretic union of the $C_ i$ we see that $X_{\overline{k}}$ is the scheme theoretic union of the $C_{i, \overline{k}}$. Hence $X_{\overline{k}}$ is a reduced connected proper scheme of dimension $1$ over $\overline{k}$ with $\dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}}) = 1$. Also, by Varieties, Lemma 33.30.3 and the above we still have

\[ \dim _{\mathbf{F}_\ell }(\mathop{\mathrm{Pic}}\nolimits (X_{\overline{k}}) \geq 1 \]

By Algebraic Curves, Proposition 53.17.3 we see that $X_{\overline{k}}$ has at only multicross singularities. But since $X_ k$ is Gorenstein (Lemma 55.9.2), so is $X_{\overline{k}}$ (Duality for Schemes, Lemma 48.25.1). We conclude $X_{\overline{k}}$ is at-worst-nodal by Algebraic Curves, Lemma 53.16.4. This finishes the proof in case (B).

Example 55.16.1. Let $k$ be an algebraically closed field. Let $Z$ be a smooth projective curve over $k$ of positive genus $g$. Let $n \geq 1$ be an integer prime to the characteristic of $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Z$-module of order $n$, see Algebraic Curves, Lemma 53.17.1. Pick an isomorphism $\varphi : \mathcal{L}^{\otimes n} \to \mathcal{O}_ Z$. Set $R = k[[\pi ]]$ with fraction field $K = k((\pi ))$. Denote $Z_ R$ the base change of $Z$ to $R$. Let $\mathcal{L}_ R$ be the pullback of $\mathcal{L}$ to $Z_ R$. Consider the finite flat morphism

\[ p : X \longrightarrow Z_ R \]

such that

\[ p_*\mathcal{O}_ X = \text{Sym}^*_{\mathcal{O}_{Z_ R}}(\mathcal{L}_ R)/(\varphi - \pi ) = \mathcal{O}_{Z_ R} \oplus \mathcal{L}_ R \oplus \mathcal{L}_ R^{\otimes 2} \oplus \ldots \oplus \mathcal{L}_ R^{\otimes n - 1} \]

More precisely, if $U = \mathop{\mathrm{Spec}}(A) \subset Z$ is an affine open such that $\mathcal{L}|_ U$ is trivialized by a section $s$ with $\varphi (s^{\otimes n}) = f$ (with $f$ a unit), then

\[ p^{-1}(U_ R) = \mathop{\mathrm{Spec}}\left( (A \otimes _ R R[[\pi ]])[x]/(x^ n - \pi f) \right) \]

The reader verifies that the morphism $X_ K \to Z_ K$ of generic fibres is finite ├ętale. Looking at the description of the structure sheaf we see that $H^0(X, \mathcal{O}_ X) = R$ and $H^0(X_ K, \mathcal{O}_{X_ K}) = K$. By Riemann-Hurwitz (Algebraic Curves, Lemma 53.12.4) the genus of $X_ K$ is $n(g - 1) + 1$. In particular $X_ K$ has genus $1$, if $Z$ has genus $1$. On the other hand, the scheme $X$ is regular by the local equation above and the special fibre $X_ k$ is $n$ times the reduced special fibre as an effective Cartier divisor. It follows that any finite extension $K'/K$ over which $X_ K$ attains semistable reduction has to ramify with ramification index at least $n$ (some details omitted). Thus there does not exist a universal bound for the degree of an extension over which a genus $1$ curve attains semistable reduction.

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