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55.17 Semistable reduction in genus at least two

In this section we prove the semistable reduction theorem (Theorem 55.18.1) for curves of genus $\geq 2$. Fix $g \geq 2$.

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 $g$. Choose a prime $\ell > 768g$ different from the characteristic of $k$. Choose a finite separable extension $K'/K$ of such that $C(K') \not= \emptyset $ and such that $\mathop{\mathrm{Pic}}\nolimits (C_{K'})[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2g}$. 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 $g$, having a $K$-rational point, and with $\mathop{\mathrm{Pic}}\nolimits (C)[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2g}$ for some prime $\ell \geq 768g$ different from the characteristic of $k$. We will prove that $C$ has semistable reduction.

In the rest of this section we will use without further mention that the conclusions of Lemma 55.11.7 are true.

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 of genus $g$ (Lemma 55.11.5). By Proposition 55.7.4 we have

\[ \dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (T)[\ell ] \leq g_{top} \]

By Lemmas 55.13.3 and 55.13.4 we conclude that there is an embedding

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

By Algebraic Curves, Lemma 53.18.6 we obtain

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

By Lemmas 55.11.8 and 55.11.9 we have

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

Elementary number theory tells us that the only way these $3$ inequalities can hold is if they are all equalities. Looking at Lemma 55.11.8 we conclude that $m_ i = 1$ for all $i$. Looking at Lemma 55.11.10 we conclude that every irreducible component of $X_ k$ is smooth over $k$.

In particular, since $X_ k$ is the scheme theoretic union of its irreducible components $C_ i$ 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 $\dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}}) = g$. Also, by Varieties, Lemma 33.30.3 and the above we still have

\[ \dim _{\mathbf{F}_\ell }(\mathop{\mathrm{Pic}}\nolimits (X_{\overline{k}})[\ell ]) \geq 2g - g_{top} = \dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}}) + g_{geom}(X_{\overline{k}}) \]

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.

Comments (2)

Comment #2948 by Maciek Zdanowicz on

It seems that the two occurrences of in first paragraphs should be substituted with .

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